Modern Digital Halftoning, Second Edition (Signal Processing and Communications)

Citation preview

Modern Digital Second Edition

Daniel L. Lau and Gonzalo R. Arce

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

© 2008 by Taylor & Francis Group, LLC

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑4200‑4753‑0 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid‑ ity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti‑ lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy‑ ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For orga‑ nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Lau, Daniel L. Modern digital halftoning / Daniel L. Lau and Gonzalo R. Arce. ‑‑ 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978‑1‑4200‑4753‑0 (alk. paper) 1. Image processing‑‑Digital techniques. 2. Color computer printers. I. Arce, Gonzalo R. II. Title. TA1637.L38 2008 621.36’7‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

© 2008 by Taylor & Francis Group, LLC

2008008944

Preface Halftoning is the process of converting a continuous-tone photograph into a binary pattern of printed and not printed dots and has been the subject of over 120 years of research and development. As a means of quantifying this work, a search of the U.S. Patent and Trademark office reveals a total of 1,433 issued patents with the phrase “halftoning,” or “halftone,” inside its abstract dating back to 1976, the earliest searchable year. Searching the published patent applications reveals another 674 applications in review, 136 published in 2006 alone. So in other words even after 120 years, there are still numerous discoveries to be made, many of which are believed, at least by their inventors, to be worthy of a patent. Surprisingly, only three scholarly books have been written on the subject with this being the only one to have a second edition. The first book, Robert Ulichney’s, “Digital Halftoning,” created a revolution in the digital printing industry. At the time, digital halftoning was done primarily using amplitude modulated screening in which varying shades of gray were produced by varying the size of black dots arranged along a regular grid. While not introducing a new technique, Ulichney studied frequency modulated halftoning in which varying shades of gray are produced by varying the spacing between randomly placed, same-sized dots and showed that this alternative screening process (introduced in 1973) reproduced images with much higher spatial resolution. The result of Ulichney’s work, within the research community, was a wealth of new halftoning techniques designed to construct these ideal frequency modulated halftone patterns. One of the more profound developments was blue-noise dither arrays capable of transforming a pixel of a continuous-tone image into a binary halftone pixel with just a single comparator operation. Within the printing industry, the impact of Ulichney’s work was felt in the ink jet market which soon introduced low-cost, four-color printers. v © 2008 by Taylor & Francis Group, LLC

Sadly, ink jet printers are the only devices that have ever really taken advantage of frequency modulated screening because, at high resolutions (2,400+ dpi versus the 300 dpi common in 1991), the ability of a device to print isolated dots reliably is severely tested. Variations in the size and shape of printed dots have drastic and sometimes catastrophic impact on the resulting images. So for many devices, these distortions require the use of robust halftoning techniques that resist the effects of printer distortion. The first edition of this book addressed the need for a statistical model defining the ideal characteristics of dither patterns resistant to the effects of printer variations by developing new standards for halftone robustness as well as visibility. Furthermore, that book provided a comprehensive study of what was a new stochastic halftoning model, green-noise, including its desirable attributes for printing. It also closely examined several techniques for producing green-noise and studying them in terms of their visual pleasantness and their computational complexity. Due to the demand for color printers, that book also looked at the application of green-noise to color halftoning environments. With the current trends in halftoning being what they were, green-noise represented the next step in the evolution of digital printing, and today, several commercial implementations of green-noise dither arrays are available. Now since that first edition, there have been numerous discoveries and advancements in the field that have made the prior edition obsolete. As such, the time seemed right for a update in order to cover the additional topics of stochastic moir´e in color printing, a blue-noise model for multi-toning applications where the target printing device is capable of printing multiple shades of the same color ink, and even an update to the original blue-noise model, which was recently determined to have incorrectly characterized the statistical properties of visual pleasing dither patterns near gray-level 12 . This book even includes a discussion of the unique problems associated with lenticular printing, a means for producing a full-color holographic photograph. Chapter 1 serves as an introduction to those not familiar with the current trends in digital printing. This chapter reviews how halftoning was first performed 200 years ago and how things have changed with the introduction of low-cost, high resolution, ink jet printers. This chapter also reviews the pros and cons of the various halftoning techniques. In Chapter 2, we cover the details associated with constructing amplitude modulated dither arrays as a means of reviewing the alternate approach vi © 2008 by Taylor & Francis Group, LLC

to halftoning that was barely referenced in the prior edition. In Chapter 3, the spatial and spectral metrics used to study binary, aperiodic, dither patterns are introduced. These metrics offer a fundamental understanding of the inter-point relationships that may exist for a given point distribution. These metrics will later serve to compare the visual quality of a given halftoning technique. Integrated into this chapter are metrics associated with color halftoning. In Chapter 4, we cover the details associated with various human visual models that have been introduced over the years in the study of digital halftoning. In the prior edition, only a simple spectral model was described. Chapter 5 is a detailed review of the blue-noise model. Using the metrics of Chapter 3, Chapter 5 describes the ideal spatial and spectral characteristics of blue-noise and also compares several techniques for producing blue-noise halftone patterns. Through illustration, the many benefits of blue-noise halftoning are presented, showing that blue-noise is the optimal halftoning method in terms of detail rendition. Now due to the performance gain in computational efficiency by employing a blue-noise dither array, Chapter 6 reviews the use and construction of blue-noise dither arrays, adding the technique of direct binary search as a means of producing said arrays. Closing out our coverage of blue-noise halftoning, a modified blue-noise model is introduced in Chapter 7 intended to correct an error in the original formulation that had dismissed the use of hexagonal sampling grids as being inferior to rectangular when, in other areas of image processing, hexagonal grids were considered superior. This new model proves that, in fact, hexagonal grids are superior for producing dither patterns that minimize halftone textures. In Chapter 8, we detail the technique of direct binary search as a means of directly trying to produce dither patterns that optimize the distribution of printed dots according to the human visual models described in Chapter 4. In the prior edition of this book, direct binary search was only referenced in passing. Beyond blue-noise, Chapter 9 offers a detailed study of the printed dot. Looking at the printed dots of both laser and ink jet printers under high magnification, variations in dot size and shape are shown to have devastating effects on the resulting tone. Printed images, therefore, suffer from great degrees of tonal distortion – appearing, in many cases, far too light or far too dark compared with their original. This chapter also shows that when the resulting tone is predictable, the effects of distortion can be minimized through compensation techniques, but compensation is impractical when variations from printed vii © 2008 by Taylor & Francis Group, LLC

dot to printed dot are high. In these instances, it is only through clustering that distortions can be minimized. Chapter 10 presents green-noise as the halftoning technique that addresses distortions in the printed dot. By clustering pixels in a random fashion, green-noise is able to preserve the benefits afforded by stochastic dot patterns and, at the same time, ensure predictability in the printed tones. In presenting green-noise, a relationship between the gray-level, the number of pixels per cluster, and the average distance between clusters is established. Relying on existing algorithms for producing green-noise, this chapter analyzes several techniques in terms of their spatial and spectral characteristics; furthermore, an algorithm for producing green-noise dither arrays is presented. Chapter 11 introduces a means by which to construct gree-noise dither arrays, while color halftoning with green-noise is presented in Chapter 12. Here, a new framework for clustering pixels of like color is presented. This new framework also offers a way of regulating the overlapping of pixels of different colors. Also presented is a technique for constructing color dither arrays. Moving to Chapter 13, we look at the problem of stochastic moir´e, which characterizes the appearance of low-frequency color noise introduced by superimposing aperiodic dither patterns where, prior to this discovery, moir´e was only associated with the periodic patterns of AM halftoning. In this chapter, we predict when stochastic moir´e is its most visible and avoid by introducing varying degrees of clustering in the component dither patterns, showing that there are reasons for using green-noise even in devices that one would otherwise expect to use blue. The topic of multi-toning is described in Chapter 14 where multiple shades of gray are used to represent a continuous-tone image instead of just black and white pixels. Specifically, this chapter describes a blue-noise model for these multi-level dither patterns that includes details on how to characterize a visually pleasing pattern. This chapter also introduces several means by which optimal multi-tone patterns can be generated. Most important, this chapter derives the optimal distribution of dots across the available levels for producing a given shade of gray. This book is concluded with Chapter 16, but only after reviewing the unique process of lenticular printing in Chapter 15, where multiple images are column-wise spliced together and printed onto the flat side of a lenticular lens array such that, when viewed through the lenticules, viii © 2008 by Taylor & Francis Group, LLC

produces a full-color holographic display. The most well-recognized instance of lenticular printing may be baseball cards of the 1970s which flipped between one of two images as the top of the card was tilted toward or away from the viewer. At that time, images were printed onto paper and then glued to the back of the lens array. Today, digital presses print directly onto the plastic sheets, producing images of much higher visual quality. In total, this new edition includes over 250 pages of new material over its predecessor.

Daniel L. Lau Gonzalo R. Arce

ix © 2008 by Taylor & Francis Group, LLC

Acknowledgement In completing the first edition of this text, we would like to acknowledge the assistance of Dr. Shaun Love and his research group at Lexmark International, Dr. Jan P. Allebach at Purdue University, and Murat Meese and his advisor Dr. P. P. Vaidyanathan at the California Institute of Technology. We would also like to acknowledge the National Science Foundation’s financial support for much of the research that went into this book under grant CDA-9703088 and the financial support of Lexmark International, Inc. A great deal of thanks is also directed to Dr. Neal C. Gallagher for his professional guidance at various stages of this work. In developing the new material for this second edition, we would like to thank Robert Ulichney whose insights into the development of the original blue-noise model were an invaluable contribution to the creation of a modified blue-noise model, which we describe in this book in relation to hexagonal sampling grids. Also, we wish to thank Alvaro Gonzalez for his work with alpha-stable human visual models and Jan Bacca Rodrguez for his work with digital multi-toning and the introduction of a blue-noise model tailored to dither patterns composed of more than two pixel colors. Lastly, we would also like to thank David Roberts and Trebor Smith for their valuable insights into the world of lenticular printing.

xi © 2008 by Taylor & Francis Group, LLC

Contents 1 Introduction 1.1 AM Digital Halftoning . . . . . . . . . . 1.2 FM Digital Halftoning . . . . . . . . . . 1.3 AM-FM Hybrids . . . . . . . . . . . . . 1.3.1 Why Modern Digital Halftoning? 2 AM 2.1 2.2 2.3 2.4 2.5

Halftoning Dot Shape . . . . . . . . . Screen Angles and Moir´e . Screen Frequency . . . . . Supercells . . . . . . . . . Zero-Angle Dither Arrays

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

3 Stochastic Halftone Analysis 3.1 Point Processes . . . . . . . . . . . . . . 3.2 Spatial Statistics . . . . . . . . . . . . . 3.2.1 Pair Correlation . . . . . . . . . . 3.2.2 Directional Distribution Function 3.3 Spectral Statistics . . . . . . . . . . . . . 3.3.1 Radially Averaged Power Spectral 3.3.2 Anisotropy . . . . . . . . . . . . . 3.4 Color Halftoning . . . . . . . . . . . . . 3.4.1 RGB . . . . . . . . . . . . . . . . 3.4.2 CMY . . . . . . . . . . . . . . . . 3.4.3 CMYK . . . . . . . . . . . . . . . 3.4.4 Color Statistics . . . . . . . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . . .

. . . . . . . . . . . .

. . . .

. . . . .

. . . . . . . . . . . .

. . . .

. . . . .

. . . . . . . . . . . .

. . . .

1 4 5 9 9

. . . . .

21 21 23 31 34 35

. . . . . . . . . . . .

39 39 44 44 46 47 49 52 53 53 59 60 61

4 Halftone Visibility 63 4.1 Campbell’s CSF Model . . . . . . . . . . . . . . . . . . . 67 4.2 N¨as¨anen (Exponential) Model . . . . . . . . . . . . . . . 69 xiii © 2008 by Taylor & Francis Group, LLC

4.3 4.4

Mixed Gaussian Models . . . . . . . . . . . . . . . . . . 70 Alpha-Stable HVS Models . . . . . . . . . . . . . . . . . 72

5 Blue-Noise Dithering 5.1 Spatial and Spectral Characteristics . . 5.1.1 Spatial Statistics . . . . . . . . 5.1.2 Spectral Statistics . . . . . . . . 5.2 Error-diffusion . . . . . . . . . . . . . . 5.2.1 Eliminating Unwanted Textures 5.2.2 Edge Enhancement . . . . . . . 6 Blue-Noise Dither Arrays 6.1 Simulated Annealing . . 6.2 Void-and-Cluster . . . . 6.3 BIPPSMA . . . . . . . . 6.4 Dither Pattern Ordering

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

77 77 79 80 82 83 113

. . . . . .

. . . . . .

. . . .

129 . 131 . 144 . 150 . 163

7 Direct Binary Search 181 7.1 Halftoning by DBS . . . . . . . . . . . . . . . . . . . . . 181 7.2 Efficient DBS Algorithm . . . . . . . . . . . . . . . . . . 183 7.3 Effect of HVS model . . . . . . . . . . . . . . . . . . . . 196 8 Hexagonal Grid Halftoning 8.1 Spectral Aliasing . . . . . . . . . . . 8.2 Modified Blue-Noise Model . . . . . . 8.3 Hexagonal Sampling Grids . . . . . . 8.3.1 Hexagonal Grid Dither Arrays 9 Printers: Distortions and Models 9.1 Printer Distortion . . . . . . . . . 9.1.1 Dot-Gain . . . . . . . . . 9.1.2 Dot-Loss . . . . . . . . . . 9.2 Dot Models . . . . . . . . . . . . 9.2.1 Physical Models . . . . . . 9.2.2 Statistical Models . . . . . 9.3 Corrective Measures . . . . . . . 9.3.1 Tone Correction . . . . . . 9.3.2 Mode-Based Halftoning . . 9.3.3 Clustering . . . . . . . . . xiv © 2008 by Taylor & Francis Group, LLC

. . . . . . . . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

. . . . . . . . . .

. . . .

209 . 211 . 223 . 226 . 233

. . . . . . . . . .

239 . 239 . 240 . 244 . 246 . 248 . 260 . 273 . 273 . 273 . 275

10 Green-Noise Dithering 10.1 Spatial and Spectral Characteristics . . 10.1.1 Spatial Statistics . . . . . . . . 10.1.2 Spectral Statistics . . . . . . . . 10.2 EDODF . . . . . . . . . . . . . . . . . 10.2.1 Eliminating Unwanted Textures 10.2.2 Edge Enhancement . . . . . . . 10.2.3 Adaptive Hysteresis . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

11 Green-Noise Masks 11.1 BIPPCCA . . . . . . . . . . . . . . . . . . . 11.1.1 Pattern Robustness Using BIPPCCA 11.1.2 Constructing the Green-Noise Mask . 11.2 Optimal Green-Noise Masks . . . . . . . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

283 . 283 . 290 . 292 . 293 . 302 . 335 . 363

. . . .

371 . 372 . 375 . 404 . 409

12 Color Printing 423 12.1 Generalized Error-diffusion . . . . . . . . . . . . . . . . . 423 12.2 Multi-Channel Green-Noise Masks . . . . . . . . . . . . . 456 12.2.1 Color BIPPCCA . . . . . . . . . . . . . . . . . . 457 13 Stochastic Moir´ e 13.1 Spatial Analysis of Periodic Moir´e . . 13.2 Spatial Analysis of Aperiodic Moir´e . 13.3 Spectral Analysis of Aperiodic Moir´e 13.4 Minimizing Stochastic Moir´e . . . . . 13.5 Stochastic Moir´e and Green-Noise . .

. . . . .

. . . . .

471 . 474 . 479 . 485 . 501 . 503

14 Multi-Tone Dithering 14.1 Spectral Statistics of Multi-tones . . . . . . . . . . . 14.2 Multi-Tone Blue-Noise Model . . . . . . . . . . . . . 14.3 Blue-Noise Multi-Toning . . . . . . . . . . . . . . . . 14.3.1 Blue-Noise Multi-Toning with Error-Diffusion 14.3.2 Multi-Toning with DBS . . . . . . . . . . . . 14.4 Optimization . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

515 517 521 525 532 532 535

15 Lenticular Halftoning 15.1 Model-Based Error-diffusion . . . . 15.2 Iterative Tone Correction . . . . . . 15.2.1 Single-Pass Tone Correction 15.2.2 Correlated Columns . . . .

. . . .

. . . .

559 562 568 572 575

xv © 2008 by Taylor & Francis Group, LLC

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . .

15.2.3 Detecting Gamut Instabilities . . . . . . . . . . . 578 16 Conclusions

585

Bibliography

589

List of Figures

602

Index

643

xvi © 2008 by Taylor & Francis Group, LLC

Chapter 1 Introduction Digital halftoning refers to the process of converting a continuous-tone image or photograph into a pattern of black and white picture elements (Fig. 1.1) for reproduction by a binary display device such as an ink jet printer, which can only choose to print or not print dots. The human visual, acting like a low-pass filter, blurs these printed and not printed dots together to create the illusion of continuous shades of gray. Depending on the specific manner in which dots are distributed, a given display device can produce varying degrees of image fidelity with more or less graininess. According to the human visual system, randomly arranged and isolated dots, properly distributed, should produce images with the highest quality, maintaining sharp edges and other fine details. But at the same time, certain display and printing devices are incapable of reproducing isolated dots consistently from dot to dot and, consequently, introduce printing artifacts that greatly degrade those same details that the dot distribution is designed to preserve. For this reason, many printing devices produce periodic patterns of clustered dots, which are easier to produce consistently across the printed page. So in studying halftoning, the principal goal is to determine what is the optimal distribution of dots for that device and then to produce these patterns in a computationally efficient manner. When analog halftoning was perfected in 1880, continuous-tone, monochrome photographs were reproduced as line drawings authored by highly skilled craftsmen, usually on scratch board. But with halftoning, newspapers and magazines could cheaply reproduce photographs in their publications, making photography a lucrative industry and resulting in a technical revolution in photographic equipment. In terms of photo-lithography, the early halftoning processes (Fig. 1.2) involved 1 © 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

2

Figure 1.1: Gray-scale image reproduced as an analog halftone.

© 2008 by Taylor & Francis Group, LLC

3 LITHOGRAPHIC FILM (BEFORE DEVELOPING)

SILK SCREEN

PROJECTOR

LITHOGRAPHIC FILM (AFTER DEVELOPING)

Figure 1.2: The binary halftone obtained by projecting the negative of the original continuous-tone image through a fine silk screen. projecting light from the negative of a continuous-tone photograph through a mesh screen, such as finely woven silk, onto a photo-sensitive plate. Bright light, as it passes through a pin-hole opening in the silk screen, would form a large, round spot on the plate. Dim light would form a small spot. Light sensitive chemicals coating the plate would then form insoluble dots that varied in size according to the tones of the original photographs. After processing, the plate would have dots where ink was to be printed raised slightly above the rest of the plate. Later versions of the halftoning process employed screens made of glass that were coated, on one side, by an opaque substance [116]. A mesh of parallel and equidistant lines were scratched in the opaque surface. A second mesh of parallel and equidistant lines were then scratched in the opaque surface running perpendicular to the original set. Screens would then differ in the number of lines per inch that had been scratched. While finer screens created better spatial resolutions (detail), the quality of the printing press would limit the fineness of the mesh. Later still, the glass plate mesh was replaced altogether with

© 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

4

HALFTONE CELL DOT SHAPE

SCREEN ANGLE (SCREEN FREQUENCY) -1

Figure 1.3: The screen frequency, dot shape, and screen angle for an analog halftone pattern. a flexible piece of processed film, placed directly in contact with the unexposed lithographic film [28]. This contact screen had direct control of the dot structure (Fig. 1.3) being able to control the screen frequency (the number of lines per inch), the dot shape (the shape of dots as they increase in size from light to dark), and the screen angle (the orientation of lines relative to the positive horizontal axis).

1.1

AM Digital Halftoning

Today, printing is a far more advanced process with the introduction of non-impact printing technologies and the emergence of desktop publishing. Brought on by advancements in the digital computer [28], the photo-mechanical screening process introduced in 1880 has, in many instances, been replaced by digital imagesetters. In some instances, printing is no longer binary as continuous-tone dye-sublimation printers are now readily available but due to their speed and material requirements (special papers and inks), have not reached the wide-spread acceptance of color ink jet or electro-photographic (laser) printers, although dye-sublimation has made a major come back with a consumer market for 4 inch by 6 inch photograph printers. In these digital printers, the halftoning process of projecting a continuous-tone original through a halftone screen has been replaced with a raster image processor (RIP) that converts each pixel of the original image from an intermediate tone directly into a binary dot based upon a pixel-by-pixel comparison of the original image with an array of thresholds (Fig. 1.4). Pixels of the original with intensities greater than their corresponding threshold are turned “on” (printed)

© 2008 by Taylor & Francis Group, LLC

1.2. FM DIGITAL HALFTONING

5

HALFTONED SUB-BLOCK ORIGINAL IMAGE

1

1

1

0

0

0

1

1

1

1

1

0

0

0

1

1

1

0

0

0

0

0

1

1

1

1

0

0

0

0

0

1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

HALFTONED IMAGE

36 32 39 17 11 9 16 15 42 33 39 26 24 21 38 31 54 34 36 38 33 39 40 34 59 50 37 35 35 37 33 41 59 60 53 42 40 41 3 48 9 58 17 27 25 15 7

1

51 56 59 60 59 60 11 61 29 60 38 46 44 36 23 5 41 47 49 55 60 58 19 57 40 58 52 58 56 50 34 13 37 40 41 44 48 48 31 46 48 44 60 64 62 54 42 21

IMAGE SUB-BLOCK

30 47 59 63 61 53 41 20 18 39 51 57 55 49 33 12 10 28 37 45 43 35 22 4 2

8 16 26 24 14 6

0

DITHER ARRAY

Figure 1.4: Digital AM halftoning. in the final halftoned image while pixels less than their corresponding thresholds are turned “off”. For large images, the threshold array is tiled end-to-end until all pixels of the original have a corresponding threshold. When first introduced, RIPs imitated the halftone patterns of contact screens by employing clustered-dot ordered dithering, where the threshold array is small (8 × 8, 12 × 12, or 16 × 16) and is composed of consecutive thresholds arranged along a spiral path radiating outward from the array’s center (Fig. 1.5). These arrangements of thresholds result in a single cluster of “on” pixels centered within each tile or cell, forming a regular grid of round dots that vary in size according to tone. These techniques are commonly referred to as amplitude modulated or AM digital halftoning due to their modulating of the size of printed dots. Like contact screens, the resulting patterns vary in their screen frequency, dot shape, and screen angle.

1.2

FM Digital Halftoning

Due to freedoms afforded by digital printers, the idea of printing isolated pixels in an effort to minimize halftone visibility (the visibility of the individual dots to a human viewer) emerged as an alternative to clustered-dot dithering. By maintaining the size of printed dots for all

© 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

6 3

9 17 27 25 15 7

1

11 29 38 46 44 36 23 5 19 40 52 58 56 50 34 13 31 48 60 64 62 54 42 21 30 47 59 63 61 53 41 20 18 39 51 57 55 49 33 12 10 28 37 45 43 35 22 4 2

8 16 26 24 14 6

0

Figure 1.5: An 8 × 8 dither array (65 gray-levels). gray-levels as individual pixels, new dispersed-dot halftoning techniques varied, according to tone, the spacing between printed dots, earning the name frequency modulated or FM halftoning. Early FM halftoning techniques were proposed by Bayer [11] and Bryngdahl [15] and produced an ordered arrangement of isolated dots. These techniques, like AM halftoning schemes, quantized each pixel independently of its neighbors (point process) according to a dither array but with consecutive thresholds dispersed as much as possible. The problem associated with these early FM techniques is that, as in the case of Bayer’s dither array (Fig. 1.6), resulting halftoned images (Fig. 1.7) suffered from a periodic structure that added an unnatural appearance [123]. For a far better approach to FM halftoning, Floyd and Steinberg [40] proposed the revolutionary error-diffusion algorithm (shown in Fig. 1.8 and covered in Chapter 5), an adaptive technique that quantized each pixel according to a statistical analysis of an input pixel and its neighbors (neighborhood process), leading to a stochastic arrangement of printed dots. While this neighborhood process had higher computational complexity, the resulting patterns had apparent spatial resolutions much higher than those achieved by clustered dots (Fig. 1.9); furthermore, as a stochastic patterning of dots, the patterns eliminated the occurrence of the moir´e that was produced by the superimposing of two or more regular patterns. By using FM halftoning schemes, printers maximize their apparent spatial resolution and are relieved of the strict tolerances on screen angles and screen registration. They can also use more and more colors to produce larger color gamuts (the set of achievable colors that can be produced by the printer) [76]. Notably, though, with its associated advantages, FM halftoning has, with few exceptions, only been employed

© 2008 by Taylor & Francis Group, LLC

1.2. FM DIGITAL HALFTONING

7

0 58 14 54 3 57 13 53 32 16 46 30 35 19 45 29 8 48 4 62 11 51 7 61 40 24 36 20 43 27 39 23 2 56 12 52 1 59 15 55 34 18 44 28 33 17 47 31 10 50 6 60 9 49 5 63 42 26 38 22 41 25 37 21

Figure 1.6: An 8 × 8 Bayer’s dither array (65 gray-levels). in ink jet printers. The problem is the increased scrutiny placed on the printer’s ability to print small, isolated dots. Noting Fig. 1.10, the ideal display produces dots that completely cover the sample area associated with a given pixel without overlapping neighboring pixels’ sample areas. By printing all pixels, perfect black can be obtained. In a real printing device, individual printed dots are round and in order to produce perfect black, must be large enough as to cover the entire sample area (Fig. 1.11). By overlapping neighboring sample areas, though, the resulting tone is darker than the fraction of all pixels that are printed. Assuming that dots are printed consistently (small variation in size and shape from printed dot to printed dot), this distortion in tone can be corrected by adjusting the intensity level of the input image before halftoning. The amount of compensation depends on the arrangement of printed dots with dispersed-dot (FM) patterns requiring greater degrees of correction than clustered (AM). Ink jet printers are such a device that prints (approximately) round dots that overlap neighboring pixels (Fig. 1.12). Being able to compensate for distortions introduced by the printing process for any arrangement of dots, ink jet printers can enjoy the benefits associated with FM halftoning. In the electro-photographic printing process, the size and shape of dots varies greatly from printed dot to printed dot (Fig. 1.13) and this variation can only be minimized when dots are grouped together to form clusters. Images printed using isolated dots tend to show severe tonal distortion and exhibit a great deal of variation in tone across the printed page, as illustrated in Fig. 1.14. For this reason, unreliable printing devices such as lithographic presses and laser printers continue to use AM halftoning schemes.

© 2008 by Taylor & Francis Group, LLC

8

CHAPTER 1. INTRODUCTION

Figure 1.7: Gray-scale image halftoned using Bayer’s dither printed at 150 dpi.

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

x [n] +

9

y [n]

Y -

+ x e[n]

b

+ Y y e[n]

Figure 1.8: The error-diffusion algorithm.

1.3

AM-FM Hybrids

Now as printers began achieving print resolutions greater than 1, 200 dpi, the limits of FM halftoning were also being reached, just as AM’s was in 1990 before the introduction of low-cost color. Researchers, therefore, begin to look at AM-FM hybrids which produce dot clusters that vary, according to tone, in both their size and spacing [6, 66, 75, 94, 110, 125, 126, 128] (Fig. 1.15). When considering the reproduction of monochrome images, AM-FM hybrids are, in general, capable of producing patterns with lower visibility (higher spatial resolution) compared to AM and, if stochastic, do so without a periodic structure adding an artificial texture to the printed image. With some amount of clustering, these halftones are easier to print reliably and with little variation in the resulting tone. In the reproduction of color, stochastic hybrids also maintain the same freedom from periodic moir´e associated with FM halftoning.

1.3.1

Why Modern Digital Halftoning?

Since its publishing in 1987, Ulichney’s Digital Halftoning [122] has proven to be the single most unifying book about halftoning research. While a large portion of that book is devoted to AM halftoning, its most significant contribution is its description of error-diffusion as a generator of blue-noise, visually pleasing halftone patterns composed of randomly placed, isolated dots. Ulichney used the term “blue” as a reference to the spectral content of the random patterns as being composed exclusively of high-frequency spectral components just as blue light is composed exclusively of the high-frequency spectral components of white light. Ulichney further showed that the average distance between same colored dots was directly related to the gray-level of the image before halftoning. In summary, Ulichney’s work tells us when

© 2008 by Taylor & Francis Group, LLC

10

CHAPTER 1. INTRODUCTION

Figure 1.9: Gray-scale image halftoned using Floyd’s and Steinberg’s error-diffusion algorithm.

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

(a)

11

(b)

(c)

(d)

Figure 1.10: Clusters of (a) one, (b) two, (c) three, and (d) four printed dots from an ideal printer with the solid lines indicating the border between neighboring output pixels.

(a)

(b)

(c)

(d)

Figure 1.11: Clusters of (a,c) one and (b,d) four printed round dots where the dots of (a) and (b) cover the entire sample area while the dots of (c) and (d) do not cover the corners. one FM halftone is better, visually, than another. It is thanks to the improved image quality produced by blue-noise that ink jet printers are so prevalent today as these printers have directly benefited from the improved spatial resolution afforded by randomly placed, isolated dots. In the decade that followed the publishing of Digital Halftoning, many technological advancements were documented directly addressing the creation of blue-noise. The most notable of which may be introduction of blue-noise dither arrays that convert the pixels of continuoustone images to binary through a pixel-wise comparison of the original pixel with a corresponding pixel in the dither array. A process identical to AM halftoning with the exception that these blue-noise dither arrays are typically much larger (128 × 128 or 256 × 256) as seen in Fig. 1.16. Being a point process operation, blue-noise dither array halfton-

© 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

12

(a)

(b)

(c)

(d)

Figure 1.12: Clusters of (a) one, (b) two, (c) three, and (d) four printed dots printed on ink jet printer.

(a)

(b)

(c)

(d)

Figure 1.13: Clusters of (a) one, (b) two, (c) three, and (d) four printed dots printed on a laser printer. ing is far less computationally complex than error-diffusion, making it a viable technique for commercial printing applications whose image sizes are orders of magnitude larger than those associated with desktop printers. Now the error-diffusion algorithm, itself, has also undergone some major improvements over the past two decade [58]. One of the more profound improvements has been the introduction of threshold modulation, where the threshold used to quantize an input pixel to either one or zero is varied in a given fashion. An early approach to threshold modulation was proposed by Ulichney [122], who suggested adding white-noise to the threshold in order to break up worm patterns and periodic textures (Fig. 1.17). This approach was later shown by Knox [55] to be equivalent to adding low-level white-noise to the original image before halftoning. Another approach to threshold perturbation was later proposed by Eschbach and Knox [34], who suggested varying the threshold by a scalar multiple of the current input pixel. This oper-

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

13

(a)

(b)

(c)

(d)

Figure 1.14: Gray-scale ramps produced by a laser printer using (a) FM halftoning and (b-d) AM halftoning with halftone cells of size 8 × 8, 12 × 12, and 16 × 16 pixels respectively.

© 2008 by Taylor & Francis Group, LLC

14

CHAPTER 1. INTRODUCTION

Figure 1.15: Gray-scale image halftoned using an AM-FM hybrid.

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

15

Figure 1.16: A 128 × 128 blue-noise mask. ation sharpened the resulting halftone pattern, thereby eliminating the need for applying an edge sharpening filter to the input image prior to halftoning. A completely different approach to halftoning, proposed since Ulichney’s book, is Analoui’s and Allebach’s [5] direct binary search (DBS) algorithm. This algorithm iteratively swaps and toggles the binary halftone pattern according to the error between the original continuous-tone image and the binary halftone image as defined by a model of the human visual system. Typically, the human visual system is modeled as a low-pass filter, and by employing it into the halftoning algorithm, causes same colored dots to be spread as far apart as possible, creating a power spectrum that models blue-noise. While far more computationally complex than even error-diffusion, the DBS algorithm’s complexity has greatly diminished since first being proposed and may one day challenge error-diffusion. Looking back at error-diffusion, another innovation or particular importance to halftoning is Pappas’ and Neuhoff’s [96] model-based halftoning, which employs error-diffusion in the traditional sense but with the printed dot modeled by a round circle, Fig. 1.11, that overlaps neighboring pixels. This phenomenon of overlapping neighboring pixels, in the printed output, is generally referred to as dot-gain and is typically corrected for by changing the input intensity levels prior to halftoning. Pappas and Neuhoff, by employing a dot model into

© 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

16

WORM PATTERNS

PERIODIC TEXTURES

FLOYD-STEINBERG

ULICHNEY

Figure 1.17: A comparison of Floyd’s and Steinberg’s original errordiffusion algorithm with Ulichney’s threshold modulated error-diffusion algorithm. the halftoning algorithm, directly addressed the problems of printer distortion, where the printer’s ability to print isolated dots reliably (with little variation in the size and/or shape of dots from printed dot to printed dot) affects and sometimes dictates which arrangements of dots, whether dispersed as in FM or clustered as in AM, can or cannot be used. This brings about another very important aspect of halftoning that has surfaced as a direct result of the blue-noise model, that is that even with ten years of technological improvements, blue-noise has made little inroads into electro-photographic printing or digital offset printing. The reason is clear being that blue-noise and FM halftones in general are expensive, if not impossible, to produce consistently across a page, from page to page, or from day to day or even hour to hour. Figure 1.18 illustrates the resulting variation in tone across a page for 7 an error-diffused halftone representing gray-level 10 produced by a laser printer set at 1,200 dpi. In this figure, the average variation in tone along the vertical axis is plot along side a picture of the page, and the average variation in tone along horizontal is plotted below the picture. In order to be reasonably efficient, a process like Pappas’ and Neuhoff’s requires that the statistical characteristics of the printed dot do not

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

17

change either over time or across the page or according to the location of other dots. So a real challenge has come to face researchers as they try to improve the image quality in these types of printers. What they are finding is that halftoning algorithms that cluster same color pixels together, in a random fashion, hold the key by creating patterns that are easier to produce consistently from page to page and by creating color halftone patterns without moir´e. When first published in 2001, the goal of this book was to introduce, in a concise and unifying framework, these modern approaches to digital halftoning. The new model introduced was that of greennoise, a statistical model that described the spatial and spectral characteristics of visually pleasing dither patterns composed of a random arrangement of clustered dots that vary with gray-level in both their spacing apart and their size and shape. The term “green” referred to the mid-frequency only content of corresponding halftone patterns as green light is the mid-frequency component of white. What made green-noise a unifying framework is that it was tunable, describing the ideal characteristics for a range of cluster sizes where small clusters were reserved for reliable printing devices that showed small variation in the size and shape of printed dots and large clusters were reserved for unreliable printing devices. In addition to describing the traits that would determine when one green-noise halftone pattern was better than another, we addressed the problem of computational efficiency by introducing the green-noise mask [63], a variation on blue-noise dither arrays for producing greennoise. Like the green-noise model, these new masks were also tunable, making them applicable to a wide range of printing devices. Having blue-noise as a limiting case of green-noise, the green-noise mask construction procedure described could even be applied to constructing the previously invented blue-noise arrays. As a last component to greennoise, the first edition of this book addressed green-noise’s application to color printing where the primary task is to reproduce color as accurately as possible without introducing visible moir´e patterns. While color printers were once very expensive and therefore rare, blue-noise has made color printing common place in both homes and offices. For any halftoning research to be noteworthy, it must address color printing, and we showed how green-noise was applied in color printers without introducing moir´e. Now in presenting a second edition of this book, we wanted to broaden the range of topics addressed by including a new chapter focus-

© 2008 by Taylor & Francis Group, LLC

CHAPTER 1. INTRODUCTION

18

10 9

vertical axis (inches)

8 7 6 5 4 3 2 1

resulting tone

0 0.6 0.7 0.8 resulting tone 0.8 0.7 0.6

0

1

2 3 4 5 6 horizontal axis (inches)

7

8

Figure 1.18: The resulting printed page produced by a laser printer at 7 1,200 dpi where the gray-level 10 is produced using error-diffusion. The average variations in tone along the horizontal axis and the vertical axis are also shown, plotted alongside and below the picture of the printed page. ing on the prior art of AM dither array construction, defining in greater detail the benefits and challenges associated with irrational screen angles. We have also included a thorough review of the original blue-noise model for hexagonal sampling grids, where we show that the original model was incorrect in its assumptions about distributing minority pix-

© 2008 by Taylor & Francis Group, LLC

1.3. AM-FM HYBRIDS

19

els in areas close to 50% gray. Specifically, it was determined that near 50% gray, the sampling grid was restricting the placement of printed dots, which was injecting periodic textures into the dot pattern. So by introducing a minimum amount of clustering, a halftone can be produced that accepts a certain amount of low-frequency graininess in exchange for reducing the periodic texture produced by the underlying sampling grid. As such, it is possible to produce visually pleasing dither patterns on hexagonal sampling grids which have a smaller maximum spatial frequency compared to the equivalent rectangular grid. In the area of color halftoning, this edition greatly expands the prior discussion of superimposing component green-noise dither patterns by focusing on the problem of minimizing stochastic moir´e, the aperiodic equivalent of the moir´e produced by superimposing AM halftones. In particular while it is generally accepted that superimposing FM halftones results in a certain amount of color noise, it is the superposition of green-noise dither patterns that results in significantly worse amounts of this noise as the frequency content of stochastic moir´e is directly related to the frequency content of the component dither patterns. Given this relationship, stochastic moir´e can be minimized by properly varying the coarseness of the component patterns across colors. Now in a problem closely related to color halftoning, a chapter has been added that focuses on the process of multi-toning and the creation of patterns composed of more than two gray-levels. Specifically, we address this new area by means of a blue-noise multi-tone model describing, not only the optimal distribution of multi-tone pixels, but also the optimal distributions of the various intensity levels between the component inks. In particular, we look at the change in halftone texture across gray-levels as the concentrations of component inks vary choosing the concentrations that minimize discontinuities in texture as being optimal. And finally, this book includes an introduction to the lenticular printing problem where multiple images are spatially multiplexed onto the back of a plastic lens array such that, when viewed through the lenses, only one component image is visible depending upon the angle at which the lens arrayed is viewed.

© 2008 by Taylor & Francis Group, LLC

Chapter 2 AM Halftoning As described in the Introduction, amplitude modulated (AM) halftoning is a process of representing varying shades of gray by a regular grid of round dot clusters that vary in size, with dark shades of gray represented by large clusters and light shades by small. This type of halftoning is also generally referred to as periodic, clustered-dot halftoning and is accomplished by means of a dither array whereby pixels are thresholded on a pixel-wise basis with the corresponding entries in the array. Perhaps the best way to describe the construction of these dither arrays for AM screening is in terms of Adobe’s PostScript page description language, which generates dither arrays according to a user defined screen frequency, angle, and spot function, as will be described in the following sections.

2.1

Dot Shape

When characterizing AM screens, dot shape refers to the specific arrangement of thresholds within the dither array. This ordering dictates how clusters vary in both size and shape according to tone. The shape of dots is most clearly recognizable at gray-level 12 (number of black dots equals the number of white dots), and the most common dot shapes are round, square, and elliptical [23]. Special effect shapes have also been introduced [12, 83]. In PostScript, dot shape is controlled through the user supplied spot function where, initially, pixels of the continuous-tone image, at the printing device’s native resolution, are divided into square tiles with side lengths equal to the inverse of the supplied screen frequency 21 © 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

22

Y

+1 +1

-1 y) enc equ n fr ree (sc

-1 y) c n ue freq n e re (sc

X

-1 -1

screen angle

Figure 2.1: Assignment of pixels to a halftone cell as a function of the screen angle and frequency. parameter and oriented at the supplied screen angle (Fig. 2.1). These tiles then form the halftone cells where, inside each cell, pixels are assigned a horizontal (x) and vertical (y) coordinate ranging from −1 at the left and bottom edges of the cell to +1 at the right and top edges in the rotated space. The origin of the cell is then located at its geometric center. Having a unique x and y coordinate, a cost value is assigned to each pixel according to the spot function, which takes a pixel’s x and y coordinates as its input. An example of such a cost function is demonstrated in Fig. 2.3, where we measured the Euclidean distance between a pixel and the cell origin as cost(x, y) = x2 + y 2 ,

(2.1)

for an inverse screen frequency of 10 pixels and a screen angle of 15 degrees. Examples of other possible spot functions include a line screen

© 2008 by Taylor & Francis Group, LLC

´ 2.2. SCREEN ANGLES AND MOIRE

23

as cost(x, y) = x,

(2.2)

cost(x, y) = max(x, y),

(2.3)

a square spot as or even a diamond spot as cost(x, y) = |x| + |y|.

(2.4)

Shown in Fig. 2.2 are demonstrations of the resulting dot shapes. Regardless of the spot function, histogram equalization is then applied to the range of cost values within the cell, such that the pixel with the lowest cost has a threshold value near 0 while the largest cost pixel has a value near 255, assuming 8 bits per pixel gray-scale. All other cost values are assigned in such a way as to have a uniform distribution from 0 to 255. In the case of two or more pixels having the same cost, it is up to the printer manufacturer to decide which pixel is assigned the lower range. In the case of Fig. 2.4, where the resulting dither array from Fig. 2.3 is shown, the angle between the subject pixel and the +x axis was used to break the ties with the smaller angles being assigned the smaller threshold.

2.2

Screen Angles and Moir´ e

The screen angle is the orientation of the screen lines relative to the horizontal axis whose optimal value is a function of the human visual system, where directional artifacts are their least noticeable when oriented along the 45 degrees diagonal [18], as demonstrated in Fig. 2.5. It follows that for monochrome printing, this screen angle should also be 45 degrees. For computational ease, screen angles are typically restricted to rational angles (rise over run equals integer over integer), where each tile is of the same size and shape. Shown in Fig. 2.6 is the division of pixels into a tiling of halftone cells of size 6 × 6 pixels at a rational angle of 9.5 degrees. Note how each tile is identically shaped. Irrational angles create a tiling that requires multiple threshold arrays due to varying tile shapes. Figure 2.7 shows the tiling of cells of size 6 × 6 pixels at an irrational angle of 16.6 degrees where tiles are not identically shaped as indicated by the cells labeled A and B. Using irrational angles requires a raster image processor (RIP) to generate

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

24

(a) round

(b) square

(c) line

(d) black square

Figure 2.2: Various dot shapes proposed for AM halftoning.

© 2008 by Taylor & Francis Group, LLC

´ 2.2. SCREEN ANGLES AND MOIRE

25

Figure 2.3: Assignment of a cost value to the pixels of a given cell where, within each pixel, an (top) x-coordinate, (center) y-coordinate, and (bottom) cost value are shown.

threshold arrays on the fly to match the size and shape of a particular tile [12]. Given the computational expense of (1) mapping image row and column coordinates m and n into halftone cell coordinates x and y, (2) calculating costs for each pixel within a cell, (3) sorting said pixels from lowest to highest cost, and then (4) equalizing and assigning thresholds within the cell, early PostScript printers required that the user supplied

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

26

212

221

243

252

205

172

146

151

165

193

231

247

233

181

139

120

96

80

108

127

179

240

195

129

92

73

51

42

61

89

134

203

224

167

110

63

35

25

16

33

70

118

170

214

153

82

44

14

7

4

23

49

94

144

148

101

54

18

0

2

11

40

77

158

210

174

113

66

28

9

21

30

59

106

162

219

198

136

87

56

37

47

68

85

125

191

238

184

122

103

75

99

115

132

186

229

250

226

188

160

155

141

177

200

245

236

217

207

255

Figure 2.4: Assignment of thresholds to the halftone cell of Fig. 2.3, where the angle between the subject pixel and the +x axis was used to break ties. screen frequency and angle parameters correspond to a rational screen angle; otherwise, the print engine would round the supplied parameters to values that do. In this way, each and every halftone cell would be of the same size and shape with identical x- and y-coordinates assigned to corresponding pixels within cells. As such, only for the very first cell would the printer derive a threshold matrix, which is then tiled end-to-end for all subsequent cells. In the case of irrational screen angles, halftone cells vary in size and shape from cell to cell, and therefore each cell requires reprocessing. Hence, it was not until the introduction of PostScript Level 2, in 1991 with the emergence of color laser printers, that Adobe added an accurate screen, boolean operator for specifying halftone screens with a value of false telling the processing engine to round off the frequency and angle to the nearest rational angle, as performed previously. A value of true tells the printer to perform the added computation necessary to maintain the requested screen angle. Now the significance of color printing, as the impetus for irrational angles, is that it is only when printing color that a restriction to rational screen angles becomes visually apparent due to the creation

© 2008 by Taylor & Francis Group, LLC

´ 2.2. SCREEN ANGLES AND MOIRE

27

(a) 0 degrees

(b) 30 degrees

(c) 45 degrees

(d) 70 degrees

Figure 2.5: The effects of varying screen angle.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

28

Figure 2.6: The tiling of 6 × 6 halftone cells for a rational screen angle of 9.5 degrees.

A

B

Figure 2.7: The tiling of 6 × 6 halftone cells for an irrational screen angle of 16.6 degrees.

© 2008 by Taylor & Francis Group, LLC

´ 2.2. SCREEN ANGLES AND MOIRE

29

(a) 5 degrees

(b) 10 degrees

(c) 15 degrees

(d) 30 degrees

Figure 2.8: The moir´e patterns created by offsetting two AM halftone patterns by (a) 5 degrees , (b) 10 degrees, (c) 15 degrees, and (d) 30 degrees. of moir´e (Fig. 2.8), the periodic interference pattern associated with overlapping two or more regular grids. In color printing, the halftone patterns of cyan, magenta, yellow, and black (CMYK) inks are superimposed with AM patterns, composed of regular grids of printed dots, exhibiting moir´e. While this interference cannot be altogether eliminated, it is through the screen angles 15 degrees , 75 degrees, 0 degrees, and 45 degrees for cyan, magenta, yellow, and black, respectively, that moir´e is minimized, creating a pleasant rosett´e pattern such as the solid

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

30

Figure 2.9: The solid spot rosett´e pattern created by setting the CMYK channels to screen angles 15 degrees, 75 degrees, 0 degrees, and 45 degrees respectively.

spot rosett´e in Fig. 2.9. Still, the computational complexity of generating a unique dither array for each and every halftone cell, for each component color, is still so high that many printer manufacturers continue to round off the requested screen angle and frequency to the nearest rational angle. But they do so over more than one halftone cell, constructing a single dither array covering an N × N block of cells such that the conglomerate is, itself, a halftone cell at a rational screen angle while the individual cells are at a potentially irrational screen angle of

screen angle =

(integer number of rows/N ) . (integer number of cols/N )

(2.5)

So while the array does not achieve the infinite precision of an irrational screen angle, it does achieve N times the precision of a rational angle while only incurring N 2 of the computational complexity and memory required to derive the rational angle threshold array. Shown in Figs. 2.10 and 2.11 are two examples of this technique for a 15 degrees screen angle and inverse frequency of 10 pixels for N equal to 2 and 3, where, in the case of Fig. 2.4 and N = 1, rounding off to the nearest rational angle resulted in a screen angle of 16.6992 degrees and inverse frequency of 10.4403 pixels. Using a 2 × 2 block results in a screen angle of 14.7436 degrees and inverse frequency of 9.8234 pixels, while a 3 × 3 block results in an angle of 15.4222 degrees and inverse frequency of 10.0277 pixels.

© 2008 by Taylor & Francis Group, LLC

2.3. SCREEN FREQUENCY

31 209

215

236

247

220

193

156

135

140

151

188

225

244

255

252

231

172

119

98

77

71

92

124

167

201

179

163

185

206

233

241

204

130

87

50

39

34

55

82

146

199

249

228

169

144

110

104

115

134

174

222

183

114

61

29

7

13

23

66

108

175

238

190

126

83

64

45

61

88

120

195

162

103

45

18

2

0

15

42

100

159

217

153

93

53

24

18

29

51

99

158

177

106

63

21

10

5

26

58

111

180

209

147

69

32

13

0

8

37

75

131

196

143

79

53

31

37

47

85

127

201

239

136

77

40

5

2

10

34

72

139

212

164

122

90

69

74

95

116

170

228

249

155

96

48

26

16

21

56

91

150

214

223

185

148

138

132

154

191

217

244

255

193

118

85

59

42

67

80

123

187

236

244

233

212

207

201

179

163

185

206

233

220

171

128

112

102

107

142

166

225

246

255

246

228

169

147

110

104

115

128

174

222

230

204

182

161

177

198

209

215

236

244

238

187

126

83

67

45

59

88

120

195

252

241

220

193

156

130

138

151

185

225

217

153

93

53

24

18

29

48

96

158

252

231

172

119

98

77

71

92

124

167

212

139

69

34

13

2

5

37

75

134

241

204

135

87

50

37

34

53

82

146

199

136

77

40

8

0

10

32

72

144

209

183

114

61

29

7

10

23

66

108

177

155

99

51

26

16

21

56

91

150

214

162

103

45

18

0

2

15

42

100

159

193

118

85

61

42

64

80

123

190

236

175

106

63

21

13

5

26

58

111

180

220

171

131

112

102

107

142

166

225

249

196

140

79

55

31

39

47

85

127

201

239

230

204

182

161

177

198

164

122

90

69

74

95

116

170

228

249

252

241

223

188

148

143

132

154

191

217

247

233

212

207

255

Figure 2.10: Assignment of thresholds to a 2 × 2 conglomerate of halftone cells with a screen angle of 14.7436 degrees and inverse frequency of 9.8234 pixels.

2.3

Screen Frequency

In AM halftoning, the screen frequency is the number of lines or rows of clustered dots per inch of the resulting halftone pattern. Like the original glass plate screens, the finer screens create patterns with higher spatial resolutions (Fig. 2.12). Depending on the resolution of the printer (measured in dots per inch), screen frequency is limited by the number of unique gray-levels the printer can represent. This relationship is

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

32

206 216 232 247 221 191 150 142 145 147 183 224 99

79

76

91

189 184 197 223 241 239 198 132

255 249 226 181 127 89

56

38

35

58

86

137 201

249 228 179 145 122 117 130 161 202 229 175 109

63

30

10

12

28

61

107 173

234 244 252 234 182 132 201 173 153 170 188 214 237 213 140

98

93

70

57

72

54

28

23

41

104 150 208 160 62

94

130 178

45

17

0

2

15

48

96

109 171 165 104

53

20

5

7

22

51

102 170

71

73

122 188 234

247 224 178 137 107

96

102 122 158 209 192 124

67

31

7

5

15

44

88

143 193 124

232 183 130

86

61

48

51

59

26

2

0

10

33

80

135 210 155 114

216 147

91

58

28

15

22

40

81

140 195 127

91

158 226 211 163 135 117 119 140 168 209 237 252

206 145

76

35

12

2

5

25

68

119 186 163 101

142

79

38

10

0

7

33

66

117 196 200 153 106

150

99

56

30

17

20

43

84

135 204 239 205 169 148 137 156 176 215 232 242 252 244 205 153 101

89

63

45

53

71

114 163 219 252 244 231 218 215 175 153 163 183 203 222 231 169 106

191 127

221 181 132 109

94

73

112 168 187 119

75

39

18

13

20

52

65

46

36

49

78

85

83

96

43

33

25

40

84

66

68

81

153

112 158 214 244

114 174 236 242 219 204 196 186 195 187 192 213 234 249

111 166 221 247 255 252 239 200 163 127 119 124 140 182 228

104 124 155 211 242 255 255 225 195 136 126 103

75

59

67

98

65

39

26

31

54

93

145

2

7

28

70

122 189

94

113 143 193 218 148

85

46

18

132 179

226 198 175 160 165 193 210 226 236 247 245 205 146

86

66

54

44

74

84

133 212 137

83

36

13

0

5

23

57

117 184

249 239 229 208 171 143 135 158 174 221 235 185 116

76

37

17

24

34

64

123 173 156

96

49

20

10

15

41

72

130 197

255 241 202 150 109

88

80

91

47

27

7

4

14

51

101 151 176 111

78

52

33

44

62

223 161 104

62

44

33

52

78

56

9

0

2

22

42

91

91

80

88

109 150 202 241 255

197 130

72

41

15

10

20

49

96

156 168 118

59

29

19

12

32

71

111 180 230 221 174 158 135 143 171 208 229 239 249

184 117

57

23

2

0

13

36

83

137 207 128

79

69

39

49

61

81

141 200 240 247 236 226 210 193 165 160 175 198 226

189 122

70

28

7

5

18

46

85

148 218 188 138 108

89

99

121 131 190 220 250 255 242 211 155 124 104

145

93

54

31

26

39

65

98

67

59

75

101 153 205 244 247 237 227 215 176 156 137 148 169 205 239 204 135

179 132

114 166 215 165

96

111 176 155 106

160 215 166 114

106 169 231 217 198 178 158 148 170 210 218 231 244 252 219 163 114

45

63

89

84

43

20

17

30

56

99

150

66

33

5

0

10

38

79

142

68

25

7

2

12

35

76

145 206

81

40

22

15

28

58

91

147 216

73

51

48

61

86

130 183 232

83

85

249 234 213 192 187 195 184 194 201 217 239 236 174 114

78

49

36

46

65

252 234 207 166 138 118 116 133 161 209 226 158

91

52

20

13

18

39

75

127 195 140

242 212 156 111

113 154 210 135

80

33

10

0

5

26

59

119 187 168 112

88

68

65

83

232 186 121

73

40

25

32

42

70

123 191 143

169 100

50

22

5

7

20

53

103 164 171 109

151

95

44

15

2

7

31

67

124 192 209 158 122 102

62

41

23

28

54

98

140 213 237 214 188 170 153 173 201

72

57

70

93

132 182 234 252 244 234

47

15

0

2

17

45

93

171 106

60

27

12

10

30

63

108 174 227 202 161 130 117 122 145 179 228 249

199 136

85

88

131 196 237 241 223 197 184 189

176 128

58

35

37

55

90

75

78

98

109 132 181 221

53

96

101 163 186 119

94

71

228 182 140 124 119 127 163 200 239 249 252 247 221 166 111

80

106 153 200 196 117

104 161 223

159 208 150 104

96

127 191

107 137 178 224 247

126 179 224 247 255

222 181 146 143 141 148 189 219 244 229 214 204 255

Figure 2.11: Assignment of thresholds to a 3 × 3 conglomerate of halftone cells with a screen angle of 15.4222 degrees and inverse frequency of 10.0277 pixels. defined as lines per inch = √

resolution of printer (in dpi) , number of unique gray-levels − 1

(2.6)

where, in order to represent 256 gray-levels, the halftone cell needs to have at least 255 pixels, which corresponds to at least a 16 × 16 = 256 pixel halftone cell. For a 600 dots per inch printer, this corresponds to 600/16 = 37.5 cell lines per inch, but this is well within the visible range of human vision imparting a strong, periodic texture onto the

© 2008 by Taylor & Francis Group, LLC

2.3. SCREEN FREQUENCY

33

(a) 20 lpi

(b) 60 lpi

(c) 100 lpi

(d) 140 lpi

Figure 2.12: The effects of varying screen frequency (lpi=lines per inch).

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

34

Figure 2.13: Two examples of moir´e introduced by the interference between the halftone cells and the sampling grid. printed image. For reference, the best printing devices produce greater than 150 lines per inch – well above what the human visual system can detect at reasonable viewing distances of 10 inches or more. So a trade-off must be made between the number of unique gray-levels and the resulting screen frequency. In combination with irrational screen angles, high line frequencies can also be problematic due to the introduction of moir´e artifacts caused by the sampling grid as demonstrated in Fig. 2.13 (left), which shows the binary dither pattern representing 25% ink coverage for a screen angle of 10 rise pixels over 39 run pixels across 10 halftone cells. What is visible in Fig. 2.13 is a repeating beat pattern created by the fluctuating cell shape. As a function of the ratio between the cell size variability and the average cell size, this moir´e pattern’s visibility is reduced by increasing the size of cells caused by reducing the screen frequency. Shown in Fig. 2.13 (right) is the equivalent binary dither pattern for an inverse screen frequency twice that of Fig. 2.13 (left), where the larger component cells create a less visible beat pattern having a smaller amplitude.

2.4

Supercells

In order to achieve high screen frequencies on moderate resolution devices, printer manufacturers will resort to halftone supercells that form

© 2008 by Taylor & Francis Group, LLC

2.5. ZERO-ANGLE DITHER ARRAYS

E

I

D F A

35

B H

G C

Figure 2.14: Ordering between sub-cells used to generate a 3 × 3 supercell. conglomerates of halftone sub-cells with consecutive threshold levels assigned across these sub-cells in a periodic or psuedo-random fashion. Specifically, the traditional technique of assigning a cost measure to each pixel inside a sub-cell is performed as it would be performed for a halftone cell in the prior technique in order to determine the order to which thresholds will be assigned in increasing order. But when assigning specific thresholds to pixels inside sub-cells, the construction algorithm will assign the first threshold level inside sub-cell A, the second threshold inside sub-cell B, the third threshold into sub-cell C, and so on, until each sub-cell has been assigned its first threshold. The construction algorithm will then start over, assigning a second threshold to each sub-cell, not necessarily in the same order as during the first round. This process is then repeated until every pixel of every sub-cell is associated with a threshold level. The resulting dither array is then histogram equalized, resulting in an array composed of a uniform distribution of threshold levels spanning the range from 0 to 255 where each sub-cell has a uniform distribution but with N 2 the spacing. Using the sub-cell ordering in Fig. 2.14, Fig. 2.15 shows a halftone supercell where we have divided the cell into a 3 × 3 array of equally sized sub-cells.

2.5

Zero-Angle Dither Arrays

Finally, in the application of AM halftone screens in PostScript and other laser printers, manufacturers will minimize the computational

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

36

241

162

103

172

231

245

166

107

176

235

182

83

44

73

152

240

161

102

171

230

186

88

48

78

157

123

24

4

34

93

181

82

43

72

151

127

28

9

38

97

132

54

14

63

201

121

23

3

33

92

137

58

19

68

206

211

192

113

142

221

131

53

13

62

200

216

196

117

147

225

251

243

164

105

174

233

210

190

112

141

220

255

237

159

100

169

228

184

85

46

76

154

250

242

163

104

173

232

178

80

40

70

149

125

26

7

36

95

183

84

45

74

153

119

21

1

31

90

135

56

16

66

204

124

25

5

35

94

129

50

11

60

198

213

194

115

144

223

134

55

15

65

202

208

188

109

139

218

253

239

160

101

170

229

212

193

114

143

222

247

244

165

106

175

234

179

81

42

71

150

252

236

158

98

167

227

185

86

47

77

155

120

22

2

32

91

177

79

39

69

148

126

27

8

37

96

130

51

12

61

199

118

20

0

30

89

136

57

18

67

205

209

189

111

140

219

128

49

10

59

197

215

195

116

146

224

248

207

187

108

138

217

254

246

Figure 2.15: Assignment of thresholds to a 3 × 3 supercell using the order of Fig. 2.14. complexity associated with screening even further by employing strictly square dither arrays, with no screen angle. As such, the mapping of the input image pixel [m, n] to the dither coordinate [r, s], within an R × R dither array, is achieved by means of a modulo operation r = mod(m, R), and s = mod(n, R).

(2.7) (2.8)

To achieve this goal while still creating halftone cells that mimic a desired screen angle and frequency, a new square dither array can be built from any rational angle halftone cell with a rise of dM pixels over a run of dN . In accordance with Fig. 2.16, the rational angle cell forms a periodic pattern of threshold arrays along both the horizontal and vertical axis, where the bottom left pixel of each period is dN cells along the x axis of the rotated space and −dM cells along the

© 2008 by Taylor & Francis Group, LLC

2.5. ZERO-ANGLE DITHER ARRAYS



ENCY

REQU

ENF SCRE

37

 





SCREENFREQUENCY 

Figure 2.16: Alignment of a (left) rational angle halftone cell with a screen angle of 4 pixels raised to 16 pixels run to a (right) zero-angle dither array of size 16 cells by 16 cells. y axis. The resulting square dither array will have side lengths equal to the square of the inverse screen frequency, measured in pixels. The same is also true when dividing dM and dN by their greatest common denominator such that, in Fig. 2.16, a 16 × 4 shift in cells could also have been a 4 × 1, 8 × 2, or 12 × 3 shift. So, by extracting the square region from Fig. 2.16, it is only this one period that needs to be derived and stored in a printer’s memory. In most cases, this is the dither array that a PostScript printer will use in its default halftoning mode where no halftoning commands have been supplied by the user. Of course, for some screen angles and frequencies, the conglomerate dither array may be quite large, and for this reason, several commercial screen companies have developed sometimes complex tiling schemes involving the intertwining of two or more non-square, non-equal matrices [49] as, for example, depicted in Fig. 2.17.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 2. AM HALFTONING

38

2 dN

2 dM dM+dN dN-dM

dM+dN

dN-dM

Figure 2.17: Tiling pattern used to represent a rational angle dither array with a dM -pixel rise across a dN -pixel run.

© 2008 by Taylor & Francis Group, LLC

Chapter 3 Stochastic Halftone Analysis Stochastic geometry is the area of mathematical research interested in complex geometrical patterns [115]. Some of the problems addressed in this field include characterizing the growth of cells in a layer of organic tissue, calculating the average area covered by randomly placed discs of constant size and shape, and characterizing the location of individual trees within a forest [82]. This last problem is an example of a spatial point process [30], which is typically described using point process statistics–metrics developed to describe the location of points in space. While many of the statistics were developed for characterizing points in continuous-space, they are perfectly suited for the study of digital halftone patterns [66] such as those found in FM halftoning, where minority pixels are randomly distributed. In this chapter, we develop the theory of point process statistics and extend them to discrete-space for use in characterizing the spatial arrangement of minority pixels within a binary dither pattern. We will also discuss halftone visibility, showing how a model of the human visual system is used to measure the differences seen by a human viewer between an original continuous-tone image and a binary halftone.

3.1

Point Processes

The point process Φ is a stochastic model governing the location of events, or points xi , within the space 2 [25]. φ is a sample of Φ and will be written as φ = {xi ∈ 2 : i = 1, . . . , N }. Furthermore, φ(B) is a scalar quantity defined as the number of xi s in the subset B of 39 © 2008 by Taylor & Francis Group, LLC

40

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

q

\

xi

B q(B)=6 Figure 3.1: Diagram illustrating the point process Φ, a sample φ of the process, and the scalar quantity φ(B) as a function of the subset B of 2 . 2 . Figure 3.1 illustrates these concepts of a point process. We assume that the point process Φ is simple, meaning that i = j implies xi = xj , which further implies 

lim φ(dVx ) =

dVx →0

1 0

for x ∈ φ else,

(3.1)

where dVx is the infinitesimally small volume around x. In terms of a discrete dither pattern, Φ represents a stochastic halftoning process with Ig representing the binary dither pattern after applying Φ to a discrete-space monochrome image of constant gray-level g. In this framework, φ represents the set of minority pixels1 such that φ[n] = 1, for pixel index n, indicates a minority pixel at location n. From its definition in (3.1), φ(xi ) is a scalar random variable that can be characterized in terms of its moments. We start with the first-order moment, the intensity I(x) E{φ(dVx )} . dVx →0 dVx

I(x) = lim

(3.2)

For a point process to be stationary, the statistical characteristics of Φ must be invariant to translation, and if a process is stationary, then 1 The pixel Ig [n] is a minority pixel if Ig [n] = 1 when 0 ≤ g < when 12 ≤ g ≤ 1.

© 2008 by Taylor & Francis Group, LLC

1 2

or Ig [n] = 0

3.1. POINT PROCESSES

41

the intensity is constant for all x ∈ 2 , where I(x) = I is the expected number of points per unit area. Returning to discrete-space, I[n] is related to the gray-level as 

I[n] =

g 1−g

for 0 ≤ g < 12 for 12 ≤ g ≤ 1

(3.3)

and represents the unconditional probability that the sample at location n is a minority pixel. Additional insight into Φ can be gained by conditioning the probability distribution of Φ given that a point lies at a given location. The result is a conditional distribution referred to as the Palm distribution [115]. A particular measure under the Palm distribution of Φ is the quantity K(x; y) E{φ(dVx )|y ∈ φ} , dVx →0 E{φ(dVx )}

K(x; y) = lim

(3.4)

the ratio of the expected number of points in dVx under the condition y ∈ φ to the unconditional expected number of points in dVx . Referred to as the reduced second moment measure, K(x; y) may be thought of as the influence at location x of the point y. That is, is a point at x more or less likely to occur because a point exists at y? The value K(x; y) > 1 indicates that a point is more likely to occur while K(x; y) < 1 indicates that a point is less likely. For stationary processes, K(x; y) may be written as K(r, θ), where r is the distance between x and y while θ is the direction to x from y. For a point process to be isotropic, the statistical characteristics of Φ must be invariant to rotation; therefore, if Φ is also isotropic, then K(x; y) = K(r). In discrete-space, the reduced second moment measure is represented by K[n; m] such that K[n; m] =

E{φ[n]|φ[m] = 1} , E{φ[n]}

(3.5)

the ratio of the conditional expectation that sample n is a minority pixel given that sample m is a minority pixel to the unconditional expectation that n is a minority pixel. To estimate K[n; m] for a stationary point process from a given sample dither pattern, a set of regions of interest or windows surrounding each minority pixel within the dither pattern (Fig. 3.2) is extracted with these windows combined through addition in a pixel-wise fashion. Dividing each resulting pixel by the

© 2008 by Taylor & Francis Group, LLC

42

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

undefined region

region of interest

minority pixel

Figure 3.2: A binary dither pattern used to estimate K[n; m] with minority pixels near the edges (light gray) of the pattern excluded from the estimate due to undefined regions (dark gray) in their corresponding windows. ˆ m], with the total number of windows then creates the estimate K[n; center pixel representing the minority pixel m. Because minority pixels near edges of the dither pattern have windows with undefined regions, these minority pixels are left out of the estimate. To illustrate the usefulness of K[n; m], Fig. 3.3 shows three 128 × 128 dither patterns taken from the point processes Φ1 , Φ2 , and Φ3 . Shown alongside each window is its corresponding reduced second moment measure estimate with the black pixel at each estimate’s center indicating the location of the reference minority pixel. In the first example, pattern φ1 is a completely random point process with E{φ[n]|φ[m] = 1} = E{φ[n]} for all m and n, that is, the probability of any pixel n being a minority pixel independent of any other pixel, m. The result is a reduced second moment measure that is flat (K1 [n; m] = 1) for all n and m. The second example, φ2 , referred to as a hard core process [115], is one in which there exists some distance r > 0 such that for no two points, o and p (where φ2 [o] = 1 and φ2 [p] = 1), is |o − p| < r. This property results in K2 [n; m] = 0 for |m−n| < r, where, in this instance, r = 3.45 pixels. The final example, φ3 , is a clustering process, where the occurrence of a minority pixel at sample m increases the likelihood

© 2008 by Taylor & Francis Group, LLC

3.1. POINT PROCESSES

43

φ1

K1 [n; m]

φ2

K2 [n; m]

φ3

K3 [n; m]

Figure 3.3: The (left) sample patterns and the (right) corresponding reduced second moment measures for completely random, hard core, and clustering point processes.

© 2008 by Taylor & Francis Group, LLC

44

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

Figure 3.4: The (left) annular rings with center radius r and radial width Δr used to extract the pair correlation R(r) from the reduced second moment measure K[n; m] and (right) the pixels, shown crosshatched, of K[n; m] that form the annular ring for r = 3 14 pixels and Δr = 12 pixels. of a minority pixel at sample n when |m−n| is small. How small |m−n| needs to be depends on the specific process, and the resulting K3 [n; m], in this case, is one in which K3 [n; m] > 1 for |m − n| < 3.64 pixels.

3.2

Spatial Statistics

From K[n; m], we can derive two spatial domain statistics by partitioning the spatial domain into a series of annular rings Ry (r) (Fig. 3.4 (left)) with center radius r, width Δr , and centered around pixel m. In this book, the annular ring Ry (r) will be the set {n : r − Δr /2 < |n − m| ≤ r + Δr /2}, where Δr = 1/2 pixels. Demonstrated in Fig. 3.4 (right) are the pixels of K[n; m] that form the annular ring for r = 3 14 pixels.

3.2.1

Pair Correlation

The first statistic for stationary and isotropic Φ is the pair correlation R(r), defined as E{φ(Ry (r))|y ∈ φ} R(r) = , (3.6) E{φ(Ry (r))}

© 2008 by Taylor & Francis Group, LLC

PAIR CORRELATION

3.2. SPATIAL STATISTICS

45

2

1

0

0

20

PAIR CORRELATION

RADIAL DISTANCE

2

1

0

0

3.45

20

PAIR CORRELATION

RADIAL DISTANCE

2

1

0

0

3.64

20 RADIAL DISTANCE

Figure 3.5: The pair correlations corresponding to the samples of the point processes (top) Φ1 , (middle) Φ2 , and (bottom) Φ3 .

the influence that the point at y has at any x in the annular ring Ry (r). Note that for a stationary point process, the unconditional expected number of points in the ring Ry (r) is I · N (Ry (r)) (the intensity times the area of Ry (r)). R(r) is also the average value of K(x; y) in the ring Ry (r), and its usefulness can be seen in the interpretation that maxima of R(r) indicate frequent occurrences of the inter-point distance r while minima of R(r) indicate an inhibition of points at r [115].

© 2008 by Taylor & Francis Group, LLC

46

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

Figure 3.5 shows that the pair correlation is a strong indicator of the inter-point relationships for a given dither pattern. Referring back to the point processes Φ1 , Φ2 , and Φ3 of Fig. 3.3, Fig. 3.5 shows the resulting pair correlations for each process. Being a completely random point process, Φ1 has a pair correlation R(r) = 1 for all r, indicating no correlation between minority pixels. In the second example, Φ2 , R(r) = 0 for r < 3.45 is a consequence of the inhibition of points within a distance 3.45 of each other while in Φ3 , the frequent occurrence of the inter-point distances r < 3.64 is indicated by R(r) > 1 in these areas. But as R(r) clearly identifies the clustering behavior of points in Φ3 , its makes no indication of the directional artifacts that occur along the diagonal. This occurs because R(r) assumes, by its definition, isotropy in Φ. The pair correlation may, therefore, be insufficient, by itself, at describing binary dither patterns uniquely when patterns are allowed and even preferred to be anisotropic.

3.2.2

Directional Distribution Function

As a supplement to R(r) for characterizing stationary but not isotropic Φ, K[n; m] can be used to investigate anisotropy in Φ by defining the directional distribution function Dr1 ,r2 (a) as Dr1 ,r2 (a) =

E {φ(Γam )|y ∈ φ} /N (Γam ) , E {φ(Γm )|y ∈ φ} /N (Γm )

(3.7)

the expected number of points per unit area in a segment, Γam , of the ring Γm = {n : r1 ≤ |n − m| < r2 , m ∈ φ} centered around the point m ∈ φ such that a ≤  (n − m) < a + Δa (Fig. 3.6) to the expected number of points per unit area in Γm itself. Note that for isotropic point patterns, Dr1 ,r2 (a) = 1.0 for all a, and that Dr1 ,r2 (a) > 1.0 indicates a favoring of points at angles near a while Dr1 ,r2 (a) < 1.0 indicates an inhibition of points. Furthermore, the parameters r1 and r2 allow us to look at point distributions for various ranges from a point m, most notably, a near distribution for the range r ∈ [0, r1 ), a far distribution for r ∈ [r1 , ∞), and an overall distribution for r ∈ [0, ∞). In Fig. 3.5, the directional distributions for the near range, r ∈ [0, 4.5), and the intermediate range, r ∈ [4.5, 8.0), are given for each process Φ1 , Φ2 , and Φ3 , with the near range indicated in light gray. The isotropic nature of Φ1 and Φ2 is clearly visible as Dr1 ,r2 (a) = 1 for all a in both ranges. The diagonal artifacts in Φ3 are also clearly visible with D0,4.5 (a) measuring the directional distribution of points within a

© 2008 by Taylor & Francis Group, LLC

3.3. SPECTRAL STATISTICS

K

47

K

Figure 3.6: The (left) division of the spatial ring Γm into the segment Γam and (right) the pixels, shown cross-hatched, of K[n; m] that form the segment for r1 = 2 pixels, r2 = 4 12 pixels, a = 2π , and Δa = 2π . 32 16 cluster and D4.5,8.0 (a) measuring the directional distribution of points relative to neighboring clusters.

3.3

Spectral Statistics

Spectral analysis of a point process was first introduced to describe onedimensional processes by Bartlett [9] and then again by Bartlett [10] to study two-dimensional processes. While statistics have been developed based on Bartlett’s work [87], statisticians have relied almost entirely on spatial statistics, such as the pair correlation, due to the intuitive characterizations they offer in describing spatial point processes. Within the signal processing community, though, spectral analysis is a commonly used tool for studying random processes. Spectral analysis was first applied to stochastic dither patterns by Ulichney [123] to characterize dither patterns created via errordiffusion. To do so, Ulichney developed the radially averaged power spectra along with a measure of anisotropy. Both rely on estimating the power spectrum through Bartlett’s method of averaging periodograms, the magnitude-square of the Fourier transform of the output pattern divided by the sample size, to produce the spectral estimate Pˆ (f ) such

© 2008 by Taylor & Francis Group, LLC

48

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS / /2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

2

0

/ /2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

2

0

/ /2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

2

0

Figure 3.7: The directional distribution functions corresponding to the samples of the point processes (top) Φ1 , (middle) Φ2 , and (bottom) Φ3 for the near range, r ∈ [0, 4.5) (indicated by light-gray bars), and the intermediate range, r ∈ [4.5, 8.0) (indicated by dark-gray bars).

© 2008 by Taylor & Francis Group, LLC

3.3. SPECTRAL STATISTICS

49

Figure 3.8: The estimated power spectrums for point processes (left) Φ1 , (center) Φ2 , and (right) Φ3 with spectral components with high power shown in white. that

K 1  |DFT2D (φi )|2 Pˆ (f ) = , K i=1 N (φi )

(3.8)

where DFT2D (φ) represents the two-dimensional discrete Fourier transform of the sample φ, N (φ) is the total number of pixels in the sample φ, and K is the total number of periodograms being averaged to form the estimate. In this book, all estimates will be formed using 10 samples of size 256 × 256 pixels.

3.3.1

Radially Averaged Power Spectral Density

Although anisotropies of a dither pattern can be qualitatively observed by studying three-dimensional plots of Pˆ (f ) (Fig. 3.8), partitioning the spectral domain into a series of annular rings R(fρ ) of width Δρ leads to two useful one-dimensional statistics. The first statistic is the radially averaged power spectrum density (RAPSD) P (fρ ), defined for discrete Pˆ (f ) as the average power in the annular ring with center radius fρ P (fρ ) =

 1 Pˆ (f ), N (R(fρ )) f ∈R(fρ )

(3.9)

where N (R(fρ )) is the number of frequency samples in R(fρ ). Given a rectangular sampling grid where the distances between samples along the horizontal and vertical axes are both equal to some distance D (Fig. 3.9 (left)), the spectral estimate Pˆ (f ) has a maximum horizontal/vertical frequency of (1/2)D−1 pixels per inch (Fig. 3.9 (right)).

© 2008 by Taylor & Francis Group, LLC

50

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

Figure 3.9: The (left) sampling grid where the distances between samples along the horizontal and vertical axes are both equal to D and (right) the maximum frequencies are along the vertical and horizontal axes of the spectral estimate Pˆ (f ). √ −1 The maximum radial frequency is then ( – leading to partial √ 2/2)D −1 −1 spectral rings for (1/2)D < fρ ≤ ( 2/2)D as shown in Fig. 3.10 (left). For hexagonal sampling grids, the maximum radial is √ frequency −1 −1 (2/3)D with partial spectral rings in the range (1/ 3)D < fρ ≤ (2/3)D−1 as shown in Fig. 3.10 (right). As will be described in Chapter 8, this issue of partial annuli plays a significant role in the generation of digital halftones for hexagonal sampling grids. And for this reason, the partial spectral rings are included in plots of P (fρ ) and will, in this book, be indicated along the horizontal axis. Returning to rectangular grids, Fig. 3.11 shows the RAPSDs of the point processes Φ1 , Φ2 , and Φ3 of Fig. 3.3. Pattern Φ1 , having no correlation between minority pixels, has a power spectrum that is flat for all fρ . It is due to this spectrum, having equal amounts of all frequencies, that Φ1 is referred to as a white-noise process since white light contains equal amounts of all frequencies of the visible light spectrum. White-noise halftoning is accomplished by thresholding a continuous-tone input image with uniformly distributed, uncorrelated noise [123]. The strength of P (fρ ) for Φ1 , σg2 = g(1 − g), is equal to the variance of a single pixel, Ig [n], of the binary dither pattern treated as a random variable (a Bernoulli process [123]). The process Φ2 has a power spectrum that is composed almost exclusively of high-frequency noise. As blue light is the high-frequency

© 2008 by Taylor & Francis Group, LLC

3.3. SPECTRAL STATISTICS

51

PARTIAL ANNULI REGIONS LARGEST COMPLETE ANNULUS

fρ

fρ

0 Δ

1 2

1 2

fρ

0 Δ

1 2 3 3

fρ

Figure 3.10: The annular rings used to divide the power spectrum P (f ) to form P (fρ ) and A(fρ ) for a (left) rectangular and a (right) hexagonal sampling grid. component of the visible light spectrum, halftone patterns like φ2 are referred to as blue-noise dither patterns. Studied in great detail by Ulichney [123, 122], these patterns were originally created using errordiffusion, but due to the desirable attributes of blue-noise, several less computationally complex techniques have since been introduced. In Chapter 5, we review many of the benefits of this model. The final process, Φ3 , was first referred to by Lau et al. [66] as a green-noise dither pattern due to its mid-frequency only content. While blue-noise has several advantages over green with respect to their visual appearances, the failure of some printers to print isolated dots reliably makes green-noise a reasonable alternative to blue. The specifics of this model are presented in Chapter 10. What Fig. 3.11 shows is that the three processes Φ1 , Φ2 , and Φ3 each have unique spectral characteristics. Using the RAPSD, we can identify these characteristics as, in fact, the RAPSD maintains an intimate link with the spatial domain’s pair correlation. In particular, a frequent occurrence of the inter-point distance r, indicated by maxima in R(r), implies a peak in P (fρ ) for radial frequency fρ = 1/r proportional in magnitude to the peak in R(r), meaning that a larger peak in R(r) leads to a larger peak in P (fρ ). A drawback of this relationship between P (fρ ) and R(r), though, is the failure of P (fρ ), like R(r), to

© 2008 by Taylor & Francis Group, LLC

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

52

RAPSD (m2)

3

1 0

0

0.5

0.7071

0.5

0.7071

0.5

0.7071

RADIAL FREQUENCY

RAPSD (m2)

3

1 0

0 RADIAL FREQUENCY

RAPSD (m2)

3

1 0

0 RADIAL FREQUENCY

Figure 3.11: The RAPSDs corresponding to the point processes (top) Φ1 , (middle) Φ2 , and (bottom) Φ3 . identify directional artifacts.

3.3.2

Anisotropy

As a second statistic, introduced by Ulichney [123], is the anisotropy A(fρ ), which is defined as 



2 Pˆ (f ) − P (fρ )  1 A(fρ ) = , N (R(fρ )) − 1 f ∈R(fρ ) P 2 (fρ )

© 2008 by Taylor & Francis Group, LLC

(3.10)

3.4. COLOR HALFTONING

53

the relative variance or the “noise-to-signal” ratio [123] of frequency samples of Pˆ (f ) in R(fρ ). Built on the premise that aperiodic patterns generated in areas of uniform gray are most pleasant when patterns are isotropic, A(fρ ), a measure of how isotropic a dither pattern is, implies that the pattern with lower anisotropy is the visually more pleasing of two if both patterns have identical P (fρ )s. To demonstrate, Fig. 3.12 shows the anisotropy measures in units of decibels (db) for the three point processes. Ulichney has shown that the anisotropy measure of a perfectly isotropic dither pattern is 1 ; therefore, as in the case of Fig. 3.12 and all subsequent plots A(fρ ), K 1 or −10 db should be considered “background noise” (indicated by a 10 dotted line) and not until A(fρ ) > 0 db should directional components be considered strong or noticeable to the human eye. In Fig. 3.12, Φ1 and Φ2 are isotropic while the diagonal artifacts of Φ3 are indicated, in A(fρ ), by strong components at fρ = 0.05, 0.25, and 0.35. Its purpose, to measure the strength of directional artifacts, A(fρ ), does not indicate the direction. It is therefore possible for a dither pattern with strong horizontal artifacts to be indistinguishable according to P (fρ ) and A(fρ ) from a pattern with strong diagonal artifacts. The human eye, being less sensitive to diagonal correlations, may find the difference in appearance far from indistinguishable – making the metrics P (fρ ) and A(fρ ) insufficient, by themselves, at describing anisotropic dither patterns uniquely. We therefore combine all four metrics to offer a more complete description of a stochastic halftone pattern than previously offered.

3.4

Color Halftoning

In understanding the process of color reproduction, it is important to have an understanding of the human visual system and how the eye sees color. Detailed descriptions of this material can be found in Hunt [48], while the following is a brief description of the important aspects of color reproduction.

3.4.1

RGB

Human color vision begins in the eye, where, within the retina, there are two types of light sensitive cells (photo-receptors) responsible for vision. Named for their shapes, these cells are either rods or cones. The rods

© 2008 by Taylor & Francis Group, LLC

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

ANISOTROPY

54

10 0 ï10 0

0.5

0.7071

0.5

0.7071

0.5

0.7071

ANISOTROPY

RADIAL FREQUENCY

10 0 ï10 0

ANISOTROPY

RADIAL FREQUENCY

10 0 ï10 0 RADIAL FREQUENCY

Figure 3.12: The anisotropies corresponding to the point processes (top) Φ1 , (middle) Φ2 , and (bottom) Φ3 . are in much higher abundance (125 million versus 7 million) and are responsible for detecting light intensities at low levels of illumination (colorless night vision). The cones are responsible for color vision and can be divided into three subsets: (i) red, (ii) green, and (iii) blue. Shown in Fig. 3.13 are estimates of the relative sensitivity for each subset of cones versus the wavelength of incoming light. The blue cones derive their name from the fact that they are the most sensitive, of the three, to blue light (λ ≈ 440 nm) while green cones are most sensitive to green light (λ ≈ 510 nm) and red most sensitive to red

© 2008 by Taylor & Francis Group, LLC

3.4. COLOR HALFTONING

55

h

h

b

h

g

r

`(h) l(h)

a(h)

h

Figure 3.13: Approximations to the relative sensitivity for the red (ρ(λ)), green (γ(λ)), and blue (β(λ)) cones versus wavelength. light (λ ≈ 650 nm). When the incoming light is focused onto the retina by the eye lens, the light is transformed by the rods and cones into nerve impulses that are sent, via the optic nerve, to and interpreted by the brain. Colors are interpreted by the brain as the sum total of the three cone responses. For a mathematical framework, C will be the continuous vector representing the composition of incoming light, where C(λ) is the strength of light with wavelength λ such that 0 ≤ C(λ) ≤ 1. A special case of C is white light (C = W), where C(λ) = 1 for all λ. Given the relative sensitivity curves, β(λ), γ(λ), and ρ(λ), the cone response to incoming light, C, will be the three-dimensional vector c such that ⎡

⎤

βC ⎥ c = ⎢ ⎣ γC ⎦ , ρC where

(3.11)



βC = γC = ρC =

© 2008 by Taylor & Francis Group, LLC

β(λ)C(λ) dλ,

(3.12)

γ(λ)C(λ) dλ,

(3.13)

ρ(λ)C(λ) dλ,

(3.14)

λ λ λ

56

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

are the responses of each subset of cones to the incoming light C. In theory, the brain can be fooled into seeing any color by a linear combination of just the three colors red (R such that R(λ) = 1 for λ = λr , 0 else), green (G such that G(λ) = 1 for λ = λg , 0 else), and blue (B such that B(λ) = 1 for λ = λb , 0 else).2 That is, given the color C with cone response c, an equivalent response can be generated from a linear combination of R, G, and B such that c = rR + gG + bB ⎡ ⎤⎡ ⎤ βR βG βB r ⎥⎢ ⎥ = ⎢ ⎣ γR γG γB ⎦ ⎣ g ⎦ , b ρR ρG ρB where

⎡

⎤

⎡

⎤−1 ⎡

r βR βG βB ⎢ ⎥ ⎢ ⎥ ⎣ g ⎦ = ⎣ γR γG γB ⎦ b ρR ρG ρB

(3.15)

⎤

βC ⎢ ⎥ ⎣ γC ⎦ ρC

(3.16)

is the tri-stimulus RGB value of the color C. Color Triangle Shown in Fig. 3.14 are the r, g, and b values plotted versus λ corresponding to the tri-stimulus values for imitating the cone responses when the incoming light is composed exclusively of the single wavelength λ with unit strength. The curves here have each been scaled separately such that colorless white light is composed of equal amounts of each component (r = g = b). In order to obtain a perfect match between the incoming light and the combination of red, green, and blue light, negative values of r, g, or b in Fig. 3.14 are interpreted as adding an equivalent amount of that component to the incoming light. The data of Fig. 3.14 are also shown in Fig. 3.15, where the tri-stimulus values for each wavelength are shown in three dimensions. As a linear combination of R, G, and B components, increasing the strength of the incoming light by a factor of k increases the tristimulus values such that k c = krR + kgG + kbB.

(3.17)

2 The exact values of λr , λg , and λb depend on the specific light sources for each color component and will, therefore, be left in general terms.

© 2008 by Taylor & Francis Group, LLC

3.4. COLOR HALFTONING

57

g(h) r(h)

b(h)

h

Figure 3.14: The r, g, and b values versus λ for imitating the cone responses when the incoming light is composed exclusively of the single wavelength λ with unit strength.

0.4 470 0.3 485 400

BLUE

0.2

500 0.1

650 620

0

530 590 560

ï0.1 ï0.1

ï0.1 0

0 0.1

0.1 0.2

0.2 0.3

0.3 0.4

0.4

RED

GREEN

Figure 3.15: The r, g, and b values for imitating the cone responses when the incoming light is composed exclusively of the single wavelength λ with unit strength.

© 2008 by Taylor & Francis Group, LLC

58

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

0.4

0.3 k=1

BLUE

0.2

k=0

0.1 k=2 0

ï0.1 ï0.1

ï0.1 0

0 0.1

0.1 0.2

0.2 0.3

0.3 0.4

0.4

RED

GREEN

Figure 3.16: Varying the parameter k is equivalent to moving along a line, projected from the origin, through some point c (indicated by k = 1).

In three dimensions, varying the parameter k is equivalent to moving along a line projected from the origin through the point c (Fig. 3.16). The color c can, therefore, be identified uniquely by the tri-stimulus values

r˜ = r/(r + b + g), g˜ = g/(r + b + g), and ˜b = b/(r + b + g).

(3.18) (3.19) (3.20)

Geometrically, this new point is obtained by projecting, from the origin, the point c onto the plane r + g + b = 1 (Fig. 3.17). Doing so creates the RGB (red, green, blue) color triangle of Fig. 3.18, where each color of the visual spectrum is plotted versus r˜ and g˜. The third component, ˜b, is obtained as 1 − r˜ − g˜. Indicated by the triangle in Fig. 3.15 is the color gamut or the set of obtainable colors achieved by adding positive amounts of R, G, and B. Colors outside this gamut, which require negative amounts of either component, cannot be imitated.

© 2008 by Taylor & Francis Group, LLC

3.4. COLOR HALFTONING

59

400 470 1.2 485

1

BLUE

0.8

495

0.6 0.4 0.2 650

0

590

530

560

ï0.2 0

0 0.5

0.5 1

1 RED

GREEN

Figure 3.17: Projecting, from the origin, the point c onto the plane r + g + b = 1 creates a two-dimensional mapping of all colors of the visible light spectrum.

3.4.2

CMY

The system of colors formed by combining light (Fig. 3.16) of colors red, green, and blue is referred to as an additive system with the colors R, G, and B referred to as the additive primaries. In digital printing, white paper reflects all wavelengths of the incident light. Assuming the incident light is white, combining the inks of cyan, magenta, and yellow, can imitate any color by taking away particular wavelengths from this incident light. This system is referred to as a subtractive system with cyan, magenta and yellow the subtractive primaries. The reason for using cyan (Cn ), magenta (M), and yellow (Y) is that they are the complement colors to red, green, and blue such that, ideally Cn = W − R, M = W − G, Y = W − B,

(3.21) (3.22) (3.23)

where the tri-stimulus CMY values are derived from the RGB values as cn = 1 − r,

© 2008 by Taylor & Francis Group, LLC

(3.24)

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

60

G

530

560 495

485

W

590

470

R B Figure 3.18: The RGB color triangle. m = 1 − g, y = 1 − b.

(3.25) (3.26)

We say “ideally” because we are assuming that our inks are the exact complements to red, green, and blue. So by printing cyan ink, we take away, from the incident light, a proportional amount of red light.

3.4.3

CMYK

The color black (K) is, in the ideal case, obtained by combining equal amounts of Cn , M, and Y. As inks are never the perfect complement to R, G, and B, perfect black cannot be obtained. So, by adding a fourth ink, black, to cyan, magenta, and yellow, we obtain the set of

© 2008 by Taylor & Francis Group, LLC

3.4. COLOR HALFTONING

61

process colors CMYK where the conversion from CMY to CMYK is done (in the ideal case) as k = cn,new = mnew = ynew =

min(cn,old , mold , yold ), cn,old − k, mold − k, yold − k.

(3.27) (3.28) (3.29) (3.30)

The exact conversion is dependent on the exact RGB values of each ink. The reader is directed to Hunt [48] for further details. By using this fourth component, printers can produce images with greater contrast and less expense (financially) as black inks are less expensive than combining the three inks cyan, magenta, and yellow to form black [24].

3.4.4

Color Statistics

In the case of a color halftone, the monochrome model Ig that must be revised as a dither pattern is now composed of C colors, where, for generality, the quantity C is an arbitrary integer. For RGB and CMYK, where images are composed of the additive colors red, green, and blue or the process colors cyan, magenta, yellow, and black, C = 3 and 4, respectively. So for color images, the halftone pattern Ig is now composed of the monochrome binary dither patterns Ig1 , Ig2 , . . . , IgC , where gi is the gray-level of pattern Igi and φi is the corresponding point process. In this new framework, the quantity Ki,j [n; m] is the reduced second moment measure between colors such that Ki,j [n; m] =

E{φgi [n] | φgj [m] = 1} E{φgi [n]}

(3.31)

is the ratio of the conditional probability that φgi [n] is a minority pixel given that a minority pixel exists at sample m of φgj to the unconditional probability that φgi [n] is a minority pixel [67]. Similar to K[n; m], Ki,j [n; m] = 1 indicates that the locations of minority pixels in colors i and j are uncorrelated. The pair correlation between colors gi and gj follows as E{φgi (Rm (r)) | φgj [m] = 1} Ri,j (r) = , (3.32) E{φgi (Rm (r))}

© 2008 by Taylor & Francis Group, LLC

62

CHAPTER 3. STOCHASTIC HALFTONE ANALYSIS

the ratio of the expected number of minority pixels of color gi located in the ring Rm (r) = {n : r < |n − m| ≤ r + dr} under the condition that φgj [m] is a minority pixel to the unconditional expected number of minority pixels with color gi located in Rm (r).

© 2008 by Taylor & Francis Group, LLC

Chapter 4 Halftone Visibility When reproducing continuous-tone images, halftone visibility refers to the visibility or “notice-ability” of the binary dots when viewing the printed image. The ideal halftoning algorithm is the one that minimizes this visibility. Halftone visibility metrics that give it a scalar cost have, therefore, been employed for comparative studies between halftoning algorithms [22, 81], and in some cases [5, 90] they have even been employed within the halftoning algorithm to decide where and where not to print a dot. Because the eye sees distortions at particular spatial frequencies more so than at others, developing a halftone visibility metric begins with a model of the human visual system (HVS). Having a model of the HVS allows us to measure the distortion seen by a human viewer when representing a continuous-tone image with black and white dots. The distortion as seen by a human viewer between an original continuous-tone image, X, and its binary halftone, Y , is calculated as the mean squared error between the HVS filtered images X and Y (Fig. 4.1). As a demonstration, Fig. 4.2 shows the modeled visual response to the point process samples Φ1 , Φ2 , and Φ3 , where, for this particular HVS model, the visual costs compared to an image composed of a flat shade of gray are 0.0022, 0.0003, and 0.0134 respectively – indicating that Φ2 creates the most accurate illusion of the continuous shade of gray. Relatively simple models of the HVS have proven to be quite successful when applied to algorithms that search for the best possible dot distribution. Digital halftoning techniques including screening algorithms, error-diffusion algorithms, and iterative halftoning methods all use, either implicitly or explicitly, a model of the human visual system. In fact, even those methods that are not classified as model-based, 63 © 2008 by Taylor & Francis Group, LLC

CHAPTER 4. HALFTONE VISIBILITY

64

HVS +

Σ

X

−

Y H

| |2

E

HVS

Figure 4.1: The distortion introduced by halftoning is measured as the mean squared error between the HVS filtered input image, X, and its corresponding HVS filtered halftoned image, Y . because they do not include an explicit HVS block within their block diagram (for example, Bayer’s dither [11]), nevertheless agree with a model that treats the HVS as a low-pass filter. And not only is a HVS model crucial for the design of almost every halftoning technique, but the shape of the HVS model can also be tuned to yield better texture quality in the obtained dither patterns. Thus, the performance of a given halftoning algorithm can be maximized by properly designing improved HVS models. The selection of HVS models used in digital halftoning leads to different output performance, in terms of smoothness, homogeneity, coarseness, and in the appearance of artifacts. One of the objectives in designing an HVS model is to quantify these differences. In particular, one can look at how the bandwidth and tail characteristics of a particular HVS model affect spectral metrics like the radially averaged power spectral density (described in Chapter 3) of the resulting dither patterns. As such, HVS models have been proposed to offer a quantitative measure of image distortion as seen by a human viewer. In this regard of measuring halftone texture, an HVS model is a linear shift invariant filter based on the point spread function (PSF) or the contrast sensitivity function (CSF) of the human eye. When the HVS model is proposed based on the eye’s response to stimuli in the spatial domain, the function obtained is the PSF and is derived by the fact that, as stated by Whesteimer [133], under no circumstances are point objects ever actually imaged as points because of several physical and geometrical optical factors. As a result, a point object gives rise to a retinal light distribution that is bell-shaped in cross-section. This distribution is precisely the PSF of the eye and has significance not only when the object is a point but whenever it

© 2008 by Taylor & Francis Group, LLC

65

Figure 4.2: The point processes (top) Φ1 , (middle) Φ2 , and (bottom) Φ3 as seen by a human viewer for a given HVS model. is necessary to know the light distribution for a target more complex than a point source of light, since any visual object can be thought of as made up of points. The experimental determination of the eye’s PSF is carried out with the use of instruments designed to measure the entire refractive error of the eye, called “wavefront sensors” or “aberrometers.” Now in the frequency domain, an HVS model is based upon the modulation transfer function (MTF) or the CSF of the human eye, which attempts to characterize a subject’s responsiveness to particular spatial frequencies. As illustrated in Fig. 4.3, associated experiments consist of a sinusoidal grating where the modulation or contrast of the grating can be measured as the ratio between the difference of the maximum and minimum amounts of light reflected by the grating to the sum of the two. The frequency of the grating is varied from lower to higher frequencies in each experiment. For equally spaced particular

© 2008 by Taylor & Francis Group, LLC

CHAPTER 4. HALFTONE VISIBILITY

66

VIEWING ANGLE

1 CYCLE

VIEWING DISTANCE EYE GRATING

Figure 4.3: A sinusoidal grating placed at varying distances from the human eye. frequencies, a subjective measure of the contrast sensed by the human viewer is taken, and the ratio between the real contrast of the image and the contrast resolved by the viewer is used as the contrast sensitivity of the eye for that exact frequency. The sensitivity of the eye is high (near unity) for low frequencies, but as the frequency is increased, the eye starts to fail in detecting the real contrast of the grating, and the sensitivity decays to zero. Taken together, the PSF and the CSF form a Fourier pair, denoted h(x, y) (x, y in inches) in the spatial domain and H(u, v) (u, v in cycles/degree) in the frequency domain. The inverse Fourier transform ¯ x, y¯), with x¯, y¯ in degrees. To convert these anguof H(u, v) yields h(¯ lar units to units on the printed page, notice that a length of x inches when viewed at a distance D inches will subtend an angle of x¯ degrees satisfying x¯ = (180/π)(arctan(x/D)) ≈ (180x)/(πD) (4.1) for x D. Therefore, assuming a printer with resolution R (in dots per inch, dpi), the discrete filter characterizing the HVS model in the spatial domain will be given by



1802 ¯ 180m 180n h [m, n] = 2 2 h , . π D πRD πRD

(4.2)

The term S ≡ RD in (4.2) is called the scale parameter. Kim and Allebach [54] experimented with different values of S in HVS models, as part of their research with direct binary search (an iterative halftoning algorithm described in Chapter 7) and demonstrated why this parameter, which in theory should be determined precisely by the intended viewing distance and printer resolution, in reality serves more as a free

© 2008 by Taylor & Francis Group, LLC

4.1. CAMPBELL’S CSF MODEL

67

Figure 4.4: The contrast sensitivity function of the human visual system. parameter that can be adjusted to yield halftone textures of the desired quality. The behavior of the scale parameter is such that, when the distance or print resolution is increased, the viewer’s visual response sensitivity is reduced by a fixed amount of cycles per degree; therefore, the bandwidth of the HVS filter in the frequency domain is decreased by an amount proportional to the distance (or resolution) increase. In this scenario, if the viewer observes a printed page from a greater distance, or if the resolution is larger, it is expected that the eye will perceive a better overall impression of the image at the expense of losing the ability to discern the details of the printed dots on the page. That is, the image will appear to the viewer as being more homogeneous. The contrary effect occurs if the distance or the resolution is decreased with the HVS bandwidth increasing and the halftone textures becoming more apparent.

4.1

Campbell’s CSF Model

From the experiments of Fig. 4.3, Campbell et al. [17] proposed modeling the contrast sensitivity function using the one-dimensional equation CSF (fρ ) = k (exp(−2πfρ α) − exp(−2πfρ β))

(4.3)

plotted in Fig. 4.4, where α and β are arbitrary constants, later set by Analoui and Allebach [5] to 0.012 and 0.046, respectively. The parameter k is a constant proportional to the average illumination and is typically set such that maxfρ CSF (fρ ) = 1 (at fρ = fmax = ln(α/β)/[2π(α − β)]). The drop in sensitivity, proposed by this model

© 2008 by Taylor & Francis Group, LLC

CHAPTER 4. HALFTONE VISIBILITY

68

Figure 4.5: The two-dimensional contrast sensitivity function CSF2D (fρ , fθ ) of the human visual system that accounts for the reduced sensitivity of the eye along the diagonal. near DC, corresponds with a sinusoidal grating changing so slowly as to have an imperceptible gradient. For halftoning purposes, this drop is usually ignored with the CSF modeled by CSF  (fρ ) defined as  

CSF (fρ ) =

CSF (fρ ) , CSF (fmax )

for fρ ≥ fmax

1, else,

(4.4)

which mimics the prior equation in the high-frequency range. Under the assumption that the human visual system is not isotropic, the orientation of the grating should play a role in the sensitivity of the eye to its sinusoidal frequency [18], namely, near odd multiples of 45 degrees. It has been hypothesized that this reduced sensitivity is a consequence of our adaptation to the predominantly horizontal and vertical structures in the natural and man-made worlds. Regardless, a two-dimensional model proposed by Daly [27] of the HVS that describes less sensitivity along diagonals is shown in Fig. 4.5 and is defined by the equation CSF2D (fρ , fθ ) = CSF  (

fρ ), 0.15 cos(4fθ ) + 1.35

(4.5)

where fθ is the angle in the spectral plane from the positive horizontal axis to the polar coordinate (fρ , fθ ). Now in contrast to the theory of reduced sensitivity along the diagonals, Allebach [2] reports that, in his own work with direct binary

© 2008 by Taylor & Francis Group, LLC

¨ ANEN ¨ 4.2. NAS (EXPONENTIAL) MODEL

69

search, this practice results in a diagonal texture structure that is undesirably visible. Similar results were reported by Lau and Ulichney [73], who found diagonal correlations problematic at gray-levels near 12 in error-diffused halftones, an issue to be discussed in Chapter 8. These observations agree with visual studies that theorize that the reduced 45 degrees sensitivity is based upon visual evaluations where the subject was presented with a single-frequency sinusoidal grating. Mullen [88] goes further and demonstrates that the near-DC behavior of (4.4) is invalid because, in the associated experiments, only a very small number of sinusoidal wave periods were displayed to the viewers at the lowest frequencies. And a number of bars below four or five is known to reduce sensitivity to these kinds of gratings. As such, it is expected that these models are inadequate in representing the HVS near DC. As such, there is no visual masking where certain spatial frequencies block the HVS’s ability to see other frequencies. In fact, no visual model has been developed that fully explains the HVS’s response to all possible stimuli. Clearly, the HVS is low-pass in nature, but the CSF model does not characterize the HVS completely as it is seen in halftoning.

4.2

N¨ as¨ anen (Exponential) Model

As an alternative model of the human visual system that assumes radial symmetry, the N¨as¨anen model, demonstrated in Fig. 4.6, is a radially symmetric exponential function of the radial frequency, ρ, that has been used extensively in the context of digital halftoning. This model is defined as H (ρ) = exp (−kρ) k 1 h (r) =  , 2 (2π) k + r2 3/2

(4.6) (4.7)

2π

where L = 11cd/m2 is the average luminance, and c = 0.525 and d = 3.91 are constants that make the model fit the experimental data [91]. Shown in Fig. 4.7 (top) is a plot of the tail of the frequency response of the N¨as¨anen low-pass model that, due to its radially symmetric shape, is only shown along the diagonal cross-section of the two-dimensional CSF. For this and the following HVS models, Fig. 4.7 shows each model using a scale factor S = RD = 300 dpi × 9.5 in. The logarithmic view of the N¨as¨anen filter in Fig. 4.7 (bottom) shows the tails decaying

© 2008 by Taylor & Francis Group, LLC

CHAPTER 4. HALFTONE VISIBILITY

70

Figure 4.6: An (left) error-diffused halftone image and (right) its corresponding modeled HVS response using the N¨as¨anen exponential model. −3 linearly down √ to approximately 10 at the maximum radial frequency of ρ = 1/ 2. Defining a filter’s bandwidth as the frequency at which the filter has decayed to 50% of its maximum amplitude, the N¨as¨anen filter has a bandwidth of 0.078.

4.3

Mixed Gaussian Models

Although the N¨as¨anen model has proven to be an adequate approximation to the HVS model for iterative halftoning methods like direct binary search, Kim and Allebach [54] proposed a new and richer class of HVS models that offer better halftoning results. These models are based on mixed Gaussian functions whose functional form is 



H 2 (ρ) = 2πκ1 σ12 exp −2π 2 σ12 ρ2 + . . . 

. . . + 2πκ2 σ22 exp −2π 2 σ22 ρ2 

(h ∗ h) (r) = κ1 exp −

© 2008 by Taylor & Francis Group, LLC

r2 2σ12







+ κ2 exp −



r2 , 2σ22

(4.8) (4.9)

4.3. MIXED GAUSSIAN MODELS

71

0.12

0.10

0.08

0.06 Nasanen Mixed Gaussian 1

0.04

0.02

Subgaussian Subgaussian 2

0

Mixed Gaussian 2 1/3

1/2

1/sqrt(2)

RADIAL FREQUENCY 10 0

Mixed Gaussian 1 10 −1

Subgaussian 10 −2 Subgaussian 2 Nasanen Mixed Gaussian 2 10 −3 0

1/2

1/sqrt(2)

RADIAL FREQUENCY

Figure 4.7: Frequency response of N¨as¨anen, mixed Gaussian, and α-SG filters (top) and their logarithmic view (bottom). where ∗ denotes convolution and the constants κ1 , κ2 , σ1 , and σ2 are determined experimentally. The advantage of these models is that their frequency response in terms of bandwidth and tail weight can be optimized by varying a set of parameters. Such flexibility is not available in the N¨as¨anen exponential model. The diversity attained with mixed Gaussians, however, comes at a penalty: overparametrization. A total of four parameters characterize this model, making the tuning process a rather delicate task. Kim and Allebach could not find one single model for direct binary search that worked best for

© 2008 by Taylor & Francis Group, LLC

CHAPTER 4. HALFTONE VISIBILITY

72

Figure 4.8: The modeled HVS responses using the mixed Gaussian models with parameters (left) (κ1 , κ2 , σ1 , σ2 ) = (43.2, 38.7, 0.02, 0.06) and (right) (19.1, 42.7.0.03, 0.06). all tones; therefore, they suggested the use of two different Gaussian models that complement each other. The 1st mode has parameters (κ1 , κ2 , σ1 , σ2 ) = (43.2, 38.7, 0.02, 0.06) producing smooth textures but with checkerboard artifacts. Alternately, the 2nd model has parameters (κ1 , κ2 , σ1 , σ2 ) = (19.1, 42.7.0.03, 0.06) and produces a more homogeneous texture, especially in the midtone areas, that is, however, rougher than the previous. Shown in Fig. 4.7 are plots of the spatial and spectral cross-sections of these two parameter sets along with the modeled HVS responses in Fig. 4.8.

4.4

Alpha-Stable HVS Models

The use of alpha-stable functions for HVS modeling was introduced by Gonzalez et al. [45] using the functions for describing alpha-stable random variables [7]. These models are richer than mixed Gaussian models (Gaussian random variables are a sub-family of alpha-stable random variables) and are simpler to characterize having fewer parameters. No-

© 2008 by Taylor & Francis Group, LLC

4.4. ALPHA-STABLE HVS MODELS

73

tably, empirical approximations to the PSF (obtained by measuring the response of the eye to spatial stimuli) were found to have the analytical form of the characteristic function of alpha-stable random variables [133]. Thus, stable models fit well with empirical measurements and, at the same time, offer unique mathematical characteristics that ultimately render visually pleasant halftones. As mentioned before, the work of Whesteimer [133] models the shape of the PSF according to 







h(r) = 0.952 exp −2.59 |r|1.36 + 0.048 exp −2.43 |r|1.74 ,

(4.10)

which is close in form to the characteristic function of the alpha-stable distributions. If the PSF of the HVS model has the form of (4.10), its CSF will have algebraic tails [7], hence it might be a good idea to use models whose tails are heavier than those obtained with exponential and mixed Gaussian models. This fact provides the physiological foundation for the use of alpha-stable models to characterize the HVS. Stable distributions describe a rich class of processes that allow heavy tails and skewness in their functions [7]. The class was characterized by L´evy in 1925 [77] and is described by four parameters: an index of stability, α ∈ (0, 2]; a dispersion parameter, γ > 0; a skewness parameter, δ ∈ [−1, 1]; and a location parameter, β ∈ R. The stability parameter α measures the thickness of the tails of the distribution and provides this model with the flexibility needed to characterize a wide range of impulsive processes. The dispersion γ is similar to the variance of the Gaussian distribution. When the skewness parameter is set to δ = 0, the stable distribution is symmetric about the location parameter β. Symmetric stable processes are also referred to as symmetric alpha-stable or simply SαS. In HVS modeling, bivariate SαS distributions are used for characterization, focusing on their subclass, the so-called alpha-sub-Gaussian (α-SG(R)) random vectors [16], whose characteristic function is of the form

  1 T α/2 φ(t) = exp − t Rt , (4.11) 2 where t = [x, y]T is a two-dimensional column vector and the matrix R is positive definite. This characteristic function is used to model the PSF of the HVS, where, in order to guarantee radial symmetry for the filters, the matrix R is of the form 

R=

© 2008 by Taylor & Francis Group, LLC

γ 0 0 γ



,

(4.12)

CHAPTER 4. HALFTONE VISIBILITY

74

Hrholog

10

10

0

−1

alpha 0.95 alpha 1.05

10

alpha 1.15

−2

alpha 1.25 -1/sqrt(2)

-1/2

0

1/2

1/sqrt(2)

RADIAL FREQUENCY

Figure 4.9: Logarithmic view of the frequency response of α-SG filters. where γ > 0 is the dispersion parameter for the model. With this form of the matrix R and expanding the index of the exponential, the PSF in (4.11) becomes



 α/2 1 φ(x, y) = exp − γ α/2 x2 + y 2 . 2

(4.13)

To simplify (4.13) further, the rectangular coordinates can be converted to polar, resulting in

1 φ(r) = exp − γ α/2 rα 2



for r ≥ 0,

(4.14)

with r2 = x2 + y 2 and where α ∈ (0, 2] is the index of stability that determines the heaviness of the model’s tails. For simplicity in the notation, the term 21 γ α/2 in (4.14) is replaced by only one term, which is called γ  . Figure 4.9 shows the CSF of alpha-sub-Gaussian HVS models with different values of α. The CSF of these models is obtained by sampling the PSF in (4.14), truncating it, and finding its inverse discrete Fourier transform (IDFT), a procedure that carries no computational burden. Notice that this approximation, based on the IDFT, could not be applied if an alpha-stable probability density function were being sought.

© 2008 by Taylor & Francis Group, LLC

4.4. ALPHA-STABLE HVS MODELS

75

Figure 4.10: The modeled HVS responses using sub-Gaussian models with parameters γ  = 27, D = 9.5 in, R = 300 dpi and (left) α = 0.95 or (right) α = 1.05. In alpha-stable models, the parameter γ  (as well as the scale parameter S = RD) determines the bandwidth of the filter and consequently the homogeneity of the halftones. The parameter α controls the heaviness of the tails of the filter and, hence, the smoothness/coarseness of the dither patterns. Gonzalez et al. [45] found experimentally that a model with a bandwidth around 0.08 produced the homogeneity they sought. They also found that a good point to start, in terms of heaviness of tails, is to have an initial model that satisfies H(ρ)|ρ=1/√2 ≈ 10−2 . In the end, they proposed the two models shown in Fig. 4.7, where the starting model is labeled as “α-SG, α = 0.95.” It was generated with constants α = 0.95, γ  = 27, D = 9.5 in, and R = 300 dpi. The initial size of the model was N = 101; although, it was possible to truncate it further to obtain a final N = 31. From various visual studies relating to halftoning, a second rectified filter, plotted in Fig. 4.7 and labeled as “α-SG, α = 1.05,” was found by setting α = 1.05. Both of these filters are demonstrated in Fig. 4.10.

© 2008 by Taylor & Francis Group, LLC

Chapter 5 Blue-Noise Dithering Blue-noise halftoning, demonstrated in Fig. 5.1, is characterized by a distribution of binary pixels where the minority pixels are spread as homogeneously as possible [123]. Distributing pixels in this manner creates a pattern that is aperiodic, isotropic (radially symmetric) and does not contain any low-frequency spectral components. From our understanding of the human visual system (Chapter 4), it’s not surprising that blue-noise creates the visually optimal arrangement of dots. Halftoning a continuous-tone, discrete-space, monochrome image with blue-noise produces a pattern that, as Ulichney [123] describes, is visually “pleasant” and “does not clash with the structure of an image by adding one of its own, or degrade it by being too ‘noisy’ or uncorrelated.”

5.1

Spatial and Spectral Characteristics

Blue-noise, when applied to an image of constant gray-level g, spreads the minority pixels of the resulting binary image as homogeneously as possible, as shown in Fig. 5.2, such that the pixels are separated by an average distance λb 

λb =

√ D/ √ g, for 0 < g ≤ 1/2 D/ 1 − g, for 1/2 < g ≤ 1,

(5.1)

where D is the minimum distance between addressable points on the display. The parameter λb is referred to as the principal wavelength of blue-noise, with its relationship to g justified by several intuitive properties 77 © 2008 by Taylor & Francis Group, LLC

78

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.1: Gray-scale image halftoned using blue-noise shown at 150 dot per inch (dpi).

© 2008 by Taylor & Francis Group, LLC

5.1. SPATIAL AND SPECTRAL CHARACTERISTICS

79

Figure 5.2: The distribution of minority pixels in a blue-noise pattern an average distance of λb apart. 1. As the gray value approaches perfect white (g = 0) or perfect black (g = 1), the principal wavelength approaches infinity. That is, the minority pixels become spread infinitely far apart. 2. The principal wavelength decreases symmetrically with equal deviations from black and white toward middle gray (g = 1/2). 3. The square of the wavelength is inversely proportional to the number of minority pixels per unit area. Again we note that the distribution of minority pixels is assumed to be stationary and isotropic.

5.1.1

Spatial Statistics

In light of the nature of blue-noise to isolate minority pixels, we can begin to characterize blue-noise halftones in terms of the pair correlation R(r) by noting that 1. Few or no neighboring pixels lie within a radius of r < λb . 2. For r > λb , the expected number of minority pixels per unit area approaches I with increasing r. 3. The average number of minority pixels within the radius r increases sharply near r = λb .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

80

(c) (b)

(a)

/

Figure 5.3: The (top) pair correlation of the ideal blue-noise pattern with principal wavelength λb and the (bottom) directional distribution function of the ideal blue-noise pattern in the near range r ∈ [0, 1.5λb ). The resulting pair correlation for blue-noise is therefore of the form in Fig. 5.3 (top), where R(r) shows: (a) a strong inhibition of minority pixels near r = 0, (b) a decreasing correlation of minority pixels with increasing r (limr→∞ R(r) = 1), and (c) a frequent occurrence of the inter-point distance λb , the principal wavelength, indicated by a series of peaks at integer multiples of λb . The principal wavelength is indicated in Fig. 5.3 by a diamond located along the horizontal axis. As an isotropic point process, the ideal Dr1 ,r2 (a) is shown in Fig. 5.3 (bottom) for the near range of r ∈ (0, 1.5λb ], which indicates the directional distribution between nearest neighbors.

5.1.2

Spectral Statistics

Turning to the spectral domain, the spectral characteristics of bluenoise in terms of the radially averaged power spectrum, Pρ (fρ ), are shown in Fig. 5.4 and can be described by three unique features: (a) lit-

© 2008 by Taylor & Francis Group, LLC

5.1. SPATIAL AND SPECTRAL CHARACTERISTICS

81

(c) (b) (a)

Figure 5.4: The (top) RAPSD of the ideal blue-noise pattern with principal frequency fb , and the (bottom) anisotropy of the ideal bluenoise pattern. tle or no low-frequency spectral components; (b) a flat, high-frequency (blue-noise) spectral region; and (c) a spectral peak at cutoff frequency fb , the blue-noise principal frequency, such that  √ g/D, for 0 < g ≤ 1/2 (5.2) fb = √ 1 − g/D, for 1/2 < g ≤ 1. As will be the convention in this book, the principal frequency is indicated in Fig. 5.4 (top) by a diamond located along the horizontal axis. Also note that P (fρ ) is plotted in units of σg2 = g(1 − g), the variance of an individual pixel in Ig . The sharpness of the spectral peak in P (fρ ) at the blue-noise principal frequency is affected by the separation between minority pixels, which should have some variation. The wavelengths of this variation should not be significantly longer than λb , as this adds low-frequency spectral components to the corresponding dither pattern Ig [123], causing Ig to appear more white than blue. This process is referred to as spectral whitening. Being isotropic, A(fρ ) should be minimized at all fρ , as shown in Fig. 5.4 (bottom), where, based on the estimation of P (f ) (Sec. 3.3), −10db is considered background noise [123].

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

82

Y Y

Figure 5.5: The error-diffusion algorithm.

5.2

Error-diffusion

In error-diffusion (Fig. 5.5), the output pixel y[n] is determined by adjusting and thresholding the input pixel x[n] such that: 

y[n] =

1, if (x[n] + xe [n]) ≥ 0 0, else,

(5.3)

where xe [n] is the diffused quantization error accumulated during previous iterations as xe [n] =

M  i=1

with y e [n] = y[n]−(x[n]+xe [n]) and (5.4) becomes

bi · y e [n − i] M

i=1 bi

(5.4)

= 1. Using vector notation,

xe [n] = bT ye [n]

(5.5)

where b = [b1 , b2 , . . . , bM ]T and ye [n] = [y e [n − 1], y e [n − 2], . . . , y e [n − M ]]T . Shown in Fig. 5.6 are the error weights originally proposed by Floyd and Steinberg [40], and using a left-to-right raster scan, Figs. 5.7 and 5.8 show the resulting monochrome image and gray-scale ramp produced using Floyd’s and Steinberg’s error filter. For a statistical analysis, Figs. 5.9–5.11 show the corresponding spatial and spectral 1 1 metrics for intensity levels I = 32 , 16 , 18 , 14 , 38 , and 12 . While these results exhibit distinct blue-noise properties, Floyd’s and Steinberg’s error filter produces disturbing texture shifts or banding at multiples of the 13 and 14 gray-levels [40], where error-diffusion “locks” into a regular, stable pattern [56]. The brief transition in these instances occurs as a consequence of the sudden breakup in the regular pattern. The Floyd and Steinberg error filter also creates disturbing hysteresis artifacts or worm patterns at extreme gray-levels g = 0 and g = 1.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

3/16

83 • 7/16 5/16 1/16

Figure 5.6: Floyd’s and Steinberg’s proposed error filter.

5.2.1

Eliminating Unwanted Textures

Since Floyd’s and Steinberg’s first paper, many modifications to the original error-diffusion algorithm have been introduced that address the unwanted artifacts of the original algorithm. But while these modifications eliminate disturbing artifacts at certain gray-levels, many do so at the expense of other levels [56]. A typical problem that occurs is spectral whitening, described earlier, where the variation in average separation distance between minority pixels becomes so great that the pattern starts to resemble the halftone pattern created by white-noise. Modified Filter Weights In an effort to break up worm patterns in error-diffusion, both Jarvis et al. [50] and Stucki [117] introduced 12-element error filters, as shown in Figs. 5.12 and 5.13, respectively. Shown in Figs. 5.15–5.24 are their corresponding gray-scale ramps, images, and metrics. From their grayscale ramps (Figs. 5.15 and 5.20), it is apparent that both filters break up worms at extreme gray-levels, but both do so with spectral whitening in the mid-tones [56]. The Jarvis et al. filter performs especially poorly in the range 1 ≤ I ≤ 12 , where error-diffusion creates patterns with a frequent oc4 currence of the inter-point distance 2D, the principal wavelength for I = 14 . This behavior is clearly indicated in the power spectrums of Figs. 5.18 and 5.19, where the RAPSD shows a strong peak at fρ = 12 D−1 . For intensities near 14 , the banding of minority pixels occurs along diagonals, but as the intensity increases toward 12 , minority pixels begin to cluster, forming a coarse pattern at g = 12 . Stucki’s filter has several problems of its own, although not as severe. Here, dither patterns also exhibit strong spectral components below the principal frequency in the range 14 ≤ I ≤ 12 , with minority pixels exhibiting strong correlations along the diagonals. As the intensity approaches 12 , Stucki’s weights maintain this correlation, resulting in the g = 12 checkerboard pattern also found using Floyd’s and Steinberg’s weights.

© 2008 by Taylor & Francis Group, LLC

84

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.7: Gray-scale ramp halftoned using Floyd’s and Steinberg’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

85

Figure 5.8: Gray-scale image halftoned using Floyd’s and Steinberg’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

86

2 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.9: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a left-to-right raster scan for intensity levels (left) 1 1 I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

87

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.10: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a left-to-right raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

88

RAPSD

4

RAPSD

4

1 0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.11: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a left-to-right raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

89

• 7/48 5/48 3/48 5/48 7/48 5/48 3/48 1/48 3/48 5/48 3/48 1/48 Figure 5.12: Jarvis’s, Judice’s, and Ninke’s proposed error filter. • 8/42 4/42 2/42 4/42 8/42 4/42 2/42 1/42 2/42 4/42 2/42 1/42 Figure 5.13: Stucki’s proposed error filter.

1/16

1/16

2/16

• 8/16 4/16 0/16

Figure 5.14: Shiau’s and Fan’s proposed error filter. A fourth approach to filter weight selection is the error filter of Shiau and Fan [111], shown in Fig. 5.14. Here, the filter is restricted to just two rows but has an extraordinarily long tail. The result is a gray-scale ramp, shown in Fig. 5.25, without the worm patterns near g = 0 and g = 1 but also with little difference in performance from Floyd’s and Steinberg’s filter in the mid-tones. Algorithms that swap error filters according to the gray-level of the current input pixel have also been proposed [32, 134]. In Eschbach’s scheme [32], the error filter is swapped between different sized arrangements, with the larger arrangement used near extreme graylevels where large filters typically perform better while the smaller is used in the mid-tones where small filters typically perform better. To minimize artifacts or contours created at the point of the filter swap, the smaller filter is chosen as a subset of the larger.

© 2008 by Taylor & Francis Group, LLC

90

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.15: Gray-scale ramp halftoned using Jarvis’s, Judice’s, and Ninke’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

91

Figure 5.16: Gray-scale image halftoned using Jarvis’s, Judice’s, and Ninke’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

92

2 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.17: Spatial and spectral statistics using Jarvis’s, Judice’s, and Ninke’s filter weights with a left-to-right raster scan for intensity levels 1 1 (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

93

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.18: Spatial and spectral statistics using Jarvis’s, Judice’s, and Ninke’s filter weights with a left-to-right raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

94

RAPSD

4

RAPSD

4

1 0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.19: Spatial and spectral statistics using Jarvis’s, Judice’s, and Ninke’s filter weights with a left-to-right raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

95

Figure 5.20: Gray-scale ramp halftoned using Stucki’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

96

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.21: Gray-scale image halftoned using Stucki’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

97

4 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.22: Spatial and spectral statistics using Stucki’s filter weights 1 with a left-to-right raster scan for intensity levels (left) I = 32 and 1 (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

98

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.23: Spatial and spectral statistics using Stucki’s filter weights with a left-to-right raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

99

RAPSD

4

RAPSD

4

1 0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.24: Spatial and spectral statistics using Stucki’s filter weights with a left-to-right raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

100

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.25: Gray-scale ramp halftoned using Shiau’s and Fan’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

101

Figure 5.26: Gray-scale image halftoned using Shiau’s and Fan’s filter weights with a left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

102

RAPSD

2

RAPSD

4

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

1

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.27: Spatial and spectral statistics using Shiau’s and Fan’s filter 1 weights with a left-to-right raster scan for intensity levels (left) I = 32 1 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

103

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fb

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.28: Spatial and spectral statistics using Shiau’s and Fan’s filter weights with a left-to-right raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

104

RAPSD

8

RAPSD

8

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.29: Spatial and spectral statistics using Shiau’s and Fan’s filter weights with a left-to-right raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

105

left-to-right

serpentine Figure 5.30: Two commonly used raster scans for error-diffusion shown using the Floyd and Steinberg filter weights. Raster Scanning Path The scanning path is the order to which input pixels are processed by the error-diffusion algorithm. Typically, as is the case with Floyd and Steinberg, the scanning path is a left-to-right raster scan (Fig. 5.30 (top)), where pixels are processed moving left to right along consecutive rows. Another approach, and perhaps the next most common, is a serpentine raster scan (Fig. 5.30 (left)), where the input image is processed along consecutive rows with the first row processed left-to-right and the next row right-to-left. Shown in Figs. 5.31–5.35 are the resulting ramps, images, and metrics using Floyd’s and Steinberg’s error filter with the serpentine raster scan. More creative approaches to scanning paths have also been proposed based on space filling curves – connected paths that visit each and every input pixel exactly once [125]. Shown in Fig. 5.36 are illustrations showing the Peano and Hilbert paths, but while these paths break up worms, they also introduce a significant amount of noise through the introduction of new patterns [56]. Knuth [59] offered an interesting twist to scanning paths when he introduced dot-diffusion, a halftoning algorithm based loosely on error-

© 2008 by Taylor & Francis Group, LLC

106

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.31: Gray-scale ramp halftoned using Floyd’s and Steinberg’s filter weights with a serpentine raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

107

Figure 5.32: Gray-scale image halftoned using Floyd’s and Steinberg’s filter weights with a serpentine raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

108

4 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.33: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a serpentine raster scan for intensity levels 1 1 (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

109

RAPSD

8

RAPSD

4

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.34: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a serpentine raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

110

RAPSD

8

RAPSD

8

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.35: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with a serpentine raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

111

Figure 5.36: The (left) Peano and (right) Hilbert space filling curves.

= processed input pixel = unprocessed input pixel

Figure 5.37: The diffusion of quantization error for pixels within an 8 × 8 block using Knuth’s dot-diffusion.

diffusion that halftones the input image in independently processed 8 × 8 blocks. In this new algorithm, pixels of each block are processed along a disconnected path specified by the consecutive thresholds of an 8 × 8 dither array such as that used in clustered-dot ordered dither (Sec. 1.1) or Bayer’s dither. For a particular iteration of dot-diffusion, the current input pixel is quantized with the quantization error-diffused equally to the neighboring, unprocessed input pixels within the block, as shown in Fig. 5.37. While Knuth’s algorithm leads to strong periodic patterns as in Fig. 5.38 (left), where pixels are processed in the order of clustered-dot ordered dithers, Mese and Vaidyanathan [84] show, as in Fig. 5.38 (right), that the order can be optimized for 16 × 16 blocks to produce results of comparable visual quality to error-diffusion.

© 2008 by Taylor & Francis Group, LLC

112

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.38: The halftone images created using Knuth’s dot-diffusion with pixels processed in (left) 8×8 blocks using an ordering specified by Bayer’s dither and in (right) 16× 16 blocks with the ordering optimized to imitate blue-noise. Filter and Threshold Perturbations Perturbation refers to adding noise or randomness to the error weights or quantization threshold at each iteration of the error-diffusion algorithm. An early approach to filter perturbation was proposed by Ulichney [122], who suggested grouping filter coefficients of similar weights into pairs and then moving a portion of one weight to the other. The size of the portion was chosen randomly at each iteration and, in order to avoid negative weights, was limited to a maximum percentage of the smaller of the two weights. So, given a filter pair (a1 , a2 ), where a1 > a2 , the filter weights used at any given iteration would be determined as a1 = a1 + U (T · a2 ) a2 = a2 − U (T · a2 ),

(5.6) (5.7)

where U is a uniformly distributed random number from the range [−1, 1] and T is a pre-specified tolerance in the range [0, 1]. Using the Floyd-Steinberg filter weights, Ulichney showed that perturbing each of the two pairs (7/16, 5/16) and (3/16, 1/16), created a good blue-noise process. In particular, adding 50% noise to each pair appears to optimize the trade-off between graininess and stable texture

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

113

[123]. Figs. 5.39–5.43 show the resulting gray-scale ramps, gray-scale images, and the spatial and spectral characteristics using this approach. Perturbing the quantization threshold presents an advantage over filter weight perturbation, having a single, scalar quantity to adjust. An obvious approach to threshold perturbation is to add either a uniformly distributed random number from the range [−T, T ] or a Gaussian distributed random number with variance T . In both cases, T is determined empirically. Shown in Fig. 5.44 (right) is the resulting halftone image using the Gaussian random number where T = 0.01. The drawback of using a perturbed threshold, as seen in Fig. 5.44, is that the randomness added to the threshold whitens the halftone with noisy artifacts most noticeable in the mid-tones. Feedback Error-diffusion with feedback refers to algorithms that take into account previous output pixels during the quantization of the current input pixel. An early approach to feedback was proposed by Fawcette and Shrack [38], who modulated the quantization threshold according to the binary arrangement of dots in the previous five output pixels. Levien [75] offered an alternative approach that suggested modulating the threshold according to the distance between the current input pixel and the nearest minority pixel. If the nearest minority pixel is at a distance d, then the threshold is decreased by (d2 − 1/x[n]) · c for 0 < g < 12 or increased by (d2 − 1/(1 − x[n])) · c for 12 ≤ g < 1, where c is a scalar constant giving the user the ability to regulate how much of an impact d has on y[n]. From visual inspection of the gray-scale ramp of Fig. 5.45, it is clear that Levien’s offset on the quantization threshold has lesser impact in the mid-tones, where 1/x[n] or 1/(1 − x[n]) is minimized relative to the extremes and 1/x[n] or 1/(1 − x[n]) is maximized. In the same spirit as Levien, Eschbach [33] suggested that the threshold offset should be a superposition of offsets derived from all neighboring minority pixels and not just the nearest.

5.2.2

Edge Enhancement

In his book, Ulichney [123] wrote that the measure of a particular halftoning algorithm was its performance in DC regions and that its performance near edges or in areas of high-frequency image content

© 2008 by Taylor & Francis Group, LLC

114

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.39: Gray-scale ramp halftoned using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

115

Figure 5.40: Gray-scale image halftoned using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

116

2 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.41: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan 1 1 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

117

4 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.42: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

118

RAPSD

8

RAPSD

4

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.43: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

119

Figure 5.44: The halftone images created using Floyd’s and Steinberg’s filter weights with a left-to-right raster with (left) no threshold perturbation and (right) a Gaussian distributed (V AR = 0.01) threshold perturbation. can be manipulated through pre-filtering the image prior to halftoning. So the remedy for the apparent blurring of edges caused by the error-diffusion algorithm was to apply an edge sharpening filter prior to halftoning such that xsharp [n] = x[n] − β(ψ[n] ∗ x[n]),

(5.8)

where ψ[n] is a digital Laplacian filter (Fig. 5.50), ∗ denotes convolution, and β is a scalar constant (β ≥ 0) regulating the amount of sharpening with larger β leading to a sharper image xsharp . Shown in Fig. 5.51 is a comparison of two halftones produced using Ulichney’s perturbed error weight scheme with and without sharpening (β = 2). For another approach to edge enhancement, Eschbach and Knox [34] have shown that modulating the quantization threshold at each iteration by a scalar multiple of the current input pixel sharpens the resulting apparent image. To understand why requires a spectral analysis of the error-diffusion algorithm itself. To this end, Weissbach et al. [132] showed that the power spectrum, Y , of the output image of error-diffusion was related to the power spectrums, X and Xe , of the input and error images as Y = X + F Xe ,

© 2008 by Taylor & Francis Group, LLC

(5.9)

120

CHAPTER 5. BLUE-NOISE DITHERING

Figure 5.45: Gray-scale ramp halftoned using Floyd’s and Steinberg’s filter weights with Levien’s threshold offset (c = 0.005) and a left-toright raster scan.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

121

Figure 5.46: Gray-scale image halftoned using Floyd’s and Steinberg’s filter weights with Levien’s threshold offset (c = 0.005) and a left-toright raster scan.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

122

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.47: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with Levien’s threshold offset (c = 0.005) and a 1 left-to-right raster scan for intensity levels (left) I = 32 and (right) 1 I = 16 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

123

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.48: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with Levien’s threshold offset (c = 0.005) and a left-to-right raster scan for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 5. BLUE-NOISE DITHERING

124

RAPSD

4

RAPSD

4

1 0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 5.49: Spatial and spectral statistics using Floyd’s and Steinberg’s filter weights with Levien’s threshold offset (c = 0.005) and a left-to-right raster scan for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

125

Figure 5.50: Two digital Laplacian filters proposed by Ulichney for edge sharpening prior to halftoning.

Figure 5.51: The halftone images created using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan (left) before and (right) after having been edge sharpened prior to halftoning (β = 2). where F = 1 − DFT2D {a}

(5.10)

is a high-pass filter derived from the two-dimensional discrete Fourier transform of the error filter a. Relating these results to threshold modulation, Knox and Eschbach [34] later showed that varying the threshold of error-diffusion by the function t[n] was equivalent to applying ordinary error-diffusion to the input image x˜, where ˜ = X + F T. X

© 2008 by Taylor & Francis Group, LLC

(5.11)

126

CHAPTER 5. BLUE-NOISE DITHERING

If t[n] = k·x[n], then (5.11) is equivalent to adding a high-pass version of x to itself prior to halftoning – leading to sharper edges in the apparent image y. Figure 5.52 shows the resulting halftone images produced by Ulichney’s perturbed error weight scheme with Knox’s and Eschbach’s threshold modulation where k = 0, 2, 5, and −2. Note that the negative value of k leads to blurring instead of sharpening.

© 2008 by Taylor & Francis Group, LLC

5.2. ERROR-DIFFUSION

127

Figure 5.52: The halftone images created using Floyd’s and Steinberg’s filter weights with 50% perturbation and a serpentine raster scan with threshold modulation where (top left) k = 0, (top right) k = 2, (bottom right) k = 5, and (bottom right) k = −2.

© 2008 by Taylor & Francis Group, LLC

Chapter 6 Blue-Noise Dither Arrays Halftoning via blue-noise dither arrays is a process whereby a continuoustone image is thresholded on a pixel-by-pixel basis with a dither array or mask. A pixel of the resulting binary image is set to one if that pixel of the continuous-tone image is greater than or equal to the corresponding pixel of the dither array; otherwise, the pixel is set to zero. These masks can also be of any size with large images halftoned using new masks formed by tiling edge-to-edge the original dither array. The blue-noise dither array derives its name from the fact that, given a continuous-tone monochrome image of constant gray-level g, the resulting dither pattern has blue-noise characteristics appropriate to g. While dispersed-dot dithering creates patterns with a “strong periodic structure that imparts an unnatural appearance to resulting images” [124], halftoning via blue-noise dither arrays typically uses dither arrays that are much larger in size than those used in disperseddot ordered dithering (256×256 versus 16×16), and while the halftoned images are not as visually appealing as those derived from adaptive techniques such as error-diffusion, the computational complexity is greatly reduced, requiring no additional storage other than for the array itself. Also, as a process that need not consider neighboring pixels, halftoning via blue-noise dither arrays is a candidate for parallel processing. The construction of blue-noise dither arrays is generally performed by first constructing a set, {Ig : 0 < g < 1}, of binary dither patterns with exactly one dither pattern, Ig , for each unique gray-level of the original, continuous-tone image. Inside of Ig , the fraction of pixels set to white (1) is equal to g, and so for an 8 bit monochrome display, this implies constructing 256 unique dither patterns with the first pattern, I0 , corresponding to the all-black pixel pattern for g = 0 and the 129 © 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

130

12 3

9 13

4 16 6

1

7 10 5 15

1 16

14 2 11 8

Figure 6.1: The (right) dither array constructed from (left) 16 binary dither patterns. last pattern, I1 , corresponding to the all-white pattern for g = 1. And in order to produce visually pleasing dither patterns for large images, the component dither patterns must satisfy a “wrap-around” property where tiling a particular pattern end-to-end does not create any discontinuity in texture moving across boundaries. Having this set, the blue-noise dither array is then constructed by assigning to each pixel a threshold according to the spatial arrangement of binary pixels within {Ig : 0 ≤ g ≤ 1}, where, in order to avoid ambiguities in the assignment of thresholds, the dither patterns, Ig , are constructed under the stacking constraint that Ik ⊂ Ig for all k < g or that, if given Ik [n] = 1, then Ig [n] = 1 for all g > k. As a consequence, the threshold assigned to each pixel of the array is defined as DA[n] = min{g : Ig [n] = 1},

(6.1)

with the minimum gray-level g such that the corresponding pixel in Ig is equal to 1. Given a blue-noise dither array of size M , the output pixel, y[n], is derived from x[n] as 

y[n] =

1, if x[modM (n)] ≥ DA[modM (n)] 0, else.

Figure 6.1 illustrates (6.1) for an array of size 4 × 4.

© 2008 by Taylor & Francis Group, LLC

(6.2)

6.1. SIMULATED ANNEALING

6.1

131

Simulated Annealing

While many techniques have been proposed for deriving the component blue-noise dither patterns [86, 3, 20, 21, 26, 80, 92, 107, 124, 127, 128] forming the set {Ig : 0 ≤ g ≤ 1}, perhaps the first to do so successfully was Sullivan et al. [119], who proposed using a set visually pleasing, binary dither patterns, {Ig : 0 ≤ g ≤ 1}, to halftone the continuoustone image, x[n], by table look-up such that y[n] = Ix[n] [modM (n)].

(6.3)

But without a stacking constraint, the patterns that formed the set {Ig : 0 ≤ g ≤ 1} could have been generated using any blue-noise halftoning algorithm such as error-diffusion or DBS on an appropriate set of input images. Sullivan et al., instead, proposed using a model of the HVS along with simulated annealing, where the 1’s and 0’s of a binary dither pattern are swapped at random, to minimize visible artifacts. Sullivan et al.’s iterative algorithm progresses by keeping any and all random swaps that improve the visual appearance of the pattern as well as a percentage of swaps that do not improve, and possibly degrade, the appearance. With each iteration, the percentage of bad swaps that are kept is reduced, and the algorithm quits after a fixed number of iterations. The steps of Sullivan et al.’s simulated annealing algorithm are described as follows 1. Create a white-noise binary dither pattern, Ig , representing graylevel g, by thresholding a continuous-tone input image with uniformly distributed, uncorrelated (white) noise. Set K, the iteration number, equal to 1. 2. Generate the next pattern in the evolution of Ig by: (a) Calculating the visual cost, Cost(Ig ), of Ig as the total power in Ig after filtering by a model of the HVS such that: Cost(Ig ) =



|DFT2D (Ig ) × HV S(f )|2 .

(6.4)

f

(b) Randomly selecting a pair of pixels, one 1 and one 0, and swap the two, making the 1 a 0 and the 0 a 1, to create the new pattern, Ig .

© 2008 by Taylor & Francis Group, LLC

132

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.2: The (left) initial and (right) final dither pattern using Sullivan et al.’s simulated annealing algorithm. (c) Calculating the visual cost, Cost(Ig ), of the new pattern using (6.4). (d) Calculating the Boltzman test statistic, β, as: 



−(Cost(Ig ) − Cost(Ig )) β = exp , 0.95K T

(6.5)

where T is a scalar constant regulating the percentage of swaps, good or bad, that will be kept during the current iteration. (e) Calculating ζ as a uniformly distributed random number from the range [0, 1]. If β > ζ, then keep the current swap by setting Ig = Ig ; otherwise, ignore the swap and leave Ig unchanged. (f) Repeating steps (2a)–(2e) a fixed number of times (e.g., 1000). 3. Set K = K + 1. If K is sufficiently high (e.g. 500), then quit with the final pattern stored in Ig ; otherwise, return to step (2). Note that because Sullivan et al. do their filtering process in the spectral domain utilizing the discrete Fourier transform, wrap-around textures are guaranteed as the DFT implicitly assumes the subject pattern represents one period of a repeating process. As a demonstration,

© 2008 by Taylor & Francis Group, LLC

6.1. SIMULATED ANNEALING

133

Cost(Ig) (db)

ï20 ï25 ï30 ï35

0

100

200

300

400

500

Iteration

Figure 6.3: The reduction in cost for the binary dither pattern using simulated annealing. Fig. 6.2 shows an initial dither pattern and its resulting output pattern where the visual cost is based on the two-dimensional HVS model of Chapter 4 for a 300 dpi print resolution and 20 in viewing distance and where T is selected to keep 80% of the swaps in the initial iteration (K = 1). Shown in Fig. 6.3 is the visual cost of the dither pattern at each instance of K showing not only the gradual decrease in overall cost but also a reduction in the variance of the visual cost as fewer and fewer bad swaps are being retained. Shown in Figs. 6.4–6.6 are the resulting spatial and spectral metrics using the simulated annealing algorithm for intensities I = 1 , 1 , 1 , 1 , 3 , and 12 . For I = 18 , 14 , 38 , or 12 , note that simulated 32 16 8 4 8 annealing leads to a significant amount of spectral energy at radial frequencies below the principal frequency of blue-noise. By looking at the RAPSD of the HVS model in Fig. 6.7, it is apparent that there is a strong correlation between the shape of the spectral content of the dither pattern and that of the HVS model with minority pixels, as in Figs. 6.5 and 6.6, not adhering to the principal wavelength separation and exhibiting some, although slight, degree of clustering. The correlation between power spectrums of the resulting dither pattern and the HVS model suggests that to improve the blue-noise characteristics of the dither patterns created by simulated annealing, the cut-off frequency of the HVS model should be manipulated according to the gray-level such that intensities near 12 have a high cut-off frequency and intensities near 0 have a small cut-off frequency. Shown in Figs. 6.8–6.10 are the resulting spatial and spectral metrics using the simulated annealing algorithm where the print resolution for the HVS model has been varied with gray-level intensities to reduce minority pixel clustering.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

134

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.4: Spatial and spectral statistics using Sullivan et al.’s simu1 lated annealing algorithm for intensity levels (left) I = 32 and (right) 1 I = 16 .

© 2008 by Taylor & Francis Group, LLC

6.1. SIMULATED ANNEALING

135

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.5: Spatial and spectral statistics using Sullivan et al.’s simulated annealing algorithm for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

136

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.6: Spatial and spectral statistics using Sullivan et al.’s simulated annealing algorithm for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

137

l

RAPSD of CSF(f )

6.1. SIMULATED ANNEALING

1 0.5 0

0

0.1

0.2 0.3 0.4 0.5 RADIAL FREQUENCY (Dï1)

0.6

0.7

Figure 6.7: The RAPSD of the two-dimensional HVS model used with simulated annealing to construct the dither patterns of Figs. 6.4–6.6. Figures 6.11 and 6.12 illustrate Sullivan et al.’s scheme for dither array design. The drawback of ignoring the stacking constraint is evident in the gray-scale ramp of Fig. 6.11, where discontinuities, caused by isolated clusters, in the texture of the dither pattern are visible along the border between unique gray-levels. Within the gray-scale image of Fig. 6.12, the results are much worse with the halftone closely modeling white-noise. From these two results, its clear that Sullivan et al.’s unconstrained dither patterns are restricted to applications limited to business graphics (pie charts, bar graphs, etc.). In order for the constructed set, {Ig : 0 ≤ g ≤ 1}, to satisfy the stacking constraint (Fig. 6.13), Sullivan et al.’s simulated annealing algorithm must be constrained in several ways. The first is that given a binary pattern, Is , the newly constructed pattern, Ig , such that g > s, must have Is as a subset. That is, Ig [n] = 1 for all n such that Is [n] = 1. This property is satisfied by specifying the initial pattern as Is with an appropriate number of pixels, equal to zero, switched to ones. The initial pattern should, after switching, represent gray-level g. The simulated annealing algorithm must also, in order to satisfy the constraint Is ⊂ Ig , consider swapping only those pixels for which Is [n] = 0. A second constraint that Sullivan et al.’s simulated annealing algorithm must be able to satisfy is that, given a pattern IS , the newly constructed pattern, Ig , such that g < S, must be a subset of IS . This property is satisfied by specifying the initial pattern as IS with an appropriate number of pixels, equal to one, switched to zeros. One must also consider swapping only those pixels for which IS [n] = 1. Given both Is and IS , the initial pattern should be specified as Is with an appropriate number of pixels, equal to zero and with corresponding

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

138

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.8: Spatial and spectral statistics using Sullivan et al.’s simu1 lated annealing algorithm for intensity levels (left) I = 32 and (right) 1 I = 16 , where the print resolution of the HVS model has been modified according to the gray-level.

© 2008 by Taylor & Francis Group, LLC

6.1. SIMULATED ANNEALING

139

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.9: Spatial and spectral statistics using Sullivan et al.’s simulated annealing algorithm for intensity levels (left) I = 18 and (right) I = 14 , where the print resolution of the HVS model has been modified according to the gray-level.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

140

8 RAPSD

RAPSD

2

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.10: Spatial and spectral statistics using Sullivan et al.’s simulated annealing algorithm for intensity levels (left) I = 38 and (right) I = 12 , where the print resolution of the HVS model has been modified according to the gray-level.

© 2008 by Taylor & Francis Group, LLC

6.1. SIMULATED ANNEALING

141

Figure 6.11: Gray-scale ramp halftoned using a dither array constructed by Sullivan et al.’s simulated annealing scheme without a stacking constraint.

© 2008 by Taylor & Francis Group, LLC

142

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.12: Gray-scale image halftoned using a dither array constructed by Sullivan et al.’s simulated annealing scheme without a stacking constraint.

© 2008 by Taylor & Francis Group, LLC

6.1. SIMULATED ANNEALING

143

Figure 6.13: Illustration of the stacking constraint on φg such that φs ⊂ φg ⊂ φS . pixels in IS equal to one, switched to ones. So swapping must now consider only those pixels that correspond to zeros in Is and ones in IS . With the simulated annealing algorithm now constrained by previously constructed patterns, the set {Ig : 0 ≤ g ≤ 1} can be constructed one-pattern-at-a-time and in any order, but while the patterns can be constructed in sequential order according to g, Sec. 6.4 will show that some orderings are better then others. So, in summary of blue-noise dither array construction, we assume that images are composed of a discrete set of N gray-levels defined by the monotonically increasing sequence {gi : i = 1, 2, . . . , N } with g1 = 0 (black) and gN = 1 (white). The steps for generating the set {Ig : 0 ≤ g ≤ 1} are then defined according to 1. Given g1 = 0 and gN = 1, define Ig1 as an all-zero matrix and IgN as an al-one matrix. 2. Define the sequence {gki : i = 1, 2, . . . , N } as a rearrangement or re-ordering of the sequence {gi : i = 1, 2, . . . , N } such that every gray-level gi appears once and only once in the sequence {gki : i = 1, 2, . . . , N } with gk1 = 0 and gk2 = 1 (k1 = 1 and k2 = N ).

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

144

Figure 6.14: The dither array, constructed by Sullivan et al.’s simulated annealing scheme employing the stacking constraint and a manipulated HVS model, with its corresponding magnitude to its Fourier transform. 3. Set j = 3. 4. Construct the dither pattern Igkj under the stacking constraint for all patterns {Igki : i = 1, 2, . . . , j − 1} (for all patterns that have been generated up to the current iteration). 5. Set j = j + 1. If j = N + 1, the process is complete; otherwise, continue at step 4. Using the above procedure, Figs. 6.14–6.16 show the dither arrays, resulting gray-scale ramps, and resulting gray-scale image when dither patterns are constructed in the order {0, 1, 14 , 34 , 12 , 18 , 38 , 58 , 78 , . . .}. Shown in Fig. 6.17 is the same gray-scale image but with Ulichney’s edge sharpening applied prior to halftoning (β = 2).

6.2

Void-and-Cluster

A popular scheme for building blue-noise dither arrays is Ulichney’s [124] void-and-cluster (VAC) technique [124], which iteratively swaps black and white pixels according to a measure of local white pixel density. As demonstrated in Fig. 6.18, VAC measures the minority pixel density, around a given pixel x[m, n], as the gray-level of the corresponding pixel in the continuous-tone image resulting from the con-

© 2008 by Taylor & Francis Group, LLC

6.2. VOID-AND-CLUSTER

145

Figure 6.15: Gray-scale ramp halftoned using a dither array constructed by Sullivan et al.’s simulated annealing scheme with the stacking constraint and a manipulated HVS model.

© 2008 by Taylor & Francis Group, LLC

146

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.16: Gray-scale image halftoned using a dither array constructed by Sullivan et al.’s simulated annealing scheme with the stacking constraint and a manipulated HVS model.

© 2008 by Taylor & Francis Group, LLC

6.2. VOID-AND-CLUSTER

147

Figure 6.17: Gray-scale image halftoned using a dither array constructed by Sullivan et al.’s simulated annealing scheme with the stacking constraint, a manipulated HVS model, and Ulichney’s edge sharpening applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

148

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.18: The current iteration’s dither pattern and the corresponding low-pass filtered image measuring the local white pixel density. volution of a low-pass filter with the current iteration’s dither pattern. Specifically, VAC generates a seed pattern, Xk=0 , as a white-noise dither pattern representing the gray-level gi . This seed pattern is then filtered by means of circular convolution with the low-pass filter, HLP , such that Y = Xk ⊗ HLP , (6.6) where pixel values, y[m, n], less than gi correspond to regions of the dither pattern with too few neighboring white pixels (x[m, n] = 1) while pixel values greater than gi correspond to regions of the dither pattern with too few neighboring black pixels (x[m, n] = 0). Because the success of the void-and-cluster algorithm relies very heavily upon the choice of low-pass filters, Ulichney recommends a Gaussian filter defined according to HLP [n] = exp(

−|n|2 ), 2σ 2

(6.7)

where σ is a scalar constant determined empirically to offer the best results at σ = 1.5D [124]. Given the relationship of the filtered image to a measure of white pixel density, the VAC algorithm identifies the white pixel of Xk with the largest concentration of white pixels (largest y[m, n] value), toggling the pixel to black. It is these regions of high, white pixel density

© 2008 by Taylor & Francis Group, LLC

6.2. VOID-AND-CLUSTER

149

Figure 6.19: Illustration of the tiled, binary dither patterns (left) before and (right) after applying VAC to a 108×108 white-noise dither pattern representing gray-level 34 . that Ulichney refers to as a cluster where, by toggling the bit, VAC attempts to create a void. This modified dither pattern is then re-filtered with the black pixel of Xk having the sparsest concentration of white pixels (smallest y[m, n] value) toggled to white. It is these regions of low, white pixel density that Ulichney refers to as a void, where VAC attempts to create a cluster. It is this process of voiding and clustering white pixels from which VAC derives its name. Having toggled a white pixel to black and then a black pixel to white, the resulting dither pattern, Xk+1 , now completes the current iteration with Xk+1 having the same total number of white and black pixels as Xk . The algorithm then repeats the process of voiding and clustering pixels until the black pixel toggled to white is the same pixel that was toggled from white to black. If this is the case, the process has converged with repeated iterations having no effect. A significant feature of VAC is that by performing the low-pass filtering by means of circular convolution, the resulting dither pattern has wrap-around characteristics such that tiling a pattern end-to-end creates no apparent discontinuities in texture. This is demonstrated in Fig. 6.19, where we show a 108 × 108 dither pattern representing gray-level 75% along with its 8-point neighboring tiles (left) before and (right) after optimizing minority pixel placement. Shown in Figs. 6.4–

© 2008 by Taylor & Francis Group, LLC

150

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

6.6 are the resulting spatial and spectral metrics using the void-and1 1 , 16 , 18 , 14 , 38 , and 12 . Like cluster with σ = 1.5D for intensities I = 32 simulated annealing with a varying HVS model, the results of void-andcluster can be improved upon if σ is allowed to vary [80] with I as in Figs. 6.23–6.25. Now, as with simulated annealing for generating subsequent gray-levels, the process of toggling bits to create voids and clusters is repeated under the stacking constraint such that the swapping of pixels is restricted to only those pixels corresponding to white pixels in previously constructed patterns of brighter gray-levels and black pixels in previously constructed patterns of darker gray-levels. In doing so, blue-noise dither arrays can now be constructed by means of void-andcluster, and shown in Fig. 6.26 are two such arrays constructed by voidand-cluster with and without tuning σ, along with the magnitudes of their Fourier transforms. Constructed in the order {0, 1, 14 , 34 , 12 , 18 , 38 , 58 , 7 , . . .}, these two dither arrays lead to the gray-scale ramps and images 8 of Figs. 6.29–6.30. Shown in Fig. 6.31 is the resulting gray-scale image using the tuned σ dither array with Ulichney’s edge sharpening applied to the input image prior to halftoning (β = 2).

6.3

BIPPSMA

In order to construct the set {Ig : 0 ≤ g ≤ 1}, Yao and Parker [136] proposed BIPPSMA (binary pattern power spectrum manipulation algorithm) as a process identical to VAC, except that they toggle multiple pixels simultaneously and, thereby, converge at a faster rate. Also, in their version, filtering is done in the Fourier domain by means of the DFT using a low-pass, two-dimensional Butterworth filter (Fig. 6.32(a)) of the form 1 D2 (f ) =  (6.8)   |f | 6 1 + c1 f b or of a Gaussian (Fig. 6.32(b)) of the form 



|f |2 D2 (f ) = exp − . c2 fb2

(6.9)

The constants c1 and c2 are chosen based on visual evaluations at each gray-level to minimize low-frequency artifacts in Ig .

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

151

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.20: Spatial and spectral statistics using Ulichney’s void-and1 cluster algorithm with σ = 1.5 for intensity levels (left) I = 32 and 1 (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

152

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.21: Spatial and spectral statistics using Ulichney’s void-andcluster algorithm with σ = 1.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

153

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.22: Spatial and spectral statistics using Ulichney’s void-andcluster algorithm with σ = 1.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

154

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.23: Spatial and spectral statistics using Ulichney’s void-and1 cluster algorithm with σ = 2.5 for intensity levels (left) I = 32 and 1 (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

155

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.24: Spatial and spectral statistics using Ulichney’s void-andcluster algorithm with σ = 2.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

156

RAPSD

8

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.25: Spatial and spectral statistics using Ulichney’s void-andcluster algorithm with σ = 2.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

157

Figure 6.26: Two dither arrays constructed using Ulichney’s void-andcluster algorithm with (top) σ = 1.5 for all intensity levels and (bottom) σ tuned to the intensity, and the magnitude of its corresponding Fourier transform.

© 2008 by Taylor & Francis Group, LLC

158

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.27: Gray-scale ramp halftoned using a dither array constructed by Ulichney’s void-and-cluster algorithm with σ = 1.5 for all intensities.

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

159

Figure 6.28: Gray-scale image halftoned using a dither array constructed by Ulichney’s void-and-cluster algorithm with σ = 1.5 for all intensities.

© 2008 by Taylor & Francis Group, LLC

160

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.29: Gray-scale ramp halftoned using a dither array constructed by Ulichney’s void-and-cluster algorithm with σ tuned to the corresponding intensities.

© 2008 by Taylor & Francis Group, LLC

6.3. BIPPSMA

161

Figure 6.30: Gray-scale image halftoned using a dither array constructed by Ulichney’s void-and-cluster algorithm with σ tuned to the corresponding intensities.

© 2008 by Taylor & Francis Group, LLC

162

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.31: Gray-scale image halftoned using a dither array constructed by Ulichney’s void-and-cluster algorithm with σ tuned to the corresponding intensities and Ulichney’s edge-sharpening applied to the input image prior to halftoning (β = 2).

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

163

An advantage to Yao’s and Parker’s algorithm is that the characteristics of the human visual system [18, 120] can be accounted for to allow increased spectral energy along the diagonals by using the twodimensional filter, D2 (f ), with an angular weighting function (Fig. 6.32 (c)) such that D2 (f ) = [1 + 0.2 cos(4θf )]D2 (f ), (6.10) where θf is the angle of the point f from the horizontal axis in the frequency plane. Shown in Figs. 6.33–6.35 are the resulting spatial and spectral characteristics of binary dither patterns created using a 1 1 1 Gaussian filter for gray-levels g = 31 , 15 , 7 , 3 , 5 , and 12 (I = 32 , 16 , 8, 32 16 8 4 8 1 3 1 , , and ). 4 8 2

6.4

Dither Pattern Ordering

While the component dither patterns forming a blue-noise dither array may, in most cases, be constructed in any order, numerous studies have shown that, because of the stacking constraint, the ordering to which patterns are generated affects the visual quality of the resulting dither array and that constructing patterns in a random order may lead to better visual fidelity than by generating patterns in the order of increasing or decreasing gray-levels [67]. As such, the preferred orderings are the ones that maximize the spacing between consecutive gray-levels such that previously constructed patterns have the least amount of influence on the current dither pattern. In general, the best dither arrays are generated by inter-leaving gray-levels starting with the levels g = 0, 1, 14 , 34 , and 12 and then generating the four patterns corresponding to the mid-way points between these five patterns. Then, the next patterns to be generated are then ones falling mid-way between these nine levels and repeating this process of dividing in half the current ranges until all 256 patterns are generated. To see the effects of dither array construction order, Fig. 6.36 shows four 150 × 150 (256 gray-levels) dither arrays constructed by means of Yao and Parker’s BIPPSMA with array (a) constructed using 1 the sequential order {0, 1, Δ, 2Δ, . . . , 1 − 2Δ, 1 − Δ}, where Δ = 255 ; 1 1 1 1 1 array (b) constructed using the order {0, 1, 2 , 2 − Δ, 2 + Δ, 2 − 2Δ, 2 + 2Δ, . . . , Δ, 1 − Δ}; array (c) constructed using the order {0, 1, 14 , 34 , 14 + Δ, 34 − Δ, 14 + 2Δ, 34 − 2Δ, . . . , 12 − Δ, 12 + Δ, 12 , 14 − Δ, 34 + Δ, 14 − 2Δ, 34 + 2Δ, . . . , Δ, 1−Δ}; and array (d) constructed using the ordering {0, 1, 14 , 3 1 1 3 5 7 , , , , , , . . .}. Shown in Fig. 6.37 are the resulting magnitudes 4 2 8 8 8 8

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

164

1

0.5

0 1 1

0 VERTICAL FREQUENCY

0 ï1

ï1

HORIZONTAL FREQUENCY

(a) low-pass, two-dimensional Butterworth filter for BIPPSMA 1

0.5

0 1 1

0 VERTICAL FREQUENCY

0 ï1

ï1

HORIZONTAL FREQUENCY

(b) low-pass, two-dimensional Gaussian filter for BIPPSMA 1.2 1 0.8

0 1 1

0

0

VERTICAL FREQUENCY

ï1

ï1

HORIZONTAL FREQUENCY

(c) angular weighting function Figure 6.32: Spectral shaping filters used in Yao’s and Parker’s BIPPSMA [136].

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

2 RAPSD

RAPSD

2

165

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.33: Spatial and spectral statistics using Yao’s and Parker’s 1 [136] BIPPSMA with a Gaussian filter for intensity levels (left) I = 32 1 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

166

4 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.34: Spatial and spectral statistics using Yao’s and Parker’s [136] BIPPSMA with a Gaussian filter for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

RAPSD

10

RAPSD

8

fg 0.5 RADIAL FREQUENCY

0.7071

ANISOTROPY

0

1 0

10 0 ï10 0

0.5 RADIAL FREQUENCY

2

1

hg 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

PAIR CORRELATION

167

2

1

hg 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 6.35: Spatial and spectral statistics using Yao’s and Parker’s [136] BIPPSMA with a Gaussian filter for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

168

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

of the Fourier transforms of each dither array showing the blue-noise characteristics. Figures 6.38–6.41 show the resulting gray-scale ramps using each array, while Figs. 6.42–6.45 show the resulting gray-scale images. In array (d), the most recently constructed patterns are midway between two previously constructed patterns, and from the results of Figs. 6.38–6.45, this sequence seems to be the best of the four, especially near g = 0.4 and 0.6. Shown in Fig. 6.46 is the improved result of applying a blue-noise mask where Ulichney’s edge sharpening has been applied prior to halftoning (β = 2).

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

169

(a)

(b)

(c)

(d)

Figure 6.36: Four dither arrays constructed using Yao’s and Parker’s [136] BIPPSMA with Gaussian filters.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

170

(a)

(b)

(c)

(d)

Figure 6.37: The magnitudes of the Fourier transforms of the dither arrays of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

171

Figure 6.38: Gray-scale ramp halftoned using dither array (a) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

172

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.39: Gray-scale ramp halftoned using dither array (b) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

173

Figure 6.40: Gray-scale ramp halftoned using dither array (c) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

174

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.41: Gray-scale ramp halftoned using dither array (d) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

175

Figure 6.42: Gray-scale image halftoned using dither array (a) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

176

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.43: Gray-scale image halftoned using dither array (b) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

177

Figure 6.44: Gray-scale image halftoned using dither array (c) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

178

CHAPTER 6. BLUE-NOISE DITHER ARRAYS

Figure 6.45: Gray-scale image halftoned using dither array (d) of Fig. 6.36.

© 2008 by Taylor & Francis Group, LLC

6.4. DITHER PATTERN ORDERING

179

Figure 6.46: Gray-scale image halftoned using dither array (d) of Fig. 6.36 with Ulichney’s edge sharpening applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

Chapter 7 Direct Binary Search While the blue-noise model described in Chapter 5 attempts to characterize the spatial and spectral properties of visually pleasing patterns, it does not attempt to characterize visually optimal patterns. For that, various model-based halftoning algorithms have been introduced that incorporate a human visual system (HVS) model to decide where and where not to print dots. The most well recognized of these algorithms is the direct binary search (DBS) algorithm, first proposed by Analoui and Allebach [5] to obtain the absolute best arrangement of binary dots to represent a continuous-tone original.

7.1

Halftoning by DBS

The DBS algorithm processes each pixel of the binary image, one at at time, by either swapping the current pixel with one of its eight nearest neighbors or toggling the bit from 1 to 0 or 0 to 1 according to the modeled visual cost between the binary image and the continuoustone original. If neither a swap nor a toggle reduces the overall visual cost, then the pixel is left unchanged. The algorithm quits when, after processing the entire halftoned image, no swaps or toggles occur. Being a steepest descent type optimization, the DBS algorithm is susceptible to local minimum extrema, and the quality of the final halftone image is affected by the initial or seed dither pattern. But other modifications of the algorithm, like changing the scanning path, do not result in a considerable reduction in error. Specifically, the DBS algorithm takes the continuous-tone image f [n] and a first binary rendition g[n], generated by some traditional 181 © 2008 by Taylor & Francis Group, LLC

182

CHAPTER 7. DIRECT BINARY SEARCH

halftoning means. For an ideal display device, the halftone would be reproduced as a series of same-sized black and white squares accurately located on a squared sampling grid. But a real printing mechanism will introduce some level of distortion to the printed image, as will be described in Chapter 9. This distortion can be modeled by a continuous linear filter p(x). In consequence, the printed image can be represented as  g(x) = g[n]p(x − Xn). (7.1) n

And using a model of the human visual system (HVS) defined by the continuous space linear filter h(x), the image perceived by a human viewer can then be calculated by filtering g(x) to obtain the modeled output  g˜(x) = g(x) ∗ h(x) = g[n]˜ p(x − Xn), (7.2) n

where p˜(x) = p(x) ∗ h(x). The function g˜(x) is intended to be a close approximation of the image perceived by a human observer, and it is the quality of this image that determines if the halftone is an accurate representation of the initial, continuous-tone, digital image. The question now is to determine how the quality of g˜(x) can be measured. Since halftoning is inherently a quantization operation, the quality of the result can be measured by the quantization error introduced by binarization in the observed image space, g˜(x). To approximate this error, we suppose that the printing device is able to print continuous-tone where the output of the printer and the image observed by the viewer are calculated as f (x) =



f [n]p(x − Xn) and

(7.3)

f [n]˜ p(x − Xn),

(7.4)

n



f˜(x) =

n

with the resulting error image given by e˜(x) = f˜(x) − g˜(x).

(7.5)

The magnitude of the cumulative error is then given by

E=

x

|˜ e(x)|2 ≈

 m

where e˜[m] is a sampled version of e˜(x).

© 2008 by Taylor & Francis Group, LLC

|˜ e[m]|2 ,

(7.6)

7.2. EFFICIENT DBS ALGORITHM

183

Having defined an error metric, a method of iteratively improving the halftone is then achieved by visiting each pixel of the halftone image along a pre-defined raster path. At each pixel, the effect of swapping that pixel’s value with any of its eight nearest neighbors or of toggling its value is evaluated in terms of the overall error measure. Of the eight swaps and one toggle, the one that results in the largest reduction in overall error is kept. If none of the changes reduces the error, then no change is made and g[n] stays the same. This process is illustrated in Fig. 7.1, where the pixel m0 is at the center of a 3 × 3 running window. The pixel m0 can be swapped with each one of the pixels m1 , m2 , m4 , and m7 or toggled to white. The results of a trial swap with m1 and a toggle are shown in the figure. Once the swap/toggle operation is completed on m0 , the window is moved to the next pixel in the image (m5 in the example) and the process is repeated until the last pixel in the image is reached. If visiting each and every pixel within the image results in at least one change, then the entire process is repeated starting from the first pixel and following the same path until all pixels are processed again. If no swaps or toggles occur for a given pass of the DBS algorithm, then the algorithm has converged with the current binary image to give the final result. Shown in Figs. 7.2 and 7.3 are a gray-scale ramp and violin image using this technique.

7.2

Efficient DBS Algorithm

The DBS algorithm, as presented above, demands significant computational resources since (7.6) needs to be recalculated for every trial swap or toggle, for each pixel of the image, in all iterations. Since the number of accepted swaps and toggles diminishes with each iteration, a large amount of calculations performed results in no improvement of the image and, therefore, constitutes a waste of resources. So, in order to reduce the computational burden of the DBS algorithm, Analoui and Allebach [5] proposed a method to reduce the computational complexity of the algorithm based on the local effect of a swap or toggle. The gain in efficiency is based on the principle that when a pixel value is changed, it affects the values of the pixels in the observed image and halftone in an area equal to the size of the filter, p˜(x), centered around the changed pixel. To demonstrate the principal of a small region of influence, we can assume that the effect of a trial swap between pixels m0 and m1 , in

© 2008 by Taylor & Francis Group, LLC

CHAPTER 7. DIRECT BINARY SEARCH

184

m1 m2 m3 m4 m0 m5 m6 m7 m8

m1 m2 m3

m1 m2 m3

m4 m0 m5 m6 m7 m8

m4 m0 m5 m6 m7 m8

Trial swap of m0 and m1

Trial toggle of m0

Figure 7.1: The DBS algorithm. Fig. 7.4, needs to be evaluated. Noting that the filter p˜(x) is represented by an M × M matrix, then changing the value of the pixel m0 from a 1 to a 0 will only alter p˜(x) within an M × M window surrounding m0 , indicated in Fig. 7.4 by a light gray frame. The effect in the filtered image is that of subtracting the values of the filter coefficients to the highlighted section of the image such that g˜ [m] = g˜[m] − p˜[m − m0 ].

(7.7)

If the original value of the pixel were a 0 toggled to a 1, then the effect will be equivalent of adding the same values to the original. When a swap of pixels m0 and m1 is performed, the resulting observed image will suffer the effect of two toggles in those locations. So, to summarize the effect of all possible changes, the new observed image can be calculated as g˜ [m] = g˜[m] + a0 p˜[m − m1 ] + a1 p˜[m − m0 ], (7.8) where a0 and a1 are defined by 

a0 =

−1 if g[m0 ] = 1 1 if g[m0 ] = 0

© 2008 by Taylor & Francis Group, LLC



and a1 =

−a0 if swap 0 if toggle.

(7.9)

7.2. EFFICIENT DBS ALGORITHM

185

Figure 7.2: The DBS halftone of a gray-scale ramp using the N¨as¨anen exponential HVS model.

© 2008 by Taylor & Francis Group, LLC

186

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.3: The DBS halftone of a gray-scale image using the N¨as¨anen exponential HVS model.

© 2008 by Taylor & Francis Group, LLC

7.2. EFFICIENT DBS ALGORITHM

187

m1

M

m0

M

Figure 7.4: Effect of a swap in the observed halftone. Recalculating the error with the updated image results in E =

 m

||f˜[m] − g˜ [m]||2 =

 m

f˜[m] − g˜ [m]

2

,

(7.10)

since f˜ and g˜ are real valued. Replacing (7.8) in (7.10) and applying (7.5) we obtain E =

 m

=

 m

f˜[m] − g˜[m] − a0 p˜[m − m1 ] − a1 p˜[m − m0 ]

(˜ e[m] − a0 p˜[m − m1 ] − a1 p˜[m − m0 ])2 .

2

(7.11)

Expanding and rearranging the terms results in E



=



2

e˜ [m] +

m

+a1

 m





a20

+

a21

 m



2

p˜ [m] + 2 a0

e˜[m]˜ p[m − m1 ] − a0 a1 

 m

 m

e˜[m]˜ p[m − m0 ]

p˜[m − m0 ]˜ p[m − m1 ]

= E + a20 + a21 cp˜p˜[0] +2 (a0 cp˜e˜[m0 ] + a1 cp˜e˜[m1 ] − cp˜p˜[m1 − m0 ]) ,

© 2008 by Taylor & Francis Group, LLC



(7.12)

CHAPTER 7. DIRECT BINARY SEARCH

188 

where cf g = n f [n]g[n + m] is the correlation between f and g. The parameters cp˜p˜ and cp˜e˜ are the autocorrelation function of the HVS model and the cross-correlation of the HVS with the observed error image. According to (7.12), the change in the error due to a change in the halftone can be expressed as ΔE = E − E  

=

(7.13)



1 + a21 cp˜p˜[0] − 2 (a0 cp˜e˜[m0 ] + a1 cp˜e˜[m1 ] − cp˜p˜[m1 − m0 ]) ,

where if ΔE is positive, the change in the halftone increased the error measure and should be discarded. If ΔE is negative, the change should be compared with the results obtained with the other trial changes for m0 . The one resulting in the largest reduction of the error (largest absolute value among the negative ΔEs) is kept. If no trial changes result in a negative ΔE, then the halftone remains unchanged and the process moves on to the next pixel. Using (7.13), the change in the error produced by a trial swap or toggle can be calculated with just a few additions as long as the correlation matrices cp˜p˜ and cp˜e˜ have been precalculated and stored in memory. The term cp˜p˜ is constant during all the process. On the other hand, cp˜e˜ must be updated every time the halftone is updated. Again, the fact that a change in a pixel affects just a section of the image is used to avoid the calculation of the whole correlation matrix, updating just the affected terms instead. As such, the result of a change in the halftone on cp˜e˜ can be easily calculated as cp˜e˜[m] = =

 n

 n

p˜[n]˜ e [n + m] =

 n

p˜[n]˜ e[n + m] − a0





p˜[n] f˜[n + m] − g˜ [n + m]

 n

p˜[n]˜ p[n − m0 ] − a1

 n

p˜[n]˜ p[n − m1 ]

= cp˜e˜[m] − a0 cp˜p˜[m − m0 ] − a1 cp˜p˜[m − m1 ].

(7.14)

So the whole process can be summarized as follows 1. Generate a starting halftone, calculate, and store cp˜p˜, and initialize cp˜e˜. 2. For each pixel in the halftone evaluate the effect of all possible trial changes using (7.13). 3. Choose the lowest ΔE. 4. If that value is negative, update the halftone and cp˜e˜.

© 2008 by Taylor & Francis Group, LLC

7.2. EFFICIENT DBS ALGORITHM

189

5. When this procedure is finished for the last pixel in the image, count the number of accepted trial changes. 6. Stop when the number of accepted trial changes in a whole iteration is zero. Examples of results obtained with the DBS algorithm are shown in Figs. 7.5–7.9, where the resulting halftones for a gray-scale ramp and a gray-scale image are shown for different seed patterns. Specifically, Figs. 7.5 and 7.6 are the results of the DBS algorithm when FloydSteinberg error-diffusion is applied to generate the starting pattern. Here, the DBS algorithm is able to eliminate most of the worms and geometric artifacts characteristic of Floyd-Steinberg halftones with the error metric reduced approximately 500 times in 22 to 23 iterations. Shown in Figs. 7.7 and 7.8 are the DBS results obtained when the starting points are generated using Ulichney’s perturbed filter weight scheme employing a serpentine raster and 50% perturbation of the Floyd and Steinberg error filter. In this case, the noisy textures appreciated in the originals in Figs. 5.39 and 5.40 are smoothed and the edges are sharper. The error measure for the final results are again 500 times better than the original after 20 iterations. Finally, Figs. 7.9 and 7.10 show the output of the DBS algorithm when the initial halftone is generated with a blue-noise mask as described in Chapter 6. In this instance, the DBS algorithm has a similar effect as using Ulichney’s perturbed error-diffusion scheme, sharpening the edges and smoothing noisy areas, but since the initial halftones are better, the DBS mask results in an error measure that is reduced by only a factor of 10 in 15 iterations. For a more quantitative analysis of the halftones produced by the DBS algorithm, Figs. 7.11–7.13 show the spatial and spectral statistics used for characterizing blue-noise from Chapter 5. In general, the patterns shown do not present worms, clustering, or geometric artifacts (except for a few small checkerboard patches for I = 12 ). The spectral and spatial characteristics of the patches represented in the plots of the RAPSD, anisotropy, pair correlation, and directional distribution show that the output of the DBS algorithm resemble the ideal characteristics of blue-noise in the range from 0 to 25% intensity levels, but beyond 25%, the principal frequency of the patterns stalls as the selected HVS filter fails to push the spectral energy of the pattern into the corners of the baseband where the blue-noise model would otherwise define √ the principal frequency up to 1/ 2. In fact, the DBS algorithm even

© 2008 by Taylor & Francis Group, LLC

190

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.5: The DBS halftone of a gray-scale ramp seeded by Floyd and Steinberg error-diffusion.

© 2008 by Taylor & Francis Group, LLC

7.2. EFFICIENT DBS ALGORITHM

191

Figure 7.6: The DBS halftone of a gray-scale image seeded by Floyd and Steinberg error-diffusion.

© 2008 by Taylor & Francis Group, LLC

192

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.7: The DBS halftone of a gray-scale ramp seeded by Ulichney’s perturbed weight error-diffusion halftone.

© 2008 by Taylor & Francis Group, LLC

7.2. EFFICIENT DBS ALGORITHM

193

Figure 7.8: The DBS halftone of a gray-scale image seeded by Ulichney’s perturbed weight error-diffusion halftone.

© 2008 by Taylor & Francis Group, LLC

194

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.9: The DBS halftone of a gray-scale ramp seeded by a DBS blue-noise dither array.

© 2008 by Taylor & Francis Group, LLC

7.2. EFFICIENT DBS ALGORITHM

195

Figure 7.10: The DBS halftone of a gray-scale image seeded by a DBS blue-noise dither array.

© 2008 by Taylor & Francis Group, LLC

196

CHAPTER 7. DIRECT BINARY SEARCH

introduces a small degree of clustering, in this range responsible for the stalling of the principal frequency. This behavior is much more consistent with a modified model of blue-noise, which is presented in Chapter 8. Finally, an interesting example that shows the power of the DBS algorithm is the following. Recall that the DBS algorithm starts with a given halftone g[n] and improves it during the iteration process. The algorithm could actually start with an all-white image or a halftone of a different image; yet, the DBS algorithm will still produce a halftone of the target image. Figure 7.14 shows the process followed by the DBS algorithm when the starting image is an all-white image. Results obtained after 1, 3, 6, and the final iteration (12) are shown. Notice how the algorithm starts by delimiting the elements in the figure and then proceeds to accentuate the details progressively.

7.3

Effect of HVS model

Like the initial seed pattern, the selection of the HVS filter plays a significant role in shaping the spatial and spectral properties of the resulting halftones produced by the DBS algorithm. Using the models described in Chapter 4, Figs. 7.15–7.18 show the resulting halftone images when using Kim and Allebach’s mixed Gaussian HVS models with model parameters (κ1 , κ2 , σ1 , σ2 ) = (43.2, 38.7, 0.02, 0.06) and (19.1, 42.7.0.03, 0.06). Shown in Figs. 7.19–7.22 are the corresponding halftones using a sub-Gaussian, alpha-stable model with parameter values (α, γ  , D, R) = (0.95, 27, 9.5 in, 300 dpi). For reference, the halftones in Figs. 7.2 and 7.3 were produced using the N¨as¨anen exponential model. In conclusion, the characteristics of the HVS model in a digital halftoning algorithm play an important role in its performance. In the particular case of the DBS algorithm, the frequency response of the filter determines the quality of the output in terms of homogeneity, coarseness, smoothness, and the appearance of artifacts. The characteristic function and the probability density function of bi-variate, α-SG random variables provide powerful tools for modeling the PSF and the CSF of the HVS, respectively. These functions can be easily tuned to yield an HVS model that, when used in the DBS algorithm, produces pleasant results for all levels of gray.

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

197

4 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

10 RADIAL DISTANCE

20

fb 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 7.11: Spatial and spectral statistics using the DBS algorithm with the N¨as¨anen exponential HVS model for intensity levels (left) I = 1 1 and (right) I = 16 . 32

© 2008 by Taylor & Francis Group, LLC

CHAPTER 7. DIRECT BINARY SEARCH

198

2 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 7.12: Spatial and spectral statistics using the DBS algorithm with the N¨as¨anen exponential HVS model for intensity levels (left) I = 1 and (right) I = 14 . 8

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

199

2 RAPSD

RAPSD

2

1

1

fb 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fb 0

0.5 RADIAL FREQUENCY

2

1

hb 0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

hb 0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 7.13: Spatial and spectral statistics using the DBS algorithm with the N¨as¨anen exponential HVS model for intensity levels (left) I = 3 and (right) I = 12 . 8

© 2008 by Taylor & Francis Group, LLC

200

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.14: The resulting halftones produced by the DBS algorithm starting on a white canvas. Results after the first iteration (top left), the third (top right), the sixth (bottom left), and the 12th (bottom right).

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

201

Figure 7.15: The DBS halftone of a gray-scale ramp using the mixed Gaussian 1 HVS model with parameters (κ1 , κ2 , σ1 , σ2 ) = (43.2, 38.7, 0.02, 0.06).

© 2008 by Taylor & Francis Group, LLC

202

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.16: The DBS halftone of a gray-scale image using the mixed Gaussian 1 HVS model with parameters (κ1 , κ2 , σ1 , σ2 ) = (43.2, 38.7, 0.02, 0.06).

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

203

Figure 7.17: The DBS halftone of a gray-scale ramp using the mixed Gaussian 2 HVS model with parameters (κ1 , κ2 , σ1 , σ2 ) = (19.1, 42.7.0.03, 0.06).

© 2008 by Taylor & Francis Group, LLC

204

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.18: The DBS halftone of a gray-scale image using the mixed Gaussian 2 HVS model with parameters (κ1 , κ2 , σ1 , σ2 ) = (19.1, 42.7.0.03, 0.06).

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

205

Figure 7.19: The DBS halftone of a gray-scale ramp using the subGaussian 1 HVS model with model parameters (α, γ  , D, R) = (0.95, 27, 9.5 in, 300 dpi).

© 2008 by Taylor & Francis Group, LLC

206

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.20: The DBS halftone of a gray-scale image using the subGaussian 1 HVS model with model parameters (α, γ  , D, R) = (0.95, 27, 9.5 in, 300 dpi).

© 2008 by Taylor & Francis Group, LLC

7.3. EFFECT OF HVS MODEL

207

Figure 7.21: The DBS halftone of a gray-scale ramp using the subGaussian 1 HVS model with model parameters (α, γ  , D, R) = (0.95, 27, 9.5 in, 300 dpi).

© 2008 by Taylor & Francis Group, LLC

208

CHAPTER 7. DIRECT BINARY SEARCH

Figure 7.22: The DBS halftone of a gray-scale image using the subGaussian 1 HVS model with model parameters (α, γ  , D, R) = (0.95, 27, 9.5 in, 300 dpi).

© 2008 by Taylor & Francis Group, LLC

Chapter 8 Hexagonal Grid Halftoning This chapter looks at the application of blue-noise halftoning to hexagonal sampling grids; where hexagonal grids differ from rectangular in that every other row is offset one-half a pixel period. In particular, hexagonal sampling grids are well recognized for allowing a more natural, radially symmetric sampling of two-dimensional space – preserving a circular band-limited signal with only 86% of the total number of samples used by rectangular grids. Hexagonal sampling grids also increase the robustness to changes in aspect ratio. The aspect ratio is the horizontal period divided by the vertical period, and Ulichney [122] defines the covering efficiency as the ratio of pixel area divided by the circumscribing circle area. The best case for rectangular grids occurs for square grids, but what is particularly interesting, in this figure, is the wide range of aspect ratios where hexagonal grids outperform this best case. This is important because it allows for resolution to be increased asymmetrically while still enjoying superior radial symmetry of pixel coverage. It is very often easier to increase resolution in only one dimension, and using hexagonal grids would allow us to take advantage of this. Given the super-high dot addressability of modern digital printers, the implementation of hexagonal grid halftoning is easily implemented, and given the general advantages to using hexagonal sampling grids, one may wonder why hexagonal grids have not received more attention by the research community with respect to stochastic halftoning. To date, only a handful of published articles exist on the topic of hexagonal sampling grid halftoning. The first such paper was by Stevenson and Arce [114], which lead to the Stevenson-Arce error filter. Ulichney [122] later extended his perturbed filter weight scheme in or209 © 2008 by Taylor & Francis Group, LLC

210

CHAPTER 8. HEXAGONAL GRID HALFTONING

der to improve the visual pleasantness of the resulting hexagonal grid dither patterns. More recently, Jodoin and Ostromoukhov [51] followed the lead of Eschbach [32] and others by using a tone-dependent error filter. The idea was to have a separate error filter for each unique gray-level. Instead of keeping a database of 256 filters, filters were only stored for a handful of unique levels distributed between 0 and 255. Interpolation was then used to derive the remaining filters from those stored in memory. The recorded error filters were derived by iteratively adjusting each weight to minimize low-frequency artifacts in the resulting dither patterns. Turbek et al. [121] looked at the problem of printing clustereddot ordered dither patterns on electro-photographic printers, where it was thought that, beyond the advantages touted by Ulichney [123], the individual clusters would suffer less distortion near gray-level 12 caused by the close proximity of neighboring clusters, being surrounded by only six neighbors as opposed to eight for rectangular grids. The investigators used spectrogram analysis to compare the uniformity in the distribution of toner for constant gray-level patterns. In a later paper, Cholewo [19] refined this spectrogram analysis as well as developed anistropic, stochastic dither arrays using the technique of minimum density variance [21]. While these dither arrays suffer from directional articifacts near gray-level 12 , they show a remarkable improvement over the artifacts found in the patterns produced by Stevenson and Arce [114] and by Ulichney [123]. Given how comparatively few papers exist on hexagonal grid halftoning, Lau and Ulichney [73] hypothesize that much of the detraction of hexagonal grids derives from the analysis of blue-noise dithering performed by Ulichney [123], who showed that only on a rectangular sampling grid is it possible to isolate minority pixels at all gray-levels. In contrast, minority pixels must begin to cluster as the gray-level approaches 1/2 on a hexagonal grid, which lead Ulichney to write that hexagonal sampling grids do not support blue-noise. Specifically being forced to cluster pixels, blue-noise isolates dots at some locations only to cluster at others, creating a wider range of frequencies in the spectral content of the dither pattern. This widening of the spectral content is referred to as “whitening,” and as a pattern becomes more and more white, it appears more and more noisy. But, as shown by the later works cited above, these early observations have since been disproven. Given the descrepancies between Ulichney’s original blue-noise

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

211

analysis and the work of Cholewo [19] in producing visually pleasing blue-noise on hexagonal grids, Lau and Ulichney introduced a revised blue-noise model that emphasizes radial symmetry even in cases where radial symmetry requires the clustering of minority pixels. In contrast, the traditional blue-noise model, described in Chap. 5, is constrained by the sampling geometry near gray-level equal to 12 , where rectangular grids force patterns into a periodic checkerboard pattern that may, in some cases, create visually disturbing artifacts. By allowing a minimum degree of clustering, the new blue-noise model makes an optimal tradeoff to acquire a sufficient degree of flexibility in where the algorithm chooses to place dots. Hexagonal grid, blue-noise halftoning should, therefore, be able to achieve higher principal frequencies for the same gray-levels as rectangular grids and, therefore, should achieve lower visual costs.

8.1

Spectral Aliasing

With regards to fb and its significance to hexagonal grids, the issue at hand is aliasing and the unwanted visual artifacts that aliasing creates. An explanation begins with Fig. 3.9 (left), where a small area near DC of the spectral plane of a rectangular sampled image is shown. As described in Chapter 3, the spectral plane is in units of inverse pixel period and is divided into spectral annuli of radial width Δ. Taking the average power within each annulus and then plotting the average power versus the center radius fρ creates Ulichney’s RAPSD measure. Note that the maximum spectral radius within each square tile is √12 . Shown in Fig. 8.1 is a diagram of the spectral domain for four blue-noise dither patterns with the black segments marking the principle wavelengths. As originally proposed, Ulichney envisioned the principal frequency as a circular wavefront eminating from the spectral DC origin and progressing outward as g approached 12 . At gray-level g = 14 , when the wavefront first makes contact with sides of the baseband entering the partial annuli region of Fig. 3.10 (left), the wave becomes √ segmented into the four corners while still progressing to fb = 1/ 2, the maximum radial frequency within the baseband of a rectangular sampling grid. In the spatial domain, this packing of energy into the corners of the baseband, as depicted in Fig. 8.1, is achieved by adding correlation between minority pixels along the diagonal, creating a pattern where neighboring minority pixels are more likely to occur along the diagonal

© 2008 by Taylor & Francis Group, LLC

212

CHAPTER 8. HEXAGONAL GRID HALFTONING

Figure 8.1: The spectral rings of blue-noise dither patterns with added diagonal correlation between minority pixels for gray-levels (top left) g = 10%, (top right) g = 26%, (bottom left) g = 42%, and (bottom right) g = 50%.

instead of side-by-side or above-and-below one another. If a particular halftoning scheme is especially successful at adding this diagonal correlation, then it is possible to create dither patterns at all gray-levels such that no two minority pixels occur adjacent to one another. Such a scheme would produce the familiar g = 12 checkerboard pattern. Floyd and Steinberg’s [40] error-diffusion is a classic example of a blue-noise generating halftoning algorithm that adds such correlation, where the error filter weights were specifically chosen because of their behavior near gray-level g = 12 . Shown in Figs. 8.2 and 8.3 are the spatial dither patterns and their corresponding power spectra as g progresses from 0 to 12 . Now, while Ulichney originally believed packing energy into the corners of the power spectrum to be the ideal behavior for blue-noise, ultimately adding diagonal correlation, especially to the degree of Floyd and Steinberg’s error-diffusion, violates the two basic characteristics of blue-noise to be radially symmetric while also aperiodic. Maintaining an average distance between minority pixels of λb near g = 12 forces the minority pixels to lock into a fixed and periodic pattern whose only saving grace, in terms of visual pleasantness,

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

213

Figure 8.2: The blue-noise dither patterns created by error-diffusion using the Floyd-Steinberg error filter for gray-levels (top left) g = 10%, (top right) g = 26%, (bottom left) g = 42%, and (bottom right) g = 50%.

is that its high spatial frequency makes it less visible than similar patterns at lesser gray-levels. As will be discussed in Sec. 8.3, maintaining a cut-off frequency of fb , by adding directional correlation, also creates a significant dilemma for hexagonal grids where the maximum spatial frequency occurs at g = 13 . Looking back at Ulichney’s original definition of blue-noise dither patterns as being radially symmetric while also having some variation in the distance between minority pixels, we wonder if there are alternative behaviors for blue-noise in this gray-level range 14 < g ≤ 34 , where the baseband constrains the placement of dots, such that dither patterns can maintain their grid-defiance illusion and not adopt a periodic or textured appearance. As a first attempt at such a blue-noise model, we can try to enforce both the principal frequency fb , defined according to (5.2), as well as radial symmetry – creating the spectral behavior depicted in Fig. 8.4, where, as g exceeds 14 , the principal frequency ring extends beyond the sides of the baseband. Spectral energy from neighboring rings will then extend into the baseband and, hence, introduce alias artifacts into the dither pattern. To see the effects of this aliasing,

© 2008 by Taylor & Francis Group, LLC

214

CHAPTER 8. HEXAGONAL GRID HALFTONING

10%

26%

42%

50%

Figure 8.3: The power spectra for the dither patterns of Fig. 8.2 for gray-levels (top left) g = 10%, (top right) g = 26%, (bottom left) g = 42%, and (bottom right) g = 50%. The superimposed black circles mark the location of the principal frequency (lines have been ommitted for g = 50%).

we can down-sample blue-noise dither patterns by a factor of two to double the radius of the spectral ring. The intensity or gray-level of the spatial dither patterns should not be affected by the down-sampling operation, and what we see in Figs. 8.5 and 8.6 is that at gray-levels beyond 6.25% ink coverage, where the corresponding principal frequencies overlap neighboring rings (after down-sampling), the resulting dither patterns will exhibit light (g = 10%) to moderate (g = 26%) and then severe (g = 50%, not shown) clustering of minority pixels, causing the pattern to take on an unpleasant appearance. For g < 6.25%, aliasing will occur due to the high-frequency spectral content (fρ > fb ) characteristic of blue-noise, resulting in a noisy appearance of its own. But this particular aliasing leads only to the high variation in the distance between minority pixels that is not so high as to cause minority pixels to touch. Clustering of minority pixels seems to occur only when the spectral rings intersect. Under the premise that aliasing of the principal frequency leads to unwanted clustering of minority pixels in the spatial domain, we can

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

215

Figure 8.4: The spectral rings of blue-noise dither patterns for graylevels (top left) g < 25%, (top right) g = 25%, (bottom left) g = 40%, and (bottom right) g = 50%.

now look at specific error-diffusion techniques known to exhibit clustering at gray-levels between g = 14 and 34 to see if the clustering that these algorithms introduce are, in fact, the product of aliasing. Some of this behavior is evident in Fig. 8.8 at gray-level g = 42% using Floyd and Steinberg’s error filter, where there is a clear correlation between the distribution of energy in the power spectrum and the spectral ring at fb . Looking at the spatial dither patterns and power spectra produced by error-diffusion using Jarvis, Judice, and Ninke’s [50] 12-weight filter in Figs. 8.7 and 8.8, clustering/aliasing behavior is visible as evidenced by the strong spectral components for g = 12 shown in close alignment with the spetral rings at radial frequency fb from neighboring replications of the baseband frequency. If we now look at the clustering found in Ulichney’s perturbed filter weight scheme [122] in Figs. 8.9 and 8.10, where the spatial dither patterns and corresponding spectra for g transitioning from 6% to 50% coverage are shown, we see some differences with both Floyd and Steinberg’s error filter and with Jarvis et al.’s. From visual inspection, one can see that, by perturbing filter weights, the resulting dither patterns better maintain radial symmetry by moving some of the spectral energy

© 2008 by Taylor & Francis Group, LLC

216

CHAPTER 8. HEXAGONAL GRID HALFTONING BEFORE

BEFORE

6% BEFORE

AFTER

18% AFTER

10% BEFORE

AFTER

26% AFTER

Figure 8.5: The spatial blue-noise dither patterns, before (top left) and after (bottom right) down-sampling by 2, representing gray-levels g = 6%, 10%, 18%, and 26%.

6%

10%

18%

26%

Figure 8.6: The power spectra corresponding to the blue-noise dither patterns of Fig. 8.5 before (top left) and after (bottom right) downsampling with the principal frequency marked in black.

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

217

Figure 8.7: The spatial dither patterns created by error-diffusion using the Jarvis et al. error filter for g = 10%, 26%, 42%, and 50% ink coverage.

10%

26%

42%

50%

Figure 8.8: The power spectra for dither patterns of Fig. 8.7 with the principal frequency marked in black.

© 2008 by Taylor & Francis Group, LLC

218

CHAPTER 8. HEXAGONAL GRID HALFTONING

inside the principal frequency ring through a small, controlled degree of clustering. That is, by allowing a small degree of clustering, Ulichney’s perturbed filter scheme is able to reduce the principal frequency of the pattern, breaking up some of the periodic textures that would otherwise form due to the added diagonal correlation. But, given that the observed clustering is only slight, we would describe the perturbed filter weight scheme as generally behaving in the manner first prescribed by Ulichney adding diagonal correlation and packing spectral energy into the corners of the baseband as g approaches 12 . If there is a disturbing artifact to be found in the patterns of Fig. 8.9, it is the discontuities in texture created by clusters within an otherwise periodic texture, leaving open the question of whether it is the clustering or the periodic textures that are most to blame for the noisy appearance. Given the disturbing artifacts created by discontinuities in textures, we can look at alternatives to error-diffusion noting that while, in theory, error-diffusion should diffuse errors in a homogeneous fashion to minimize low-frequency graininess at all gray-levels, Figs. 8.9 and 8.10 show that not all filters are created equal. Furthermore, we note that it was a trial-and-error technique used by Ulichney to discover a perturbed error filter scheme that maintained radial symmetry without aliasing artifacts near g = 12 . So, for a scheme that generates halftones in a more intuitive fashion, we can use Ulichney’s iterative void-andcluster initial pattern technique (VAC), described in Chapter 6. Because VAC iteratively swaps pixels according to an analysis of the entire local neighborhood around a subject pixel, and not just from half of the local neigborhood as in error-diffusion, VAC can more readily guarantee spatial homogeniety. And using appropriate low-pass filters, we expect VAC to maintain radial symmetry while minimizing low-frequency graininess for any gray-level. In this regard, using a Gaussian low-pass filter with variance σ 2 = 0.32, Figs. 8.11 and 8.12 show the spatial dither patterns and corresponding spectra as g transitions from 6% to 50% coverage where, from visual inspection, it is confirmed that VAC behaves very similar to the perturbed filter weight scheme of Figs. 8.9 and 8.10 in that it allows some spectral energy to exist inside the principal frequency ring for gray-levels beyond g = 14 . What VAC does, beyond perturbed filter weights, is achieve much better radial symmetry given the lack of a deterministic raster path.

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

219

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.9: The binary dither patterns for Ulichney’s perturbed filter weight scheme on a rectangular sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

220

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.10: The power spectra for Ulichney’s perturbed filter weight scheme on a rectangular sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.1. SPECTRAL ALIASING

221

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.11: The binary dither patterns for void-and-cluster patterns on a rectangular sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

222

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.12: The power spectra for void-and-cluster patterns on a rectangular sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.2. MODIFIED BLUE-NOISE MODEL

8.2

223

Modified Blue-Noise Model

Even though the amount of clustering is only slight, the resulting patterns from VAC and from Ulichney’s perturbed filter scheme offer some evidence that, perhaps, clustering of minority pixels will have desirable properties for halftoning if not done at too much of an extreme. In particular, these algorithms move spectral energy inside the radial frequency fb , creating what Lau et al. [66, 67] referred to as green-noise, to be described in Chapter 10, where the optimal halftoning schemes distribute minority pixel clusters as homogeneously as possible. Doing so to a limited extent creates a blue-noise like power spectrum with principal frequency defined according to the traditional blue-noise equation but for gray-level g˜ = g/(average cluster size). While perturbed filter weights and VAC introduced only a small degree of clustering, the green-noise model tells us that it is possible to eliminate diagonal correlation without introducing unwanted aliasing artifacts and, hence, maintain radial symmetry at all gray-levels. Specifically, aliasing can be eliminated if the amount of clustering at gray-levels 14 < g ≤ 34 is sufficiently high as to reduce the principal frequency to that for gray-level g = 14 , where the principal frequency ring is the largest complete ring that can fit inside the baseband. So in an attempt to see the effects of adding this clustering, we can repeat the experiment of Figs. 8.11 and 8.12 using VAC but where the variance of the low-pass Gaussian filter is defined as σ 2 = 0.6. In this manner, we expect to see a cut-off frequency that increases with g, as prescribed by (5.2) for 0 ≤ g ≤ 14 , but that levels off to a constant for 14 < g ≤ 34 . This is because below gray-level 14 , the energy in the Gaussian filter is strong enough that it forces the dither pattern energy to higher radial frequencies constrained by the gray-level to, at best, achieve the principal frequency as the lowest radial frequency with non-zero spectral energy. Beyond gray-level 14 , the energy in the Gaussian filter decreases to the point where the pattern feels less force moving energy to higher radial frequencies and, hence, does not achieve a cut-off frequency as high as fb . Shown in Figs. 8.13 and 8.14 are the corresponding spatial and spectral dither patterns that show this behaviour with power spectra almost identical for 14 < g ≤ 12 . Visual inspection will show that while the patterns are coarser than before, the lack of periodic texture components near g = 12 eliminates the disturbing artifacts created by discontinuities in the textures found in previous figures.

© 2008 by Taylor & Francis Group, LLC

224

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.13: The binary dither patterns for void-and-cluster patterns with σ 2 optimized to maintain radial symmetry on a rectangular sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.2. MODIFIED BLUE-NOISE MODEL

225

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.14: The power spectra for void-and-cluster patterns with σ 2 optimized to maintain radial symmetry on a rectangular sampling grid as g transitions from 6% to 50% coverage. The black lines indicate the principal frequency’s progress according to (5.2) up to 22% and then remains constant up to 50%.

© 2008 by Taylor & Francis Group, LLC

226

CHAPTER 8. HEXAGONAL GRID HALFTONING

In light of the results demonstrated in Figs. 8.13 and 8.14, Lau and Ulichney [73] propose a new model for blue-noise that places an increased emphasis on the need for maintaining radial symmetry and avoiding periodic textures by modifying the notion of the blue-noise principal frequency from being a wavefront progressing into the corners of the baseband to, instead, a wave progressing outward until gray-level g = 14 . Beyond g = 14 , the wavefront stops its progression as a complete, unbroken ring. This new model characterizes the ideal blue-noise dither patterns as having a principal frequency defined as ⎧ ⎪ ⎪ ⎨

√

g, for 0 < g ≤

1 4 3 4

1/2, for 14 < g ≤ fb = ⎪ ⎪ ⎩ √ 1 − g, for 34 < g ≤ 1.

(8.1)

Given the above property, we note that the patterns of Fig. 8.13 succeed at modeling ideal blue-noise by lacking the patchiness/clumping appearance of Fig. 8.11, especially for gray-levels above 34%. This is a significant comparison because the old theory of blue-noise would predict that the higher cut-off frequencies of Fig. 8.11 should be more visually pleasing.

8.3

Hexagonal Sampling Grids

A regular hexagonal sampling lattice is characterized by samples placed a horizontal distance apart √ equal to some sample period Dx and a vertical distance apart of 23 Dx . By shifting every second row of the lattice by half a pixel ( D2x ), a sample point is separated from its six neighboring samples by an equal distance Dx . In order to sample an image using hexagonal grids with the same number of samples per unit 1 area as a rectangular grid, the spacing Dx should be equal to ( √23 ) 2 Dr , where Dr is the sample period for the rectangular grid. Now, as described by Ulichney [122], a well-formed blue-noise dither pattern will be such that minority pixels will be separated by an average distance λb as defined in (5.1). For reasons relating to the derivation of the Fourier transform of a regular hexagonal sampling lattice, the principal frequency, fb , of a blue-noise pattern will be defined according to  √ √2 g, for 0 < g ≤ 1/2 3 √ fb = √2 (8.2) 1 − g, for 1/2 < g ≤ 1. 3

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

227

Noting from Fig. 3.9 (right) that the maximum radial frequency that fits within the corners of the baseband occurs at fb = 23 when the graylevel reaches g = 13 , it is not possible to have a dither pattern with any frequency specified by (8.2) on a hexagonal grid in the range of 1 < g ≤ 23 . 3 Based upon trial-and-error experiments with error-diffusion, Ulichney theorized that because stochastic dither patterns would always have significant energy below fb in this region for gray-levels between 13 and 2 , patterns would always have excessively large variations in the spac3 ing between dots, and under no circumstances would patterns not look noisy and uncorrelated. Ulichney found that this theoretical shortcoming agreed with the failure to produce homogeneous (blue-noise) patterns in trial-and-error experiments with hexagonal error-diffusion. We now see that this theory was wrong! Just as in the case for rectangular grids in the last section, the definition of principal frequency should not extend beyond the point where the associated spectral ring first makes contact with the walls of the baseband. In response, we are defining fb according to (8.1) as the optimal trade-off between pattern coarseness and sample grid dot placement where we allow the pattern to exhibit a minimum degree of clustering in order to maintain radial symmetry at all gray-levels. A particularly elegant property of this new blue-noise model is that for both the rectangular ((8.1)) and hexagonal ((8.2)) cases, the largest complete annuli shown occurs for g = 3D 14 and g = 3D 34 . So just as for the rectangular case, we limit the principal frequency to stall at this largest complete annulus and define the hexagonal principal frequency as ⎧ √ √2 ⎪ g, for 0 < g ≤ 14 ⎪ 3 ⎪ ⎨

fb = ⎪ ⎪ ⎪ ⎩

√2 (1/2), for 1 < g ≤ 3 4 4 3 √ 3 2 √ 1 − g, for 4 < g ≤ 1. 3

(8.3)

Because of this √23 factor, blue-noise dither patterns on a hexagonal grid have a 15.5% higher cut-off frequency than those corresponding to rectangular grids with the same number of samples per unit area at all gray-levels. As such, blue-noise dither patterns on hexagonal grids may, theoretically, be less visible than those on rectangular grids when enforcing the radial symmetry condition. Finally, in situations where clustering does occur, either purposely for minimizing the effects of printer distortions or forcibly near gray-level g = 12 , hexagonal

© 2008 by Taylor & Francis Group, LLC

228

CHAPTER 8. HEXAGONAL GRID HALFTONING

sampling grids can form pixel pairs in three directions as opposed to two, allowing for improved radial symmetry in the size and distribution of minority pixel clusters. Coarse halftone patterns should, therefore, form smoother visual textures on hexagonal grids than rectangular. So, assuming that hexagonal is the preferred sampling geometry, we are in the familiar position of trying to find a means by which to generate optimal dot distributions and to do so in a computationally efficient manner. Looking at the first published study of error-diffusion on hexagonal sampling grids, Figs. 8.15 and 8.16 show the spatial dither patterns and power spectral densities corresponding to the Stevenson and Arce [114] error filter. From visual inspection, one sees a consistent blue-noise appearance for gray-levels below g = 13 , but strong vertical artifacts, deriving from the raster scan, seem to dominate near g = 12 . Looking specifically at the power spectra, one sees the spectral lines running vertically that intersect the limits of the baseband at the same points where the spectral rings make first contact at g = 14 . Furthermore, these lines of energy do not seem to become prominant components until after gray-level g = 14 , adding credence to the claim that it is at these gray-levels where the sampling lattice begins to constrain the distribution of dots. Noting the relatively poor performance of the Stevenson and Arce filter, Ulichney proposed using the same perturbed filter weight scheme demonstrated in Figs. 8.9 and 8.10. Shown in Figs. 8.17 and 8.18 are the binary dither patterns and corresponding power spectra for this technique on hexagonal sampling grids. Like the Stevenson and Arce filter, the perturbed filter scheme produces visually pleasing patterns below gray-level g = 14 but is forced to cluster pixels as g approaches 12 . While it clearly does a better job in this range, the deterministic raster leads to strong vertical artifacts very similar to those produced by the Stevenson and Arce filter. Given the poor performance of error-diffusion, we can attempt to create dither patterns using VAC where we expect to achieve the preferred behavoir of only introducing clustering, when necessary, as to avoid aliasing. Like the discrete Fourier transform, convolution of a binary dither pattern with a linear FIR filter can be achieved using techniques for matrices from rectangular sampling grids if the subject matrices store the skewed versions of the hexagonal sampled data [122]. Shown in Figs. 8.19 and 8.20 are the binary dither patterns and corresponding power spectra generated by VAC using Gaussian low-pass filters where σ 2 = 0.32 as g transitions from 6% to 50%. This same

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

229

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.15: The binary dither patterns for error-diffusion using the Stevenson and Arce filter on a hexagonal sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

230

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.16: The power spectra for error-diffusion using the Stevenson and Arce filter on a hexagonal sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

231

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.17: The binary dither patterns for Ulichney’s perturbed filter weight scheme on a hexagonal sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

232

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.18: The power spectra for Ulichney’s perturbed filter weight scheme on a hexagonal sampling grid as g transitions from 6 to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

233

filter variance resulted in dither patterns with significant energy packed into the corners of the baseband on a rectangular sampling grid, but here, results show identical spectral distributions beyond g = 14 , where there is no room in the corners of the baseband as there was for rectangular grids. From visual inspection, it is clear that these spectral distributions are achieved through the clustering of minority pixels in the spatial domain that, near gray-level g = 12 , create worm patterns. While worm textures/patterns are traditionally thought of as disturbing artifacts, we note that the radial symmetry of these particular patterns creates a twisting and turning path from pixel to pixel. This constant spiraling creates a smooth, almost invisible, texture. We further note that the worm patterns found here are far less objectionable than the strong directional patterns created by error-diffusion in either Fig. 8.15 or 8.17. In seeing Fig. 8.19, it is hoped that deriving optimal halftoning schemes for these grids will be easier to do now that we know what an isotropic, hexagonal grid dither pattern for g = 12 looks like. Now, in light of Fig. 8.19, the obvious question is, how does it compare with the equivalent dither patterns in Fig. 8.13 for a rectangular grid? This is a very difficult comparison to make based upon visual inspection of the respective figures due to the re-scaling of the hexagonal grid to have an increased horizontal scale. So, in order to offer some degree of comparison, Fig. 8.21 shows the g = 12 patterns formed by rectangular and hexagonal VAC, where the rectangular grid pattern has been re-scaled to match the aspect ratio of the hexagonal grid. Also included for comparison is the dither pattern, from Fig. 8.11, for σ 2 optimized to minimize low-frequency energy regardless of radial symmetry. This figure should be held at arm’s length prior to viewing, and while personal preferences will certainly vary, we feel that the discontinuities in texture for the rectangular grid are more visible than the increased coarseness of the hexagonal grid. Clearly, the hexagonal grid dither pattern is superior to the diagonal packing of energy in Fig. 8.21 (left).

8.3.1

Hexagonal Grid Dither Arrays

Figure 8.22 shows a 128 × 128 cropped section of a 256 × 256 dither array along with the magnitude of its corresponding Fourier transform, showing the uniquely high-frequency components of the dither array. In comparison with an equivalently constructed, rectangular grid,

© 2008 by Taylor & Francis Group, LLC

234

CHAPTER 8. HEXAGONAL GRID HALFTONING

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.19: The binary dither patterns constructed by means of VAC on a hexagonal sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

235

6%

10%

14%

18%

22%

26%

30%

34%

38%

42%

46%

50%

Figure 8.20: The power spectra for patterns constructed by means of VAC on a hexagonal sampling grid as g transitions from 6% to 50% coverage.

© 2008 by Taylor & Francis Group, LLC

236

CHAPTER 8. HEXAGONAL GRID HALFTONING

Figure 8.21: The binary dither patterns representing gray-level (left) Fig. 8.11, (center) Fig. 8.13, and (right) Fig. 8.19.

1 2

from

Figure 8.22: The (left) blue-noise dither array for hexagonal sampling grids along with (right) the corresponding magnitude of its Fourier transform. dither array, Figs. 8.23 and 8.24 show the corresponding rectangular and hexagonal grid halftones for the image first introduced by Ulichney [122]. To present a fair comparison, both images in this figure have the same number of pixels per unit area. From visual inspection, we argue that the hexagonal grid dither array is visually more pleasing than the techniques of Figs. 8.15–8.18 in terms of maintaining radial symmetry while, simultaneously, spreading minority pixels as homogenuously as possible. Seeing both the rectangular and hexagonal grid dither arrays side-by-side, it should also be clear from visual inspection that the hexagonal dither array clearly creates the grid-defiance illusion, made noted by Ulichney [123] as an important property of blue-noise, as one cannot, without very close inspection, determine which of the images is printed on a hexagonal sampling grid.

© 2008 by Taylor & Francis Group, LLC

8.3. HEXAGONAL SAMPLING GRIDS

237

Figure 8.23: Halftone image, from [122], produced using a blue-noise dither array on a rectangular sampling grid.

© 2008 by Taylor & Francis Group, LLC

238

CHAPTER 8. HEXAGONAL GRID HALFTONING

Figure 8.24: Halftone image produced using a blue-noise dither array on a hexagonal sampling grid.

© 2008 by Taylor & Francis Group, LLC

Chapter 9 Printers: Distortions and Models Research in digital halftoning has traditionally focused on the visual appearance of binary dither patterns assuming an ideal printer model where the output patterns are composed of perfectly square black and white dots (Fig. 9.1). While results have shown that blue-noise halftoning is the optimal technique for minimizing halftone visibility [119] and maximizing the apparent spatial resolution [105], the vast majority of printing processes still rely on AM halftoning due to the poor reliability of many devices to reproduce isolated dots. Blue-noise is generally still considered to be an option only for very high-quality printing situations [12]. In this chapter, we look at some of the practical considerations involved in digital printing, such as printer distortions. These practical considerations, in many cases, determine which halftoning techniques are applicable to a specific printing process. Assisted by several proposed dot models, this chapter also looks at some corrective measures that, tailored to a specific process, improve the visual quality of resulting images. It is surprising to note that the applicability of a given halftoning process is determined not by the amount of distortion, but by the predictability of variation.

9.1

Printer Distortion

The reason that many printers rely on AM halftoning and clustered dots is that, by clustering, these patterns become resilient to the dis239 © 2008 by Taylor & Francis Group, LLC

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

240

Figure 9.1: Clusters of one, two, three, and four printed dots from an ideal printer with the solid lines indicating the border between neighboring output pixels. MECHANICAL DOT-GAIN

OPTICAL DOT-GAIN INCIDENT LIGHT

PAPER SURFACE

LIGHT REFLECTED BACK TO VIEWER

LIGHT TRAPPED UNDER DOT

BORDERS OF IDEAL DOT

Figure 9.2: Illustration depicting (left) mechanical dot-gain, where the increase in size of a printed dot is due to the physical spreading of ink in the printing process, and (right) optical dot-gain, where the apparent increase in size is due to the trapping of light underneath a printed dot. tortions of the printing process [97], a quality referred to as halftone robustness. Recent work in halftoning has focused on this problem of printer distortions and the need for robust halftoning. The following sections look at the two distortions, dot-gain and dot-loss, and discuss the impact that each has on both AM and blue-noise halftoning. The purpose of this is to show that robustness is a quality achieved through dot clustering since these are more resilient to the distortions of the printing process.

9.1.1

Dot-Gain

As stated in the introduction, dot-gain is the increase in size of the printed dot relative to its intended size. In the case of mechanical dotgain (Fig. 9.2 (left)), this increase is due to the physical spreading of ink as it is applied to paper [1]. Optical dot-gain (Fig. 9.2 (right)) refers to

© 2008 by Taylor & Francis Group, LLC

9.1. PRINTER DISTORTION

241

4 3.5

PïA Ratio

3 FM

2.5 2 8×8 12×12

1.5

16×16

1 0.5 0

0

0.1

0.2

0.3

0.4

0.5 g

0.6

0.7

0.8

0.9

1

Figure 9.3: The perimeter-to-area ratios versus gray-level g for AM halftoning with 8 × 8, 12 × 12, and 16 × 16 cells and for FM halftoning using a blue-noise dither array.

the apparent growth due to the trapping of incident light underneath the printed dot [60]. When printing black-on-white, dot-gain can cause isolated white pixels to fill or “plug” [105] with ink; furthermore, by creating patterns that are darker than the original ratio of white-toblack pixels [23], dot-gain can cause a shift in color [74]. Whether a function of the mechanical printing process or the optical properties of paper, dot-gain is present in all printing processes. It, therefore, cannot be eliminated, but assuming that the process is repeatable, dot-gain can be anticipated and controlled [23]. The major relationship between halftone patterns and the amount of dot-gain seems to be the perimeter-to-area ratio of printed dots. That is, the halftone screen having the greatest perimeter-to-area ratio of printed dots will be more susceptible to the distortions caused by dot-gain [105]. FM halftoning, having a much higher ratio compared

© 2008 by Taylor & Francis Group, LLC

242

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

RESTRICTOR

ACTUATOR CHAMBER

FLOW CHANNEL

RESEVOIR

NOZZLE

PAPER

Figure 9.4: An ink jet print head showing the primary components of the printing mechanism.

to AM (Fig. 9.3), is therefore more susceptible [137]. A printing process where dot-gain is particularly common is ink jet printing, where dots are created from small droplets of ink that are discharged from a series of ink ejection chambers stored in a print head [130], Fig. 9.4. This print head is fabricated by bonding metal plates [52] to form multiple ink jets, each composed of a reservoir, restrictor, chamber, actuator, flow channel, and a nozzle. The chamber of each ink jet contains an independently controlled ink injection mechanism (the actuator) that changes the shape of the chamber in response to an electric pulse. By changing the chamber’s shape, ink is pulled from the reservoir, with its flow regulated by the restrictor, and then forced out and into the channel, expelling an ink droplet through the nozzle. In this process, the absorbency of the paper greatly affects the visual quality of the printed image [24]. Absorbent papers minimize the amount of dot-gain as ink is allowed to diffuse down into the paper (Fig. 9.5(a)). Dense papers, on the other hand, do not absorb the ink, but instead allow ink to spread across the surface, creating a dot much greater in size than intended (Fig. 9.5(b,c)). This growth is a function of the time it takes the dot to dry, with greater drying times allowing for greater degrees of dot growth. Because of the dramatic difference in quality, many desktop printers now include interactive device drivers that allow the user to specify the type of paper being used [46] as dense papers require a greater degree of dot-gain compensation.

© 2008 by Taylor & Francis Group, LLC

9.1. PRINTER DISTORTION

243

(a) ink jet printer output (ink jet paper)

(b) Ink jet printer output (laser paper)

(c) Ink jet printer output (newsprint)

(d) Laser printer output Figure 9.5: Clusters of one, two, three, and four printed dots printed on (a–c) an HP DeskJet 1600c ink jet printer using three different qualities of paper stock and (d) an Apple LaserWriter 630 Pro laser printer under a dissecting microscope and shown here at 10 dpi.

© 2008 by Taylor & Francis Group, LLC

244

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

Noting Fig. 9.3, where the perimeter-to-area ratio for AM halftoning (using 8×8, 12×12, and 16×16 cells) and FM halftoning (blue-noise dither array) are plotted versus gray-level, Fig. 9.6 shows the measured input versus output reflectance curves for an HP DeskJet 1600c ink jet printer (300 dpi) using (a) high-quality (ink jet) and (b) low-quality (newsprint) paper stocks. From the AM plots, the correlation between the perimeter-to-area ratio and the amount of dot-gain is clearly visible as 8 × 8 cells lead to the highest degree of distortion (the difference between the input and output tone levels). The impact of paper quality with respect to absorbency is also visible as newsprint shows the most severe distortion of the two stocks. A similar relationship between the perimeter-to-area ratio of printed dots and dot-gain can also be seen in electro-photographic (laser) printers (Figs. 9.5(d) and 9.7). Here, a dot is created by a laser beam that scans across the surface of a photo-conductive drum, leaving the drum positively charged in areas where black dots are to be printed [130]. Toner particles that are negatively charged are then attracted to these positions on the drum, forming an image that is then electro-statically transfered to the paper. Through heat and pressure, the toner is then fused to the paper’s surface.

9.1.2

Dot-Loss

Printer distortion in high-resolution laser printers differs greatly from that found in ink jet printers. In addition to dot-gain, laser printers also suffer from dot-loss, the inability of a printing device to print an isolated black dot. This distortion is particularly common when toner is low. Unlike dot-gain, dot-loss results in tone levels that are much lighter than the original fraction of black pixels, and is not directly related to the perimeter-to-area ratio of printed dots for a given halftone pattern but in the fineness of dot structure [23]. For an illustration of the effects of dot-loss, Fig. 9.8 shows the resulting gray-scale ramps formed by 192 squares printed on an Apple LaserWriter Pro 630 laser printer. In Fig. 9.8(a), halftoning is done using a blue-noise dither array while Fig. 9.8(b–d) are done using AM halftoning with cells sizes 8×8, 12×12, and 16×16, respectively. With isolated dots, Fig. 9.8(a) shows very clearly the effects of dot-loss where fading has occurred along the left and rights edges of the printed page. Shown in Fig. 9.9 are the input versus output reflectance curves that correspond to the gray-scale ramps of Fig. 9.8 with blue-noise having

© 2008 by Taylor & Francis Group, LLC

9.1. PRINTER DISTORTION

245

1 0.9 0.8

output reflectance

0.7 0.6

16×16

0.5

12×12 8×8

0.4 0.3 0.2

FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.9

1

(a) ink jet paper 1 0.9 0.8 16×16

output reflectance

0.7 0.6

12×12

0.5 8×8

0.4 0.3 0.2

FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

(b) newsprint Figure 9.6: The input versus output reflectance curves for an HP DeskJet 1600c ink jet printer.

© 2008 by Taylor & Francis Group, LLC

246

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 16×16

0.6 0.5

12×12

0.4

8×8

0.3 0.2 FM 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

Figure 9.7: The input versus output reflectance curves for an HP LaserJet 4000 laser printer. a greatly distorted curve.

9.2

Dot Models

In this section, several models for imitating the printed dot for various printers are presented. Such models have been used (i) to explain phenomenons such as dot-gain, (ii) to develop visual quality metrics for printed images [81], and (iii) to determine, within the halftoning process, when and when not to print a dot [5]. These dot models can be divided into two categories. The first category, physical models, attempts to characterize the placement of ink/toner on the printed page and is typically derived from the physics involved with the printing mechanism. These models try to produce an image of the printed dot under high magnification. The second category, statistical models, is usually look-up based and attempts to predict the apparent gray-level of each printed pixel according to the local arrangement of ones and zeros in the binary image. Figure 9.10 shows an example of both a physical and a statistical model.

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

247

(a)

(b)

(c)

(d)

Figure 9.8: The gray-scale ramps produced by an Apple LaserWriter 630 Pro using (a) a blue-noise dither array and (b-d) AM halftoning with 8 × 8, 12 × 12, and 16 × 16, respectively.

© 2008 by Taylor & Francis Group, LLC

248

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 16×16 0.6 0.5 12×12 0.4 0.3

8×8

0.2 FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

Figure 9.9: The input versus output reflectance curves for an Apple LaserWriter 630 Pro laser printer.

9.2.1

Physical Models

Hard Circular Dot Model Physical models attempt to mimic the physical application of ink/toner on paper. A primitive example of a physical dot model is the hard circular dot model [106]. This model, demonstrated in Fig. 9.11, characterizes a printed dot by a round circle that overlaps its nearest neighbors. This overlap suggests that resulting halftone patterns will be darker than the original fraction of black pixels; furthermore, in accordance with the model, halftone patterns with a higher perimeter-to-area ratio of printed dots will exhibit greater degrees of tonal distortion [104]. Blue-noise patterns having the highest perimeter-to-area ratio will, therefore, be the most susceptible to the distortions of overlap while clustered-dot ordered dither patterns will be the least. Figures 9.12 and 9.13 show the resulting modeled input versus output reflectance values for an HP DeskJet 1600c ink jet printer using AM halftoning with 8 × 8, 12 × 12, and 16 × 16 cells and a blue-noise dither array. The dot radius was chosen, in these curves, to minimize the mean error between the modeled and measured curves such that the

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

249

Figure 9.10: Clusters of one, two, three, and four printed dots modeled using (top) a physical model and (bottom) a statistical model.

Figure 9.11: Clusters of one, two, three, and four printed dots modeled using the hard circular dot model with a dot radius of 2.1 pixels.

dot radii are 2.50, 2.80, 3.30, and 1.90 times the inter-pixel distance D, respectively. The difference in radii can be attributed to increased dotgain caused by greater ink spreading when minority pixels are clustered. The greater spreading is a function of the increased time it takes for the ink droplets to dry when clustered than when dispersed. This phenomenon is the same as how it takes longer for water to evaporate from a full glass than when the water is poured out onto a table or other flat surface. The difference in dot-gain is more dramatic using a more absorbent paper such as newsprint (Figs. 9.14 and 9.15) with optimized radii 3.40, 3.60, 4.20, and 2.20, respectively. Table 9.1 summarizes these results.

© 2008 by Taylor & Francis Group, LLC

250

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (8 × 8) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) AM (12 × 12) Figure 9.12: The modeled input versus output reflectance curves for an HP DeskJet 1600c ink jet printer on ink jet paper using the hard circular dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

251

1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(c) AM (16 × 16) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(d) BN Figure 9.13: The modeled input versus output reflectance curves for an HP DeskJet 1600c ink jet printer on ink jet paper using the hard circular dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

252

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (8 × 8) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) AM (12 × 12) Figure 9.14: The modeled input versus output reflectance curves for an HP DeskJet 1600c ink jet printer on newsprint paper using the hard circular dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

253

1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (16 × 16) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) BN Figure 9.15: The modeled input versus output reflectance curves for an HP DeskJet 1600c ink jet printer on newsprint paper using the hard circular dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

254

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

Stochastic Dot Model I Now, a printed dot that is not accurately modeled using the hard circular dot model is that produced by an electro-photographic printer. This printing mechanism, as described by [13] and shown in Fig. 9.16, is not unlike that of a lithographic press. Like the lithographic press, a laser printer has a toner (ink) reservoir, the toner hopper, that supplies an even layer of negatively charged toner particles to the developing roller. These toner particles are electro-statically transfered from the developer roller onto the photo-conductive drum, which has a latent halftone image “burned” into its negatively charged surface by a scanning laser. Locations on the drum’s surface that carry negative charges repel toner particles while areas that have been exposed to light by the laser, and therefore carry no charge, form a weak bond that pulls toner off the developer roller. The excess toner that is not transfered from the developer roller is then recycled back into the toner hopper. In a lithographic press, ink is moved from the developing roller to the raised surfaces of the printing plate through physical contact with the plate’s surface. Excess ink not transferred to the plate is recycled into the next pass of the developer roller over the printing plate. A positively charged transfer roller peels the negatively charged toner particles from the drum’s surface and onto the surface of the paper. Excess toner left on the photo-conductive drum is scrapped off the drum by a cleaning blade and then disposed of in the waste toner hopper with the latent image on the drum erased by a charging roller that recharges the drum’s surface with a negative charge. The printing process is complete after heat and pressure are used to fuse the toner to the surface of the paper. In the lithographic press, direct contact or impact between the printing plate and the paper transfers ink, with

Table 9.1: The optimal dot radius, r, for the hard circular dot model. HP Deskjet 1600c ink jet printer halftone radius radius (ink jet) (newsprint) AM (8x8) 2.5 3.4 AM (12x12) 2.8 3.6 AM (16x16) 3.3 4.2 BN 1.9 2.2

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

255 LASER BEAM DEVELOPER ROLLER

CHARGING ROLLER ---

WASTE HOPPER

--

---

TONER PATH

FUSING ROLLER

--

DRUM

--

TONER HOPPER

E

FUSING ROLLER

- -- - - - -- - - - ONDUCTIV -C

PHOTO

CLEANING BLADE

-

---

+++ + ++ + + + + + + + + + + ++ + + + ++ TRANSFER ROLLER

-- - -- - - - - - -- -- - - -- -- -

PAPER

Figure 9.16: The electro-photographic printing mechanism (not to scale). the process complete after the ink dries. A model proposed to imitate the printed dot of a laser printer was introduced by Lin and Wiseman [81]. Given an isolated dot printed on the photo-conductive drum centered at point x, this model (Fig. 9.17) suggests that the probability, P (y; x), of a toner particle at point y is P (y; x) = exp−α|y−x|

2 /D 2

,

(9.1)

where D is the minimum distance between addressable samples (centerto-center) of the output image and α depends on the laser beam profile. This parameter, α, is a device-dependent parameter and should, therefore, be measured experimentally for a specific printer. For multiple printed dots, toner particles are combined in an OR type fashion. This combination implies statistical independence between toner particles belonging to different dots – leading to the probability of a toner particle at y given the set of printed dots at points {xi : i = 1, 2, . . . , N } being P (y; {xi : i = 1, 2, . . . , N }) = 1 −

N  

1 − exp−α|y−x|

2 /D 2



(9.2)

i=1

where 1 − exp−α|y−x| given xi .

2 /D 2

is the probability that no toner occurs at y

© 2008 by Taylor & Francis Group, LLC

256

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

Figure 9.17: Clusters of one, two, three, and four printed dots modeled using Lin’s and Wiseman’s [81] stochastic dot model with α = 1.79. Figures 9.18 and 9.19 show the resulting input versus output reflectance values for an HP LaserJet 4000 laser printer (600 dpi) using AM halftoning with 8 × 8, 12 × 12, and 16 × 16 cells and a blue-noise dither array. The constant α was chosen to minimize the mean error between the modeled and measured curves such that α = 0.834, 0.871, 0.787, and 0.969, respectively. The difference in α is attributed by Lin and Wiseman to a deformation in the electrostatic fields on the photoconductor when dots are clustered together [81]. Table 9.2 summarizes these results. Stochastic Dot Model II A problem with Lin’s and Wiseman’s model is that it does not form the dense packing of toner seen at the centroid of the 2 × 2 cluster, as in Fig. 9.17. So as an alternative to Lin’s and Wiseman’s model, Lau [62] proposed a new stochastic dot model that defined the probability of a toner particle at point y given the set of printed dots {xi : i = 1, 2, . . . , N } as a function of the total amount of light energy collected from a laser by the point on the photo-conductive drum corresponding to y. Assuming a Gaussian beam, this total amount of light, Ly , is

Table 9.2: The optimal dot α for Lin’s and Wiseman’s [81] stochastic dot model. HP Laserjet 4000 laser printer _ halftone AM (8x8) 2.5 AM (12x12) 2.8 AM (16x16) 3.3 BN 1.9

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

257

1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (8 × 8) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) AM (12 × 12) Figure 9.18: The modeled input versus output reflectance curves for an HP LaserJet 4000 laser printer using Lin’s and Wiseman’s stochastic dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

258

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (16 × 16) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) BN Figure 9.19: The modeled input versus output reflectance curves for an HP LaserJet 4000 laser printer using Lin’s and Wiseman’s stochastic dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

259

Figure 9.20: Clusters of one, two, three, and four printed dots modeled using the second stochastic dot model with parameters {α, T1 , T2 } = {1.11, 0.23, 1.46}. defined as Ly =

N 

exp−α|y−xi |

2 /D 2

,

(9.3)

i=1

the superposition of beams corresponding to each printed dot xi . The probability of toner at y is then defined as P (y; {xi : i = 1, 2, . . . , N }) = F (Ly ) ,

(9.4)

where the transfer function F (·) determines how the total amount of light energy collected at point y maps to a probability of a toner particle occurring at y. The exact form of F (·) is a function of the printing device that, in order to specify, may require close examination of the physics involved in the process, but for the simplest form of F (·), we propose a function of the two parameters T1 and T2 such that ⎧ ⎪ ⎨

F (Ly ) =

⎪ ⎩

0, for Ly < T1 for T1 ≤ Ly < T2 1, for T2 ≤ Ly .

(Ly −T1 ) , T2 −T1

(9.5)

This form of F (Ly ) describes a printer for which an absorbed light intensity less than T1 will attract no toner to point y while an absorbed intensity greater than T2 will guarantee toner. Defining Ly = T2 as the edge of a printed dot, the difference T2 − T1 dictates how far from an edge toner may be found for a given beam profile. Figure 9.20 illustrates this new model using (9.5) with the parameters {α, T1 , T2 } = {1.11, 0.23, 1.46}. An advantage of the latter model is that by employing the superposition of beams, the probability of toner occurring for a particular dot

© 2008 by Taylor & Francis Group, LLC

260

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

is now directly affected by the concentration of dots in the surrounding area, as demonstrated in Fig. 9.20, where the modeled output now forms a sufficiently dense macro-dot when composed of four pixels. Another advantage of this model is that it is simple, being a function of only three parameters; furthermore, (9.5) can be tuned to model the distortion dot-loss – a distortion that cannot be accounted for in the previous two models, as seen in Figs. 9.21 and 9.22, where the modeled input versus output reflectance curves for the Apple LaserWriter Pro 630 using Lin’s and Wiseman’s stochastic dot model are shown. From inspection of curve (d), this model is clearly inappropriate in situations where both dot-loss and dot-gain occur. Both dot-gain and dot-loss can be modeled in (9.5) any time T2 is greater than the maximum amount of absorbed light for an isolated pixel (Ly = 1). The severity to which a dot is lost (how much toner is actually printed) is dependent on F (Ly ) for Ly < 1 with complete loss occurring when F (1) = 0 (when T1 > 1). Dot-loss is overcame when the superposition of beams causes Ly , to exceed T2 , but it is dependent upon the thresholds T1 and T2 to determine how close together two neighboring dots must be in order for toner to be printed. In the case of a cluster composed of two printed dots with α = 1.1, the maximum light energy collected will be 1.32 while a cluster of three printed dots will have a maximum light energy of 1.67. So, in instances where T2 > 1.15, only through clustering can dot-loss be overcome, and as T2 increases, the size of clusters must also increase in order to minimize the effects of dot-loss. Shown in Fig. 9.23 are modeled clusters using the parameters {α, T1 , T2 } = {1.11, 0.25, 1.25}, {1.11, 0.50, 1.50}, {1.11, 0.75, 1.75}, and {1.11, 1.0, 2.0}. While these parameters have not been optimized to mimic distortion in the Apple LaserWriter Pro 630, their reflectance curves (Figs. 9.24 and 9.24) are much closer in their general shape than that of Lin’s and Wiseman’s (Fig. 9.22(d)). These curves are also much closer, in their general shape, to the reflectance curves of lithographic screening, as reported by Coudray [23].

9.2.2

Statistical Models

A statistical model replaces each pixel of the ideal printer output with the expected gray-level produced by the modeled printer when trying to produce the same arrangement of dots. Given the ideal pattern, the expected gray-level of each pixel is the average measured reflectance

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

261

1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (8 × 8) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) AM (12 × 12) Figure 9.21: The modeled input versus output reflectance curves for an Apple LaserWriter Pro 630 laser printer using Lin’s and Wiseman’s stochastic dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

262

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

0.7

0.8

0.9

1

(a) AM (16 × 16) 1 0.9 0.8

output reflectance

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

(b) BN Figure 9.22: The modeled input versus output reflectance curves for an Apple LaserWriter Pro 630 laser printer using Lin’s and Wiseman’s stochastic dot model. The measured curves are indicated by the dashed lines.

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

263

(a) {α, T1 , T2 } = {1.11, 0.25, 1.25}

(b) {α, T1 , T2 } = {1.11, 0.50, 1.50}

(c) {α, T1 , T2 } = {1.11, 0.75, 1.75}

(d) {α, T1 , T2 } = {1.11, 1.00, 2.00} Figure 9.23: Clusters of one, two, three, and four printed dots modeled using the new stochastic dot model defined by the parameters α, T1 , and T2 .

© 2008 by Taylor & Francis Group, LLC

264

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

1 0.9 0.8

output reflectance

0.7 16×16

0.6

12×12 0.5 8×8 0.4 0.3 0.2 FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

(a) {α, T1 , T2 } = {1.11, 0.25, 1.25} 1 0.9 0.8

output reflectance

0.7 0.6

16×16

0.5

12×12

0.4

8×8

0.3 0.2 FM 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

(b) {α, T1 , T2 } = {1.11, 0.50, 1.50} Figure 9.24: The modeled input versus output reflectance curves using the new stochastic dot model defined by the parameters α, T1 , and T2 .

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

265

1 0.9 0.8

output reflectance

0.7 0.6

16×16

0.5

12×12

0.4

8×8

0.3 0.2

FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

(c) {α, T1 , T2 } = {1.11, 0.75, 1.75} 1 0.9 0.8

output reflectance

0.7 16×16

0.6

12×12

0.5

8×8

0.4 0.3 0.2

FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 input reflectance

0.7

0.8

0.9

1

(d) {α, T1 , T2 } = {1.11, 1.00, 2.00} Figure 9.25: The modeled input versus output reflectance curves using the new stochastic dot model defined by the parameters α, T1 , and T2 .

© 2008 by Taylor & Francis Group, LLC

266

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

after repeatedly printing the same arrangement of dots on the target device. The gray-level is then stored in a look-up table and is indexed any time the particular arrangement is to be printed. The size of this table, depending on the desired accuracy and the printer, can be very large and sometimes intractable [95]. A neighborhood of only 5 × 5 requires, at most, 225 different dot combinations of printed dots, but due to symmetry, only a smaller subset of these may be required. It may also be that no correlation exists between a printed dot and its neighbors 2 pixels away, for a given printer. Such a printer would only need a neighborhood of 3 × 3 that, assuming symmetry, requires only 102 of 512 patterns [95]. By redefining the problem to consider, not the pixels, but the space between, Wang et al. [129] reduce the problem further to only 16 patterns. Ink Jet Dot Model In order to take into account variations in dot-gain as well as dot placement observable in ink jet printers caused by aerodynamic turbulence between the paper substrate and the ink jet head, an alternate model for predicting dot coverage, for a binary pattern representing gray-level g, can be summarized by the equation Total Coverage(g) =

N 

αi Ci (g),

(9.6)

i=1

where Ci (g) represents the total percentage of paper that is covered by i dots at the initial point of impact by ink droplets. The parameters αi define how much a dot increases in size from the initial impact, with dot overlap increasing the amount of time it takes the ink to dry and, thereby, giving the ink more time to diffuse outward for an even larger dot. As a linear combination of Ci (g)s, the αi s can be optimized by means of least squares using a sampling of at least N gray-levels. To adapt the coverage model of (9.6) to a specific print head, we first determine the average radius of a printed dot for light gray-levels where the average distance between printed dots is sufficiently large as to preclude the possibility of printed dots overlapping. This can be achieved at gray-levels 250...254 , where 255 Measured tone of printed pattern Area of printed dot = . Area of pixel Measured tone of paper

© 2008 by Taylor & Francis Group, LLC

(9.7)

9.2. DOT MODELS

267

Figure 9.26: The (left) real and (right) modeled output patterns for gray-level 208. The corresponding dot radius is then derived as the square root of this printed dot area divided by π. Using multiple gray-levels, the optimal dot radius is taken as the average of the individual radii recovered from each tonal measurement. For large-dot print heads, the derived dot radius is 1.5 times larger than the minimum spacing between pixel centroids. Once the printed dot radius for non-overlapping dots has been measured, we need to measure the variance in dot placement where we will assume that the distribution in placement of the dot around its intended position is a circular symmetric, zero-mean, Gaussian distribution with fixed variance vr . While this may not be a wholly accurate model of dot placement variability, especially in dark gray regions where banding artifacts are clearly visible, it is thought that the optimization of the dot-gain parameters αi will appropriately account for any deviation in the dot placement models. Furthermore, the proposed model is being developed with the belief that the only device available for optimizing the dot model is a densitometer. The use of a drum scanner, as will be described, need only occur once for a given ink jet production process where our derived variance will be used for all future print heads. Dot placement variance may also be available directly from the print head manufacturer. To measure dot placement variance, we use a high resolution drum scanner to scan in a printed dither pattern representing a moderately light gray-level (g = 208), where printed dots just begin to overlap and the amount of overlap is sufficiently small as to ignore the effects

© 2008 by Taylor & Francis Group, LLC

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

268

of increased dot-gain. This scanned image is then thresholded, at graylevel 0.5, to produce a binary pattern (Fig. 9.26 (left)), where pixels set to black are considered to be areas of the page covered by ink and pixels set to white are considered to be uncovered paper. Blob analysis is then applied to this pattern to identify the centroids of blobs such that multiple overlapping dots result in a single centroid. Using the same binary pattern, we then generate a modeled output image using the dot radius calculated previously where we change the variance in dot placement until the number of dot centroids, after applying blob analysis, is approximately equal to that in the scanned image. For large-dot print heads, the derived dot variance is equal to 0.3 times the minimum spacing between pixel centroids. Having the dot radius for non-overlapping dots as well as an estimate of dot placement variability, we now use the binary dither patterns for all gray-levels between 0 and 254 to model the resulting output patterns using our previously measured fixed dot radius. We then calculate the percentage of paper coverage for multiple dots to derive the parameters C1 (g, vr ), C2 (g, vr ), C3 (g, vr ), . . . . This is demonstrated in Fig. 9.27 (left), where we show the percentage of coverage for various amounts of dot overlap as g ranges from 0 to 255 for large-dot print heads. In order to derive the optimal αi s, we arbitrarily select a set {gi : i = 1, 2, . . . , K} of at least N sample gray-levels, and the densitometer measurements for corresponding, printed, halftone patterns. We can now define the following relationship ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

C2 (g1 ) · · · CN (g1 ) C2 (g2 ) · · · CN (g2 ) .. .. ... . . C1 (gK ) C2 (gK ) . . . CN (gK ) C1 (g1 ) C1 (g2 ) .. .

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

α1 α2 .. . αN

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎦

=⎢ ⎢

⎢ ⎢ ⎣

g1 g2 .. .  gK

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(9.8)

with the least-squares solution 

α = CT C

−1

CT g  .

(9.9)

Specifically, we sampled the gray-level range at 12 equally-spaced intervals, which were then scaled according to the gray-level of the paper (g = 255) and up-sampled by means of linear interpolation to produce 256 unique gray-levels. Plotted in Fig. 9.28 (left) is the resulting tone reproduction curve where, as a smooth line, is placed the best-fit modeled tone curve using (9.9).

© 2008 by Taylor & Francis Group, LLC

9.2. DOT MODELS

269

Large Dot Ink Jet Head Coverage Curves

Small Dot Ink Jet Head Coverage Curves 70 percent coverage

percent coverage

70 60 50

C (g)

40

1

30

60 50 40

C1(g)

30

C2(g)

20 10

C (g)

20

C (g) 3

C2(g)

10

C3(g)

4

0

0

50

100 150 gray level (g)

200

0

250

0

50

100 150 gray level (g)

200

250

100

100

90

90

80

80 percent coverage

percent coverage

Figure 9.27: The initial estimates of dot coverage for the (left) largeand (right) small-dot ink jet heads.

70 60 50 40

70 60 50 40

30

30

20

20

10

10

0

0

50

100 150 gray level (g)

200

250

0

0

50

100 150 gray level (g)

200

250

Figure 9.28: The measured and modeled output levels for the large-dot ink jet heads using (left) unconstrained and (right) constrained α. For large-dot print heads, Fig. 9.28 (left) shows the tone reproduction curve of the printed page along with the the predicted tone curve where α1 = 1.5326, α2 = −0.0712, α3 = 3.0463, and α4 = −1.8287. Across the entire range of 256 gray-levels, the resulting mean-squared error between the measured and model curves is equal to 1.29%. When interpreting these results, we note that the high quality ink jet paper eliminates dot growth such that if we had an accurate dot model for predicting Ci (g)s, then the corresponding αi s would all equal 1.0. Given that this is not the case, then we can only assume that our dot placement model was not accurate and that Ci (g) = αi Ci (g) is a

© 2008 by Taylor & Francis Group, LLC

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

270

Small Dot Ink Jet Head Coverage Curves 80

70

70

60

percent coverage

percent coverage

Large Dot Ink Jet Head Coverage Curves 80

C (g) 1

50 40 30

1

50 40 30

C (g)

20

20

2

10 0

C (g)

60

10

C3(g) 0

50

100 150 gray level (g)

200

250

0

C2(g) 0

50

100 150 gray level (g)

200

250

Figure 9.29: Final, derived dot coverage curves for (left) large- and (right) small-dot ink jet heads. better model of dot coverage, as Total Coverage(g) =

N  i=1

(αi Ci (g)) =

N 

1.0Ci (g),

(9.10)

i=1

as depicted in Fig. 9.29 (left). The fact that α1 is greater than 1.0 indicates that our derived dot radius is too small, with C1 (g) being the single dot coverage corresponding to a dot area α1 times larger. Negative αi values do not have an intuitive interpretation other than to further compensate for errors in the other coverages. As such, we can repeat the least-squares optimization of α but under the constraint that all αi ≥ 0.0. The resulting modeled tone curve using this procedure is plotted in Fig. 9.28 (right), where the derived α is equal to [1.5160, 0.0807, 2.2989, 0.0000]T . Here, the total squared error is equal to 1.47%, with α2 indicating that our modeled coverages anticipate too much two-dot overlap and α4 indicating that we need not consider more than three dots of coverage. Note that the definition of C2 (g) being considered a two-dot overlap is, at this point, completely arbitrary, and as such, we will swap C2 (g) and C3 (g) such that the amount of two-dot coverage always exceeds three-dot coverage. Moving to smalldot print heads (Figs. 9.27 (right) and 9.29 (right)), we derived a new dot radius r = 1.13 as well as a new dot placement variance vr = 0.26, and what we see is that α1 = 1.7887 and α2 = 0.9203, where, due to the smaller dot size, no three dots overlap in the same place. Shown in

© 2008 by Taylor & Francis Group, LLC

271

100

100

90

90

80

80 percent coverage

percent coverage

9.2. DOT MODELS

70 60 50 40

70 60 50 40

30

30

20

20

10

10

0

0

50

100 150 gray level (g)

200

250

0

0

50

100 150 gray level (g)

200

250

Figure 9.30: The measured and modeled tone curves for (left) small dots only and (right) large and small dots. Fig. 9.30 (left) are the measured and modeled tone curves with a total squared error of 6.00%. Double Sized Dot Model From observation of the scanned, solid black pattern, it appears that the ink jet heads of large and small dots are arranged as depicted in Fig. 9.31, where rows of small dot ink jet heads are placed between rows of large dot heads. As such, we use the same described process modeling a single large-small dot pair as being a single dot whose dot area is equal to the sum of the two dots areas derived previously. We will also assume that the variation in dot placement is equal to the average of the two previously calculated variances (vr = 0.28). Accordingly, we have derived the coverage curves of Fig. 9.32 and the corresponding αi parameters as α = [0.8510, 0.7062, 1.0887, 1.081, 0.0283, 2.7426]T . Shown in Fig. 9.30 (right) is a comparison between the measured and modeled tone reproduction curves with a total squared error of 1.8%. Mixing Colors Using the above described process, predicting the resulting color composition is achieved by simply determining what percentage of ink corresponds to each color channel and then scaling this number by the predicted amount of overlap. As an example, if a pattern was printed using 30% coverage of large cyan dots and 20% coverage of large ma-

© 2008 by Taylor & Francis Group, LLC

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

272

Figure 9.31: A (left) high-resolution drum scan of the 100% large and small dot pattern and the (right) suspected ink jet head layout. Initial Large-Small Dot Ink Jet Head Coverage Curves

Final Large-Small Dot Ink Jet Head Coverage Curves 70

C1(g)

60 50 40 30

C (g)

C (g)

C (g) 3

percent coverage

percent coverage

70

C2(g)

4

5

C4(g)

C (g) 3

C (g)

30

10

10

50

C1(g)

40

20

0

5

50

20

0

C (g)

60

100 150 gray level (g)

200

250

0

2

0

50

100 150 gray level (g)

200

250

Figure 9.32: The (left) initial and (right) final derived dot coverage curves for large and small-dot ink jet heads. genta dots, then a jointly-blue distribution of dots would be the equivalent of printing 50% monochrome. Assuming 50% monochrome lead to 70% coverage, then we should expect 30%/(20 + 30)% × 70% cyan coverage and 30%/(20 + 30)% × 70% magenta coverage. Arbitrary Paper Stock Having the above described Ci (g), predicting the tonal coverage for arbitrary paper stock involves optimizing the αi s using unconstrained

© 2008 by Taylor & Francis Group, LLC

9.3. CORRECTIVE MEASURES

273

least squares for the equation Total Coverage(g) =

N 

αi Ci (g),

(9.11)

i=1

where the Ci (g)s are defined for both large- and small-dot ink jet heads as plotted in Figs. 9.32 (right).

9.3

Corrective Measures

With an understanding of some of the basic distortions, we can now look at several corrective measures that, when applied to the halftoning process, can tune the results to create the best possible images.

9.3.1

Tone Correction

Tone correction is a process where each pixel of the input image with gray-level g is replaced by a pixel1 with gray-level g  [125]. The mapping of gray-levels from g to g  is usually determined by direct measurement of the input versus output reflectance curve for a given printer [122] but can also be estimated using a printed dot model [108]. The underlying assumption is that when trying to reproduce gray-level g  , the printer will consistently produce the apparent gray-level g. The problem for tone correction occurs when the printer does not produce dot distortions consistently [57], such as in the Apple LaserWriter Pro 630 using bluenoise, where the characteristics of the printed dot change with location across the page. These characteristics make tone correction impractical, requiring a correction curve that also varies spatially.

9.3.2

Mode-Based Halftoning

Model-based halftoning refers to algorithms that take into account a model of the printed dot to decide when and when not to print a dot. Roetling and Holladay [106] offered one of the first examples of modelbased halftoning when they adjusted the thresholds of ordered dither arrays based on the dot overlap created by the hard circular dot model. Pappas [96] also used the hard circular dot when he first included a 1 In the case of halftoning via dither arrays, tone correction can be achieved by modifying the thresholds of the dither array and not the input image [106].

© 2008 by Taylor & Francis Group, LLC

274

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS DOT MODEL

Y

Y

Figure 9.33: The modified error-diffusion algorithm. dot model into error-diffusion by introduced modified error-diffusion (MED). In MED (Fig. 9.33), the output pixel, y[n], is still determined as  1, if (x[n] + xe [n]) ≥ 0 (9.12) y[n] = 0, else, but in this case, the error terms, ye [n − i] for i = 1, 2, . . . , M , are calculated at each iteration and cannot be stored in an error image buffer. That is, assuming an ideal printer means that the quantization error, ye [n], can be diffused and stored in an error buffer, e[n], such that  1, if (x[n] + e[n]) ≥ 0 (9.13) y[n] = 0, else eupdate [n] = y[n] − (x[n] + e[n]) (9.14) eupdate [n + i] = e[n + i] + ai eupdate [n]. (9.15) Using a dot model such as the hard circular dot, which affects neighboring pixels, means that the final value of y[n] is not determined until all its neighbors have been quantized. So, in MED, the error term, e[n], has to be calculated at each iteration prior to quantizing x[n] + e[n] as e[n] =

M 

ai (˜ y [n − i] − x[n − i]) ,

(9.16)

i=1

where y˜[n − i] is the modeled tone for output pixel y[n − i] assuming y[n + i] for i = 1, 2, . . . are not printed (Fig. 9.34). From eqns. (9.13) and (9.16), MED can be summarized as 

y[n] =



1, if x[n] + 0, else.

M i=1



ai (˜ y [n − i] − x[n − i]) ≥ 0

(9.17)

Shown in Fig. 9.35 are the resulting halftone images using the Jarvis, Judice, and Ninke error filter using the hard circular dot model with radii (left) r = 0.95D and (right) r = 1.1D.

© 2008 by Taylor & Francis Group, LLC

9.3. CORRECTIVE MEASURES

275

Figure 9.34: The determination of diffused quantization error for a 12-weight filter in modified error-diffusion where printed dots overlap neighboring dots.

9.3.3

Clustering

The problem with many model-based techniques is that they fail to address dot-loss, and by assuming reliability in the printing device, resulting patterns are typically expensive (computational complexity) tone-corrected blue-noise patterns [137]. The patterns are, therefore, inappropriate in devices that do not produce isolated pixels reliably, such as in laser printers [6]. Printer distortion is not, in general, considered a “bad” thing, and its occurrence, either high or low, does not limit the choice of halftoning techniques for a given printing process. So blue-noise may be just as applicable as AM halftoning for a given printer. Dot-loss can be bad if it limits the extent to which perfect black can be produced. In many instances, though, dot-loss only occurs for small, isolated black dots, and therefore does not limit the choice in halftoning techniques either. What does limit the choice, in halftoning, is the repeatability of dot-gain/loss. If a printer consistently reproduces dots with little variation, accurate tone reproduction can be achieved through tone correction [104], but in a printing process that is not repeatable, compensating for distortion is much more complicated as isolated dots are more sensitive to process variation [105]. This is clearly evident in laser printers. Looking at squares of constant gray-level g = 34 , Figs. 9.36 and 9.37 show the resulting pages produced using an Optra T610 at 1200

© 2008 by Taylor & Francis Group, LLC

276

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

Figure 9.35: The halftone images created using modified error-diffusion with (left) r = 0.95D and (right) r = 1.1D. dpi with Fig. 9.36 halftoned using blue-noise and Fig. 9.37 using the printer’s default AM screen (120 lpi at a 45 degrees screen angle). The horizontal banding in both figures is a result of charging inconsistencies on the developer roller’s surface that create an inconsistent density of toner particles across the roller’s surface prior to the toner’s transfer to the photo-conductive drum [22]. The visual effect on the output from these charging inconsistencies is a periodic banding with period ≈ 1.77 in., the horizontal distance traversed by the paper for every complete revolution of the developer roller. As shown in Figs. 9.38 and 9.39, the plots characterizing the average variation in tone for gray squares representing gray-levels g = 0.1, 0.2, . . ., and 0.9 along the vertical axis (top) clearly illustrate a periodic fluctuation caused by a charging inconsistency on the developer roll. The plots characterizing average variations in tone along the horizontal axis (bottom) show the change in toner density near the edges of the page. This distortion is due to a center mounted mirror where the laser projects onto the photo-conductive drum at an indirect angle near the edges while being directly perpendicular near the center [22], as in Fig. 9.40. At an indirect angle the laser is less intense, thereby attracting less charge and less toner. As would be expected, the mechanism noise is much more prominent with blue-noise than with the default AM

© 2008 by Taylor & Francis Group, LLC

9.3. CORRECTIVE MEASURES

277

10 9

vertical axis (inches)

8 7 6 5 4 3 2 1

resulting tone

0 0.6 0.7 0.8 resulting tone 0.8 0.7 0.6

0

1

2 3 4 5 6 horizontal axis (inches)

7

8

Figure 9.36: The gray-scale square of constant gray-level g = 0.7 produced by an Optra T610 at 1200 dpi using blue-noise. screen. For quantitative measures of the data plotted in Figs. 9.38 and 9.39, Table 9.3 lists the minimum, maximum, and range of values for each plotted line. Tone correction has not been applied prior to halftoning in these instances except to the extent that tone correction was included with the default AM screen. Noting that the AM screen produces halftone patterns with far less noticeable variation in tone, it may be more advantageous in unreliable printing devices to use a halftoning scheme that resists distortion,

© 2008 by Taylor & Francis Group, LLC

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

278

10 9

vertical axis (inches)

8 7 6 5 4 3 2 1

resulting tone

0 0.6 0.7 0.8 resulting tone 0.8 0.7 0.6

0

1

2 3 4 5 6 horizontal axis (inches)

7

8

Figure 9.37: The gray-scale square of constant gray-level g = 0.7 produced by an Optra T610 at 1200 dpi using the default AM screen. making the output more robust to variations in the printing process [66]. Robustness is usually achieved through clustering [56], as indicated by comparison of the input versus output reflectance curves between AM and blue-noise halftoning for the Apple LaserWriter Pro 630 (Fig. 9.8) and the plots of Figs. 9.38 and 9.39. Only through clustering is this printer able to achieve the consistent behavior necessary for accurate tone reproduction through tone correction. Blue-noise is, therefore, not an acceptable halftoning tech-

© 2008 by Taylor & Francis Group, LLC

9.3. CORRECTIVE MEASURES

279

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 9.38: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for halftone patterns printed by a Lexmark Optra T610 at 1200 dpi using blue-noise. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

280

CHAPTER 9. PRINTERS: DISTORTIONS AND MODELS

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 9.39: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for halftone patterns printed by a Lexmark Optra T610 at 1200 dpi using the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

9.3. CORRECTIVE MEASURES

281

PHOTOCONDUCTIVE DRUM

BEAM SHAPES

{

LASER

ROTATING MIRROR

Figure 9.40: The scanning of the laser across the surface of the photoconductive drum using a center mounted rotating mirror. nique for the Apple or Lexmark printers [6]. blue-noise is acceptable, though, in the HP 1600c ink jet printer, which, although, it has a higher degree of dot-gain compared with the Apple and Lexmark printers, prints with a much greater degree of consistency from printed dot to printed dot (Fig. 9.6(a)). Because of this consistency, the HP ink jet printer is a candidate for accurate tone reproduction through either tone correction or through model-based halftoning.

Table 9.3: Table listing gray-level statistics for the plots of Fig. 9.38.

© 2008 by Taylor & Francis Group, LLC

Chapter 10 Green-Noise Dithering Just as blue-noise is the high-frequency component of white-noise, greennoise, demonstrated in Figs. 10.1–10.3, is the mid-frequency component that, like blue, benefits from aperiodic, uncorrelated structure without low-frequency graininess, but unlike blue, exhibits clustering. The result is a frequency content that lacks the high-frequency component characteristic of blue-noise, hence the term “green.” The objective of using green-noise is to combine the maximum dispersion attributes of blue-noise with that of clustering of AM halftone patterns, and to do so at varying degrees. The motivation for using green-noise and green-noise halftoning algorithms is to produce patterns with adjustable coarseness that can be tuned to the reliability of a given printer to produce dots consistently. In a reliable printing device, green-noise would be tuned to produce patterns composed of small clusters having low halftone visibility, but in an unreliable device, green-noise would be tuned to produce large clusters that have high visibility but print consistently. Figure 10.4 summarizes green-noise’s behavior as offering the tunable halftoning scheme that bridges the gap in halftone visibility and robustness between AM and FM halftoning.

10.1

Spatial and Spectral Characteristics

Point process statisticians have long described clustering processes such as those seen in green-noise by examining the cluster process in terms of two separate processes: (i) the parent process, which describes the 283 © 2008 by Taylor & Francis Group, LLC

284

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.1: Gray-scale image halftoned using green-noise with small clusters.

© 2008 by Taylor & Francis Group, LLC

10.1. SPATIAL AND SPECTRAL CHARACTERISTICS

285

Figure 10.2: Gray-scale image halftoned using green-noise with medium clusters.

© 2008 by Taylor & Francis Group, LLC

286

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.3: Gray-scale image halftoned using green-noise with large clusters.

© 2008 by Taylor & Francis Group, LLC

10.1. SPATIAL AND SPECTRAL CHARACTERISTICS

287

4 FM 3.5

PïA Ratio

3 2.5 GREENïNOISE REGION

2 1.5 1 0.5

16×16 0

0

0.1

0.2

0.3

0.4

0.5 g

0.6

0.7

0.8

0.9

1

0.9

1

(a) ink jet paper 1 0.9 0.8 GREENïNOISE output reflectance

0.7 0.6 16×16

0.5 0.4 0.3 0.2

FM

0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 input reflectance

0.8

(b) news print Figure 10.4: The (a) perimeter-to-area ratio of printed dots using AM halftoning (16 × 16 cells), FM halftoning; and green-noise and (b) the corresponding modeled input versus output reflectance curves for an HP DeskJet 1600c ink jet printer on newsprint paper using the hard circular dot model.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

288

φg

φp

φdi

Figure 10.5: A sample, φg , of a green-noise point process, the corresponding sample, φp , of the parent process, and a corresponding sample, φdi , of the daughter process . location of the clusters,1 and (ii) the daughter process, which describes the shape of the clusters. In AM halftoning, clusters are placed along a regular lattice, and therefore, variations in AM patterns occur in the cluster shape. In FM halftoning, cluster shape is deterministic, a single pixel. It is the location of clusters that is of interest in characterizing FM patterns. Green-noise patterns, having variation in both cluster shape and cluster location, require an analysis that looks at both the parent and daughter processes. Shown in Fig. 10.5 is an illustration of how a sample, φg , of a green-noise point process is related to the parent process and the daughter process. Looking first at the parent process Φp , φp represents a single sample of the parent process such that φp = {xi : i = 1, . . . , Nc }, where Nc is the total number of clusters. For the daughter process Φd , φd represents a single sample cluster of Φd such that φd = {yj : j = 1, . . . , M }, where M is the number of minority pixels in cluster φd . By first defining the translation or shift in space Tx (B) of a set B = {yi : i = 1, 2, . . .} by x relative to the origin as Tx (B) = {yi − x : i = 1, 2, . . .}

(10.1)

and then defining φdi as the ith sample cluster for i = 1, . . . , Nc , a sample φG of the green-noise halftone process ΦG is defined as φG =

 xi ∈φp

1

Txi (φdi ) =



{yji − xi : j = 1, . . . , Mi },

(10.2)

xi ∈φp

The location being defined is the centroid of all points within the cluster.

© 2008 by Taylor & Francis Group, LLC

10.1. SPATIAL AND SPECTRAL CHARACTERISTICS

289

the sum of Nc translated clusters. The overall operation is to replace each point of the parent sample φp , of process Φp , with its own cluster φdi , of process Φd . In order to derive a relationship between the total number of clusters, the size of clusters, and the gray-level of a binary dither pattern, Ig is defined as the binary dither pattern resulting from halftoning a continuous-tone discrete-space monochrome image of constant graylevel g, and Ig [n] is defined as the binary pixel of Ig with pixel index n. From the definition of φ(B) as the total number of points of φ in B, φG (Ig ) is the scalar quantity representing the total number of minority pixels in Ig , and φp (Ig ) is the total number of clusters in Ig with φp (Ig ) = Nc . The intensity, I, being the expected number of minority pixels per unit area can now be written as φG (Ig ) = I= N (Ig )



g, for 0 < g ≤ 1/2 1 − g, for 1/2 < g ≤ 1,

(10.3)

the ratio of the total number of minority pixels in Ig to N (Ig ), the total ¯ , the average number number of pixels composing Ig . Given (10.3), M of minority pixels per cluster in Ig , is ¯ = φG (Ig ) = I · N (Ig ) , M φp (Ig ) φp (Ig )

(10.4)

the total number of minority pixels in Ig divided by the total number of clusters in Ig . Although obvious, (10.4) shows the very important relationship between the total number of clusters, the average size of clusters, and the intensity for Ig . AM halftoning is the limiting case where φp (Ig ) is held constant for varying I, while FM halftoning is the limiting case ¯ is held constant for varying I. In addition, (10.4) says that where M ¯ . For the total number of clusters per unit area is proportional to I/M isolated minority pixels (blue-noise), the square of the average separation between minority pixels (λb ) is inversely proportional to I, the average number of minority pixels per unit By determining √ area [123]. 1 the proportionality constant using λb = 2 for I = 2 , the relationship √ between λb and I is determined as λb = D/ I. In green-noise (Fig. 10.6), it is the minority pixel clusters that are distributed as homogeneously as possible, leading to an average separation (center-to-center) between clusters (λg ) whose square is inversely proportional to the average number of minority pixel clusters

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

290

Figure 10.6: The distribution of minority pixels in a green-noise pattern ¯ pixels and cluster radius rc separated forming clusters of average size M by an average distance of λg . ¯ . Using the fact that limM →1 λg = λb , the proporper unit area, I/M tionality constant can be determined such that λg is defined as ⎧ ⎨



¯ , for 0 < g ≤ 1/2 D/ (g)/M  λg = ⎩ D/ (1 − g)/M ¯ , for 1/2 < g ≤ 1,

(10.5)

the green-noise principal wavelength. This implies that the parent pro¯. cess, φp , is itself a blue-noise point process with intensity I/M

10.1.1

Spatial Statistics

If we assume a stationary and isotropic green-noise pattern, the pair correlation will have the form of Fig. 10.7 (top) given that 1. Daughter pixels, on average, will fall within a circle of radius rc ¯ (the area of centered around a parent point such that πrc2 = M the circle with radius rc is equal to the average number of pixels forming a cluster). 2. Neighboring clusters are located at an average distance of λg apart. 3. As r increases, the influence that clusters have on neighboring clusters decreases.

© 2008 by Taylor & Francis Group, LLC

PAIR CORRELATION

10.1. SPATIAL AND SPECTRAL CHARACTERISTICS

(c)

(a)

291

(b)

1

hg

r 0

c

0 RADIAL DISTANCE

/ /2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

2

0

Figure 10.7: The (top) pair correlation of the ideal green-noise pattern with principal wavelength λg and cluster radius rc and the (bottom) directional distribution function of the ideal green-noise pattern in the near range r ∈ [0, 0.5λg ) and the range r ∈ [0.5λg , 1.5λg ). The result is a pair correlation that has: (a) a non-zero component for 0 ≤ r < rc due to clustering, (b) a decreasing influence as r increases, and (c) peaks at integer multiples of λg indicating the average separation of pixel clusters. Note that the parameter rc is also indicated by a diamond placed along the horizontal axis in Fig. 10.7. In the case of stationary and anisotropic green-noise patterns, the pair correlation will also be of the form of Fig. 10.7 (top), but because clusters are not radially symmetric, blurring occurs in R(r) near the cluster radius rc . In a similar fashion, because the separation between clusters will also vary with direction, blurring will occur at each peak in R(r) located at integer multiples of λg . This blurring will also occur as the result of variations in the cluster size/shape and in the variations in the separating distance between neighboring clusters.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

292

If this variation is too high, patterns will tend to look “white” as the pair correlation begins to resemble that of a completely random, whitenoise process, but too little variation leads to orderly, periodic textures that are also disturbing to the eye. So some variation is necessary for a visually pleasing, green-noise dither pattern. While radial symmetry is a desired characteristic of visually pleasing patterns [123], it is also a desired characteristic for pattern robustness as elliptical clusters have a higher perimeter-to-area ratio of printed dots than radially symmetric clusters. The directional distribution function of the ideal green-noise pattern is of the form of Fig. 10.7 (bottom), where the distribution of minority pixels within a cluster is shown in light gray (r ∈ [0, 0.5λg )) and the distribution of minority pixels between neighboring clusters is shown in dark gray (r ∈ [0.5λg , 1.5λg )).

10.1.2

Spectral Statistics

Assuming that the variation in cluster size is small for a given Ig , the placement of clusters λg apart leads to a strong spectral peak in P (fρ ) at fρ = fg , the green-noise principal frequency ⎧ ⎨



¯ /D, for 0 < g ≤ 1/2 (g)/M fg = ⎩  ¯ /D, for 1/2 < g ≤ 1. (1 − g)/M

(10.6)

From (10.6) we make several intuitive observations: (i) as the average size of clusters increases, fg approaches DC, and (ii) as the size of clusters decreases, fg approaches fb . Figure 10.8 (top) illustrates the desired characteristics of P (fρ ) for φG showing three distinct features: (a) little or no low-frequency spectral components, (b) high-frequency spectral components that diminish with increased clustering, and (c) a spectral peak at fρ = fg . Like the pair correlation, the sharpness of the spectral peak in P (fρ ) at the green-noise principal frequency is affected by several factors. Consider first blue-noise, where the separation between minority pixels should have some variation. Here, the wavelengths of this variation, in blue-noise, should not be significantly longer than λb as this adds low-frequency spectral components to the corresponding dither pattern Ig [123], causing Ig to appear more white than blue. The same holds true for green-noise with large variations in cluster separation leading to a spectral peak at fρ = fg which is not sharp but blurred as

© 2008 by Taylor & Francis Group, LLC

293

2 g

RAPSD (m )

10.2. EDODF

(c)

1

(a) 0

fg

(b)

0

0.7017 RADIAL FREQUENCY

Figure 10.8: The (top) RAPSD of the ideal green-noise pattern with principal frequency fg , and the (bottom) anisotropy of the ideal greennoise pattern. the variation in separation adds new spectral components to Ig . This whitening effect on Ig is also created by increased variation in the size of clusters, with excessively large clusters leading to low-frequency components and excessively small clusters leading to high. In summary, the sharpest spectral peak at fg will be created when Ig is composed of round (isotropic) clusters whose variation in size is small and whose separation between nearest clusters is also isotropic with small variation. Being that isotropy is a desired property of green-noise dither patterns, the ideal green-noise pattern will have an anisotropy measure of the form of Fig. 10.8 (bottom).

10.2

EDODF

Although error-diffusion is a good generator of blue-noise, the nature of green-noise to cluster pixels makes error-diffusion inappropriate. As an alternative, Levien [75] has proposed error-diffusion with outputdependent feedback (EDODF), shown in Fig. 10.9, where a weighted

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

294

x h [n] x [n] +

Y

+

h

a

+

y [n]

Y -

+ x e [n]

+

b

Y y e [n]

Figure 10.9: The error-diffusion with output-dependent feedback algorithm. sum of the previous output pixels is used to vary the threshold – making minority pixels more likely to occur in clusters. Furthermore, the amount of clustering is controlled through the scalar constant h, the hysteresis constant, with large values of h leading to large clusters and small values of h leading to small clusters. This scheme is very different from Levien’s output-dependent feedback algorithm reviewed in Sec. 5.2.1 for reducing worm artifacts in blue-noise dither patterns. Mathematically, Levien’s algorithm is defined as follows 

y[n] =

1, (x[n] + xe [n] + xh [n]) ≥ 0 0, else,

(10.7)

where xh [n] is the hysteresis or feedback term defined as: xh [n] = h

N 

ai · y[n − i]

(10.8)

i=1



with N i=0 ai = 1 and h is the hysteresis constant. The hysteresis constant, h, acts as a tuning parameter with larger h leading to coarser output textures [75] as h increases (h > 0) or decreases (h < 0) the likelihood of a minority pixel if the previous outputs were also minority pixels (Fig. 10.10). Equation (10.8) can also be written in vector notation as xh [n] = haT y[n],

(10.9)

where a = [a1 , a2 , . . . , aN ]T and y[n] = [y[n−1], y[n−2], . . . , y[n−N ]]T . The calculation of the parameters ye [n] and xe [n] remains unchanged

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

295

0.00

1.25

2.50

h

Figure 10.10: Halftone image (g = 34 ) created by error-diffusion with output-dependent feedback where the hysteresis parameter, h, is varied from 0 to 2.5. hysteresis 1/2 1/2 • 1/2 1/2 error Figure 10.11: Levein’s error and hysteresis filter combination for EDODF. in Levien’s approach. So, in summary of Levien’s EDODF, the binary output pixel y[n] is determined as 

y[n] =

1, if (x[n] + bT ye [n] + haT y[n]) ≥ 0 0, else,

(10.10)

where ye [n] = [y e [n − 1], y e [n − 2], . . . , y e [n − N ]]T such that y e [n] = y[n] − (x[n] + xe [n]) and y[n] = [y[n − 1], y[n − 2], . . . , y[n − N ]]T . When first proposed, Levien suggested the arrangement of two hysteresis and two error filter weights shown in Fig. 10.11. Coupled with a left-to-right raster scan, EDODF yields poor results due to strong diagonal texture patterns (Fig. 10.12) – making alternate scanning paths such as the serpentine raster scan mandatory. Shown in Figs. 10.13 and 10.14 are the gray-scale ramp, and image, produced using Levien’s filter arrangement with a small (h = 0.5) hysteresis constant. Shown in Figs. 10.15–10.17 are the corresponding spatial and spectral metrics. As would be expected for a small hysteresis constant, Figs. 10.15–10.17 show distinct blue-noise characteristics and only seem to show significant power at frequencies below the blue-noise principal frequency in

© 2008 by Taylor & Francis Group, LLC

296

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.12: Gray-scale ramp halftoned using two hysteresis and two error filter weights with a normal left-to-right raster scan.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

297

Figure 10.13: Gray-scale ramp halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

298

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.14: Gray-scale image halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

299

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.15: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant 1 1 h = 0.5 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

300

RAPSD

2

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

1

2

1

rc

hg

0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.16: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

301

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.17: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

302

CHAPTER 10. GREEN-NOISE DITHERING

the mid-tones near I = 12 . From inspection of the gray-scale ramp (Fig. 10.13), it is evident that this filter arrangement produces unwanted banding artifacts at intensities near 0.15 and 0.33. A slight degree of history artifacts can also be seen near extreme intensities (I near 0). Illustrating the impact of a moderate hysteresis value (h = 1.0), Figs. 10.18 and 10.19 show the resulting gray-scale ramp, and image, using Levien’s filter arrangement. Figures 10.20–10.22 show the resulting spatial and spectral metrics, and in this instance of h, dither patterns exhibit a higher degree of clustering. Note, though, that as the intensity I approaches 12 , the clusters start to touch until individual clusters become indistinguishable from one another, and due to the side-to-side raster scan, clusters form diagonal correlations that favor the horizontal direction more than the vertical. The result is a denim-type texture. For h = 1.5 (Figs. 10.23–10.27), clusters are bigger at all graylevels compared with h = 1.0, and due to the raster, form strong directional artifacts along the horizontal. The denim texture of h = 1.0 is now a pattern composed of straight lines. At h = 2.0 and h = 2.5 (Figs. 10.28–10.31, respectively), the artifacts only get worse as clusters are becoming further and further elongated along their horizontal axis with no growth along the vertical.

10.2.1

Eliminating Unwanted Textures

For the remainder of this section, we look at modifications to Levien’s EDODF that may improve the results offered by Levien’s arrangement of two hysteresis and two error filter weights. Unlike Floyd’s and Steinberg’s error-diffusion, EDODF is still relatively new and has not had the interest of researchers that error-diffusion has had. So very little work has been done to improve the visual quality of halftones generated by EDODF for a given level of robustness. While Levien’s filter arrangement exhibits some degree of history artifacts at extreme graylevels and some banding in the mid-tones, the major artifacts that these modifications are trying to remove are the strong directional textures that occur for large h. Balanced Filter Weights The strong horizontal artifacts using Levien’s filter arrangement with large h are due, in large part, to the horizontal scanning of the serpen-

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

303

Figure 10.18: Gray-scale ramp halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

304

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.19: Gray-scale image halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

305

RAPSD

6

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.20: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant 1 1 h = 1.0 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

306

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.21: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.0 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

307

RAPSD

10

RAPSD

8

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.22: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.0 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

308

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.23: Gray-scale ramp halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

309

Figure 10.24: Gray-scale image halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.5.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

310

RAPSD

6

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.25: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant 1 1 h = 1.5 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

311

RAPSD

8

RAPSD

6

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.26: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

312

RAPSD

10

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.27: Spatial and spectral statistics using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 1.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

313

Figure 10.28: Gray-scale ramp halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

314

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.29: Gray-scale image halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

315

Figure 10.30: Gray-scale ramp halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 2.5.

© 2008 by Taylor & Francis Group, LLC

316

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.31: Gray-scale image halftoned using Levien’s filter arrangement with a serpentine raster scan and a hysteresis constant h = 2.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

317

tine path. By adjusting the proportions of the two hysteresis weights, Lau and Arce [64] balance the horizontal and vertical growth of clusters to improve the isotropy measure. Listed in Table 10.1 are the horizontal and vertical hysteresis coefficients chosen empirically by visual inspection. Figures 10.32–10.46 show the ramps, images, and metrics using balanced weights with h = 0.5, 1.0, and 2.0. Shown in Fig. 10.47 is a direct comparison of Levien’s filter arrangement versus balanced weights where a continuous-tone image of constant gray-level g = 34 has been halftoned using EDODF with a hysteresis constant that varies from 0 to 3.0. While the balanced weights clearly break up the horizontal bars of Levien’s arrangement, they also introduce long, thin vertical lines for h > 2.0 that create visual artifacts that are just as objectionable as the horizontal bars. In a real printing situation, the artifacts created by these thin lines are more noticeable since many will only partially print. Shown in Fig. 10.48 is the modeled laser printer output of a halftone produced by EDODF with balanced weights and h = 2.5 using the stochastic dot model of Sec. 9.2.1 (α = 1.11, T1 = 1.00, and T2 = 2.00). These results demonstrate that EDODF has a performance limit for generating visually pleasing dither patterns inside the range 0 ≥ h < 2.5. Modified Filter Weights As an investigation of the effects of different filter weights, Fig. 10.49 shows the combination of the Floyd-Steinberg weights [40] as the hysteresis filter and the Stucki weights [117] as the error filter. Using a small hysteresis constant (h = 1/2), Fig. 10.50 shows the resulting

Table 10.1: The optimal two hysteresis weights versus h for EDODF.

ah

av •

bh

bv

h av ah h av ah h av ah

0 0.425 0.575 1 0.556 0.444 2 0.648 0.352

0.1 0.430 0.570 1.1 0.569 0.431 2.1 0.651 0.349

0.2 0.435 0.565 1.2 0.580 0.420 2.2 0.658 0.342

0.3 0.441 0.559 1.3 0.595 0.405 2.3 0.660 0.340

© 2008 by Taylor & Francis Group, LLC

0.4 0.454 0.546 1.4 0.604 0.396 2.4 0.669 0.331

0.5 0.530 0.470 1.5 0.616 0.384 2.5 0.674 0.326

0.6 0.535 0.465 1.6 0.622 0.378 2.6 0.678 0.322

0.7 0.536 0.464 1.7 0.629 0.371 2.7 0.683 0.317

0.8 0.540 0.460 1.8 0.637 0.363 2.8 0.688 0.312

0.9 0.544 0.456 1.9 0.642 0.358 2.9 0.692 0.308

1 0.556 0.444 2 0.648 0.352 3 0.695 0.305

318

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.32: Gray-scale ramp halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

319

Figure 10.33: Gray-scale image halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

320

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.34: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 0.5 for 1 1 intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

321

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.35: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

322

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.36: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

323

Figure 10.37: Gray-scale ramp halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

324

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.38: Gray-scale image halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

325

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.39: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 1.0 for 1 1 intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

326

RAPSD

8

RAPSD

8

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.40: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 1.0 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

327

RAPSD

10

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.41: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 1.0 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

328

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.42: Gray-scale ramp halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

329

Figure 10.43: Gray-scale image halftoned using balanced weights with a serpentine raster scan and a hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

330

RAPSD

6

RAPSD

6

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.44: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 2.0 for 1 1 intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

331

RAPSD

10

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.45: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 2.0 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

332

RAPSD

10

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.46: Spatial and spectral statistics using balanced weights with a serpentine raster scan and a hysteresis constant h = 2.0 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

333

0.00

1.50

3.00

h

Levien’s arrangement

0.00

1.50

3.00

h

balanced weights Figure 10.47: Halftone image (g = 34 ) created by error-diffusion with output-dependent feedback where the hysteresis parameter, h, is varied from 0 to 3.0. gray scale ramp, which exhibits a larger degree of clustering relative to using two hysteresis and two error weights with equivalent h. Due to very small variations in cluster size, this method leads to patterns with very sharp spectral peaks in P (fρ ), as seen in Figs. 10.52–10.54, but at g = 12 , the clusters form a pattern with very distinct horizontal artifacts. Filter Perturbation Filter perturbation was first discussed in sec. 5.2.1 for breaking up regular patterns in blue-noise dither patterns. It can also be used to break up the horizontal bar artifacts that occur using EDODF with Levien’s filter arrangement. Figures 10.55 and 10.56 show the gray scale ramps, and images, where 50% error has been added to the error filter (not the hysteresis filter) for a small hysteresis constant h = 0.5. This perturbation results in dither patterns with less diagonal correlation but also with some degree of spectral whitening, as can be seen in the spatial and spectral metrics of Figs. 10.57–10.59.

© 2008 by Taylor & Francis Group, LLC

334

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.48: The printed halftone images created with EDODF using balanced weights with a hysteresis constant h = 2.5 shown modeled by (left) the ideal printer model and (right) by the stochastic dot model of Sec. 9.2.1 (α = 1.11, T1 = 1.00, and T2 = 2.00).

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

335 hysteresis 1/16 5/16 3/16 7/16 • 8/42 4/42 2/42 4/42 8/42 4/42 2/42 1/42 2/42 4/42 2/42 1/42 error

Figure 10.49: Error and hysteresis filter arrangements composed of 4 hysteresis and 12 error-diffusion coefficients. For h = 1.0, Figs. 10.60–10.61 show the gray scale ramps, and images, for Levien’s arrangement where 50% perturbation has been added to the error filter. While the diagonal denim texture at I = 12 has been broken up to form an isotropic pattern (Figs. 10.62–10.64), the resulting patterns exhibit a high degree of spectral whitening. For h = 1.5 (Figs. 10.65–10.66), 50% perturbation has again been added to the error weights, breaking up the strong horizontal artifacts that occur without perturbation. With a high hysteresis value, Figs. 10.67–10.69, the corresponding spatial and spectral metrics, show less whitening than for h = 1.0. To demonstrate the perturbation of the hysteresis filter coefficients, Figs. 10.70–10.72 show the gray scale ramps, and images, using the filter arrangement of Fig. 10.49 with h = 0.5 and 30% perturbation added to both the hysteresis and error filters separately such that the  M constraints N i=1 ai = 1 and i=1 bi = 1 are always maintained. As illustrated in the spatial and spectral metrics (Figs. 10.72–10.74), perturbation succeeds at breaking up horizontal textures in the mid-tones, creating close to ideal isotropy properties. It also does so with only minor spectral whitening.

10.2.2

Edge Enhancement

By clustering minority pixels, green-noise has most of its spectral content at frequencies below that of blue-noise. This limits green-noise’s ability to preserve fine details in the output image. Green-noise will therefore require a greater degree of edge sharpening prior to halftoning. Shown in Fig. 10.75 are four images comparing edge rendition in halftones produced by blue- and green-noise using Ulichney’s edge sharpening where β = 2.0 and 3.0. Figure 10.76 makes the same comparison but with Eschbach’s and Knox’s threshold modulation.

© 2008 by Taylor & Francis Group, LLC

336

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.50: Gray-scale ramp halftoned using 4 hysteresis and 12 error weights with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

337

Figure 10.51: Gray-scale image halftoned using 4 hysteresis and 12 error weights with a serpentine raster scan and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

338

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.52: Spatial and spectral statistics using 4 hysteresis and 12 error weights with a serpentine raster scan and a hysteresis constant 1 1 h = 0.5 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

339

RAPSD

6

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.53: Spatial and spectral statistics using 4 hysteresis and 12 error weights with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

340

RAPSD

10

RAPSD

8

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.54: Spatial and spectral statistics using 4 hysteresis and 12 error weights with a serpentine raster scan and a hysteresis constant h = 0.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

341

Figure 10.55: Gray-scale ramp halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

342

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.56: Gray-scale image halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

343

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.57: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster 1 scan, and a hysteresis constant h = 0.5 for intensity levels (left) I = 32 1 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

344

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.58: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 0.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

345

2 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

fg 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.59: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 0.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

346

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.60: Gray-scale ramp halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

347

Figure 10.61: Gray-scale image halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

348

4 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.62: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster 1 scan, and a hysteresis constant h = 1.0 for intensity levels (left) I = 32 1 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

349

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.63: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.0 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

350

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.64: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.0 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

351

Figure 10.65: Gray-scale ramp halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.5.

© 2008 by Taylor & Francis Group, LLC

352

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.66: Gray-scale image halftoned using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

353

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 ï10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.67: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster 1 scan, and a hysteresis constant h = 1.5 for intensity levels (left) I = 32 1 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

354

RAPSD

6

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.68: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

355

RAPSD

6

RAPSD

6

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.69: Spatial and spectral statistics using Levien’s arrangement with 50% perturbation added to the error weights, a serpentine raster scan, and a hysteresis constant h = 1.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

356

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.70: Gray-scale ramp halftoned using 4 hysteresis and 12 error weights with 30% perturbation added to both the error and hysteresis weights, a serpentine raster scan, and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

357

Figure 10.71: Gray-scale image halftoned using 4 hysteresis and 12 error weights with 30% perturbation added to both the error and hysteresis weights, a serpentine raster scan, and a hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

358

4 RAPSD

RAPSD

2

1

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc

hg

0 0

10 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.72: Spatial and spectral statistics using 4 hysteresis and 12 error weights with 30% perturbation added to both the error and hysteresis weights, a serpentine raster scan, and a hysteresis constant h = 0.5 1 1 for intensity levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

359

RAPSD

6

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.73: Spatial and spectral statistics using 4 hysteresis and 12 error weights with 30% perturbation added to both the error and hysteresis weights, a serpentine raster scan, and a hysteresis constant h = 0.5 for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 10. GREEN-NOISE DITHERING

360

RAPSD

10

RAPSD

6

fg 0.5 RADIAL FREQUENCY

10 0 ï10 0

PAIR CORRELATION

0.7071

ANISOTROPY

0

1 0

0.5 RADIAL FREQUENCY

2

1

rc

hg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 ï10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

//2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

hg

0 10

//2

0 2

/

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 10.74: Spatial and spectral statistics using 4 hysteresis and 12 error weights with 30% perturbation added to both the error and hysteresis weights, a serpentine raster scan, and a hysteresis constant h = 0.5 for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

361

Figure 10.75: The halftone images created using EDODF with balanced weights where Ulichney’s edge sharpening has been employed prior to halftoning with (top left) β = 0.0, (top right) β = 1.0, (bottom left) β = 2.0, and (bottom right) β = 3.0.

© 2008 by Taylor & Francis Group, LLC

362

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.76: The halftone images created using EDODF with balanced weights where Eschbach’s and Knox’s threshold modulation for edge enhancement has been employed with (top left) k = 0.0, (top right) k = 2.0, (bottom left) k = 5.0, and (bottom right) k = −2.0.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

363

a

LUT

x [n] +

Y

+

+

x h [n] y [n]

Y -

+ x e [n]

b

+ Y y e [n]

Figure 10.77: The tone-dependent hysteresis algorithm.

10.2.3

Adaptive Hysteresis

It is conceivable that a particular printer may want to use different configurations of error-diffusion with output-dependent feedback depending on the input image. In error-diffusion with output-dependent feedback, the number of variations in parameters possible is much higher, as Levien’s technique can vary in (i) the error weights, (ii) the hysteresis weights, and (iii) hysteresis parameter h. As h is a single parameter, it is computationally the most efficient. More importantly, it is this parameter that has the greatest impact on the characteristics of the resulting pattern. Unlike Wong’s technique of varying the error filter, varying h leads to varying degrees of coarseness, which, although it impacts the visual “pleasantness” of a particular pattern, it also impacts the robustness. That is, an adaptive hysteresis parameter can regulate the amount of clustering according to the input image. Tone-Dependent Hysteresis Assuming ODF creates isotropic green-noise patterns with cluster size controlled exclusively through h, optimizing ODF for a given printing process is achieved by specifying the parameter h according to the desired robustness, but as a constant, ODF may, like AM halftoning, sacrifice spatial resolution at certain gray-levels for pattern robustness at other levels. It may therefore be advantageous to employ an adaptive hysteresis parameter (Fig. 10.77) that varies according to the input gray-level. One such arrangement would be, for each gray-level, to select the minimum h such that the output tone is within a pre-specified

© 2008 by Taylor & Francis Group, LLC

364

CHAPTER 10. GREEN-NOISE DITHERING

tolerance of the input. Shown in Fig. 10.78 (left) is the optimal h using balanced filter weights that vary along with and according to h for each gray-level, where h was chosen to maximize the spatial resolution while maintaining an output intensity (modeled using the Flohr-Wiseman dot model with α = 0.97) within 50% of the input. Figure 10.78 (right) shows the resulting input versus output tone reproduction curve where the tone constraint is plotted as a dotted line. The image of Fig. 10.79 shows the resulting halftone image using tone-dependent hysteresis. Frequency-Dependent Hysteresis A second approach to adaptive hysteresis is to vary h according to the frequency content of the input image. In this scheme, the resulting halftoned image will be composed of large clusters in DC regions, where distortions are most noticeable to the human eye, and small clusters near edges where distortions are least noticeable and spatial details require small clusters in order to be preserved. Using the hysteresis values of Fig. 10.80 (white=2.5 and black=0.0), Fig. 10.81 shows the resulting halftone pattern generated with the improvement in resolution best illustrated in the close-up image of Fig. 10.82.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

365

3

HYSTERESIS LUT

2.5

2

1.5

1

0.5

0

0

0.2

0.4 0.6 INPUT TONE

0.8

1

0

0.2

0.4 0.6 INPUT TONE

0.8

1

1 0.9 0.8

OUTPUT TONE

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 10.78: (top) Look-up table values and (bottom) tone reproduction curve for tone-dependent hysteresis.

© 2008 by Taylor & Francis Group, LLC

366

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.79: Gray-scale image halftoned using the tone-dependent hysteresis algorithm with balanced weights.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

367

Figure 10.80: Hysteresis parameter values for frequency-dependent hysteresis algorithm (white=2.25 and black=0.0), where the parameter h is small near edges and high in DC regions of the input image.

© 2008 by Taylor & Francis Group, LLC

368

CHAPTER 10. GREEN-NOISE DITHERING

Figure 10.81: Gray-scale image halftoned using the frequencydependent hysteresis algorithm with balanced weights.

© 2008 by Taylor & Francis Group, LLC

10.2. EDODF

369

Figure 10.82: A close-up of the gray-scale image halftoned using the frequency-dependent hysteresis algorithm with balanced weights.

© 2008 by Taylor & Francis Group, LLC

Chapter 11 Green-Noise Masks Halftoning via green-noise masks is a process identical to halftoning via blue-noise masks (Chapter 6) with the exception that given a continuoustone, monochrome image of constant gray-level g, the resulting dither pattern has green-noise characteristics appropriate to g. Green-noise masks can, like blue, be of any size, with larger masks created by tiling edge-to-edge the original. While these masks do not offer the same flexibility as error-diffusion with output-dependent feedback using an adaptive hysteresis parameter, the stacking constraint does allow enough flexibility in the design of a mask that the size of clusters can be tuned to the reliability of a particular printing device. The green-noise mask can, therefore, be tuned to offer a much higher spatial resolution than AM halftoning but with a minimum of computational complexity; furthermore, the green-noise mask, at times, presents some advantage over error-diffusion with output-dependent feedback with respect to isotropy. Like the blue-noise mask, the green-noise mask, GN M , is constructed by first generating a set of binary dither patterns, {Ig : 0 ≤ g ≤ 1}, for each discrete gray-level g (256 levels for 8 bit gray-scale images) such that Ik ⊂ Ig for all k < g. GN M is then constructed by assigning to each pixel a threshold such that GN M [n] = min{g : Ig [n] = 1},

(11.1)

the minimum gray-level g such that the corresponding pixel in Ig is equal to 1. 371 © 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

372

11.1

BIPPCCA

The physical construction of binary dither patterns for the green-noise mask is done through BIPPCCA, the binary pattern pair correlation construction algorithm. The basic premise of BIPPCCA is to take an empty array (containing no minority pixels) and assign, to each element, a probability of that element becoming a minority pixel. BIPPCCA will then convert the most likely elements to minority pixels, one at a time, until the ratio of white to black pixels is g, the desired gray-level. The most likely element is the majority pixel with the highest probability during the current iteration, and in order for the resulting dither pattern to have the desired statistical properties (i.e. a desired pair correlation), BIPPCCA will adjust, at each iteration, the probability of each majority pixel in the array according to the current set of minority pixels. In BIPPCCA, the initial assignment of probabilities is done in an uncorrelated manner, but as each new minority pixel is added, the probabilities of all neighboring majority pixels are adjusted according to the desired pair correlation of the resulting pattern. As R(r) is a function of the radial distance between pixels, a majority pixel’s probability is increased if its radial distance from the newest minority pixel, r, corresponds to R(r) > 1 and decreased if r corresponds to R(r) < 1. As an example, consider using BIPPCCA to construct a blue-noise pattern where the pair correlation is zero for r near zero. This feature of R(r) is achieved in BIPPCCA if, with each new minority pixel, the probability of every element directly adjacent is set to zero; furthermore, as R(r) has a peak at λb , the blue-noise principal wavelength, all elements a distance λb from each new minority pixel should be increased to ensure a peak exists in the pair correlation of the final pattern. In practice, how much to increase or decrease a given probability ˜ in BIPPCCA is defined according to R(r), the pair correlation shaping ˜ function. R(r) is a user-defined function based on the desired pair correlation. Shown in Fig. 11.1 is the shaping function used by Lau et al. [67] to construct green-noise dither patterns. This function has peaks at integer multiples of λg , the green-noise principal wavelength, and valleys mid-way between. The parameter G is a tuning parameter and is shown in [67] to create visually pleasing patterns when G = 1.10. Being piecewise linear, this pair correlation shaping function is an especially simple approximation of the pair correlation of the ideal green-noise pattern for a given gray-level and cluster size, but by itself,

© 2008 by Taylor & Francis Group, LLC

SHAPING FUNCTION

11.1. BIPPCCA

373

G 1

0

0

0.5

1

1.5 2 2.5 3 3.5 RADIAL DISTANCE (hg)

4

4.5

5

˜ Figure 11.1: An isotropic shaping function, R(r), used to construct green-noise dither patterns in BIPPCCA. resulting dither patterns tend to look noisy and non-stationary. Because stationarity is a necessary property for digital halftoning [123], the most likely pixel will no longer be the majority pixel with the highest probability, but instead be the majority pixel with the highest product U [n] × CM[n], where U [n] is the probability of a given pixel and CM[n] is a function of the density of minority pixels within the surrounding area. Referred to as the concentration matrix, CM makes majority pixels more likely to become minority pixels in areas of low minority pixel concentration and less likely in areas of high. It is possible to eliminate the use of CM and still obtain stationary patterns, but to do so requires close examination of the shaping function. In BIPPCCA, the concentration of minority pixels is measured as the output after applying a low-pass filter, HLP , using circular convolution. In [67], Lau et al. construct green-noise patterns using the Gaussian filter, HLP , defined as 



−r2 HLP (r) = exp , λ2g

(11.2)

where HLP has a wide spread impulse response for large λg , where clusters are far apart and a narrow-spread impulse response for small λg , where clusters are close together. How much to increase or decrease a probability according to the minority pixel concentration is then determined by the user through a mapping of the filtered output to the concentration matrix. Figure 11.2 shows the mapping of concentration values used in [67] to determine CM, where {HLP ⊗ Ig } represents the output after filtering the binary dither pattern of the current it-

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

374

{

€

}

{

{

€

€

}

}

Figure 11.2: The mapping function used to construct the concentration matrix CM from the output after filtering Ig with the low-pass filter HLP using circular convolution. eration, Ig , with the low-pass filter HLP . In this mapping, values of {HLP ⊗ Ig } are scaled in a linear fashion such that max{HLP ⊗ Ig } → 0 and min{HLP ⊗ Ig } → 1. In summary, the steps for monochrome BIPPCCA are performed as follows, where Ig is initially an M × N array with no minority pixels 1. Create an M ×N array, U , of uniformly distributed random numbers such that U [n] ∈ (0, 1] is the probability that Ig [n] will become a minority pixel. 2. Construct the concentration matrix CM using a user-defined mapping of {HLP ⊗ Ig }, the output after filtering Ig with the low-pass filter HLP using circular convolution, and then locate the majority pixel in Ig with the highest modified probability (the majority pixel Ig [n] such that U [n] × CM[n] > U [m] × CM[m] for all m, where Ig [m] is also a majority pixel). Replace that pixel, Ig [n], with a minority pixel.

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

375

3. Given the new minority pixel, Ig [n], adjust the probability of each and every majority pixel, Ig [m], such that ˜ (U [m])new = (U [m])old × R(r),

(11.3)

where r is the minimum wrap-around distance from the majority pixel Ig [m] to the new minority pixel Ig [n]. 4. If the ratio of the total number of minority pixels to the total number of pixels in Ig is equal to g, the desired gray-level, then quit with the desired dither pattern given by Ig ; otherwise, continue at step 2. Shown in Figs. 11.3–11.23 are the resulting spatial and spectral characteristics for binary dither patterns created using BIPPCCA for intensity 1 1 levels I = 32 , 16 , 18 , 14 , 38 , and 12 with average cluster sizes of 1, 2, 4, 9, 16, 25, and 36 pixels, respectively. Noting Dr1 ,r2 (a) and A(fρ ) for all intensities and cluster sizes, BIPPCCA produces isotropic patterns with features that closely approximate the ideal green-noise model. Be˜ cause R(r) is applied solely according to the distance between pixels, isotropy is guaranteed, but BIPPCCA can create patterns which are anisotropic by applying a two-dimensional shaping function, K(r, θ), that employs the direction, θ, between pixels as well as the distance. The point process, φ3 of Fig. 3.3, demonstrates the construction of an anisotropic dither pattern.

11.1.1

Pattern Robustness Using BIPPCCA

To measure the robustness of dither patterns created by BIPPCCA, we can repeat the experiment of Sec. 9.3.3, where a dither pattern generated by BIPPCCA is printed by a Lexmark Optra T610 at 1200 dpi and the resulting variation in tone is measured and plotted. Shown in Figs. 11.24–11.28 are the plotted spatial variations in tone corresponding to gray-levels g = 0.1, 0.2, ..., 0.8, and 0.9 using average cluster sizes ¯ = 4, 9, 16, 25, and 36 pixels. Listed in Table 11.1 are the maxof M imum, minimum, mean, and variance values for each curve. Shown in Fig. 11.29 are bar graphs illustrating the total variation on a page versus cluster size where, for reference, the measured variation, for blue-noise and for AM halftoning are included on each graph. Clearly, mechanism noise becomes less apparent as the average cluster size increases [22].

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

376

2 RAPSD

RAPSD

2

1

1

fg 0

0.7071

10 0 −10 0

PAIR CORRELATION

fg 0.5 RADIAL FREQUENCY

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

10 RADIAL DISTANCE

20

0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

λg

rc 0 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.3: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 1 pixel per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

377

RAPSD

4

RAPSD

4

1

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 −10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

λg

rc 0 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.4: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 1 pixel per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

378

RAPSD

10

RAPSD

8

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

λg

rc 0 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.5: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 1 pixel per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

379

RAPSD

6

RAPSD

6

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

λg

rc 0 0

10 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.6: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 2 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

380

RAPSD

4

RAPSD

6

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 −10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.7: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 2 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

381

RAPSD

8

RAPSD

4

1

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

0

2

1

λg

rc 0 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.8: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 2 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

382

RAPSD

10

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

λg

rc 0 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.9: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 4 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

383

RAPSD

8

RAPSD

10

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

λg

rc 0 0

10 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

20

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.10: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 4 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

384

RAPSD

4

RAPSD

6

0.5 RADIAL FREQUENCY

0

0.7071

ANISOTROPY

0

10 0 −10 0

PAIR CORRELATION

1

fg

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

5 RADIAL DISTANCE

10

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

1 0

2

1

rc 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.11: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 4 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

385

RAPSD

20

RAPSD

20

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

2 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

2 0

2

1

λg

rc 0 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.12: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 9 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

386

RAPSD

20

RAPSD

20

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

2 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

15 RADIAL DISTANCE

30

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

2 0

2

1

λg

rc 0 0

15 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

30

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.13: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 9 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

387

RAPSD

10

RAPSD

16

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

1 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

2 0

2

1

rc 0

5 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 10

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.14: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 9 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

388

RAPSD

30

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

3 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

λg

rc 0 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.15: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 16 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

389

RAPSD

30

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

3 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

λg

rc 0 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.16: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 16 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

390

RAPSD

20

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

2 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

10 RADIAL DISTANCE

20

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

rc 0

10 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 20

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.17: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 16 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

391

RAPSD

30

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

3 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

rc 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.18: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 25 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

392

RAPSD

30

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

3 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

rc 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.19: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 25 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

393

RAPSD

30

RAPSD

30

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

3 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

15 RADIAL DISTANCE

30

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

3 0

2

1

rc 0

10 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 20

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.20: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 25 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

394

RAPSD

50

RAPSD

50

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

5 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

λg

rc 0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

5 0

2

1

0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

rc 0

50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.21: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 36 pixels per cluster for intensity 1 1 levels (left) I = 32 and (right) I = 16 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

395

RAPSD

50

RAPSD

50

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

5 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

5 0

2

1

rc 0

25 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 50

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.22: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 36 pixels per cluster for intensity levels (left) I = 18 and (right) I = 14 .

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

396

RAPSD

50

RAPSD

50

fg 0.5 RADIAL FREQUENCY

10 0 −10 0

PAIR CORRELATION

5 0

0.7071

ANISOTROPY

0

0.5 RADIAL FREQUENCY

2

1

rc

λg

0 0

25 RADIAL DISTANCE

50

fg 0

0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

10 0 −10

0.7071 PAIR CORRELATION

ANISOTROPY

5 0

2

1

rc 0

10 RADIAL DISTANCE

π/2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

λg

0 20

π/2

0 2

π

2

1 0 1 DIRECTIONAL DISTRIBUTION

0 2

Figure 11.23: Spatial and spectral statistics using BIPCCA to create dither patterns with an average size of 36 pixels per cluster for intensity levels (left) I = 38 and (right) I = 12 .

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

397

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 11.24: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for a halftone pattern generated by BIP¯ = 4) printed by a Lexmark Optra T610 at 1200 dpi using PCCA (M the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

398

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 11.25: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for a halftone pattern generated by BIP¯ = 9) printed by a Lexmark Optra T610 at 1200 dpi using PCCA (M the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

399

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 11.26: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for a halftone pattern generated by BIP¯ = 16) printed by a Lexmark Optra T610 at 1200 dpi using PCCA (M the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

400

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 11.27: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for a halftone pattern generated by BIP¯ = 25) printed by a Lexmark Optra T610 at 1200 dpi using PCCA (M the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

401

10

9

8

7

6

5

4

3

2

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

Figure 11.28: The resulting variations in tone along the (top) vertical and (bottom) horizontal axes for a halftone pattern generated by BIP¯ = 36) printed by a Lexmark Optra T610 at 1200 dpi using PCCA (M the printer’s default AM screen. Units of length are in inches.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

402

Table 11.1: Table listing gray-level statistics for the halftone patterns created using BIPPCCA. Levien'serror-diffusion withh=2.0 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10 BIPPCCAwith ASOC=4pixels 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10 BIPPCCAwith ASOC=9pixels 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10 BIPPCCAwith ASOC=16pixels 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10 BIPPCCAwith ASOC=25pixels 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10 BIPPCCAwith ASOC=36pixels 0.90 0.80 0.70 0.60 inputgray 0.50 levels 0.40 0.30 0.20 0.10

overall variance average 0.0002 0.9609 0.0004 0.8551 0.0012 0.6645 0.0027 0.4438 0.0057 0.1821 0.0035 0.0794 0.0015 0.0388 0.0006 0.0225 0.0006 0.0294 overall variance average 0.0001 0.9842 0.0004 0.9303 0.0029 0.7218 0.0048 0.3712 0.0046 0.1096 0.0013 0.0344 0.0005 0.0162 0.0005 0.0135 0.0006 0.0247 overall variance average 0.0001 0.9303 0.0004 0.8447 0.0012 0.7035 0.0029 0.4450 0.0050 0.1657 0.0018 0.0433 0.0008 0.0227 0.0008 0.0204 0.0006 0.0233 overall variance average 0.0001 0.9272 0.0002 0.8383 0.0007 0.7237 0.0016 0.5084 0.0034 0.2662 0.0030 0.1231 0.0022 0.0802 0.0022 0.0670 0.0017 0.0518 overall variance average 0.0002 0.9319 0.0002 0.8457 0.0004 0.7391 0.0010 0.5731 0.0020 0.3726 0.0019 0.2692 0.0022 0.1923 0.0025 0.1468 0.0018 0.0724 overall variance average 0.0001 0.9393 0.0001 0.8514 0.0005 0.7619 0.0005 0.6613 0.0010 0.5135 0.0013 0.3069 0.0018 0.2757 0.0019 0.2121 0.0013 0.1037

© 2008 by Taylor & Francis Group, LLC

min 0.9388 0.8313 0.6343 0.4084 0.0994 0.0246 0.0116 0.0080 0.0125 min 0.9712 0.9061 0.6558 0.2828 0.0466 0.0161 0.0080 0.0052 0.0105 min 0.9154 0.8289 0.6707 0.3984 0.1274 0.0256 0.0116 0.0085 0.0071 min 0.9178 0.8301 0.7082 0.4791 0.2214 0.0881 0.0585 0.0295 0.0168 min 0.9232 0.8353 0.7196 0.5446 0.3282 0.2304 0.1603 0.0925 0.0355 min 0.9303 0.8454 0.7413 0.6437 0.4824 0.2751 0.2494 0.1678 0.0680

vertical max 0.9758 0.8745 0.6907 0.4968 0.2746 0.1537 0.0955 0.0532 0.0547 vertical max 0.9978 0.9484 0.7701 0.4134 0.1465 0.0509 0.0333 0.0285 0.0512 vertical max 0.9510 0.8597 0.7346 0.4780 0.2083 0.0598 0.0369 0.0369 0.0587 vertical max 0.9478 0.8454 0.7377 0.5340 0.2974 0.1463 0.1107 0.1125 0.1230 vertical max 0.9518 0.8573 0.7474 0.5899 0.4098 0.2971 0.2131 0.1993 0.1177 vertical max 0.9562 0.8578 0.7733 0.6754 0.5394 0.3331 0.2999 0.2482 0.1541

range 0.0370 0.0432 0.0564 0.0884 0.1752 0.1291 0.0839 0.0452 0.0422

min 0.9472 0.8236 0.6119 0.3668 0.0977 0.0306 0.0120 0.0055 0.0112

range 0.0266 0.0423 0.1143 0.1306 0.0998 0.0348 0.0253 0.0233 0.0407

min 0.9750 0.9003 0.6489 0.2731 0.0354 0.0063 0.0019 0.0011 0.0066

range 0.0356 0.0308 0.0639 0.0796 0.0809 0.0343 0.0253 0.0284 0.0515

min 0.9182 0.8157 0.6594 0.3723 0.0745 0.0078 0.0032 0.0018 0.0077

range 0.0300 0.0153 0.0295 0.0549 0.0760 0.0582 0.0522 0.0830 0.1061

min 0.9170 0.8174 0.6889 0.4507 0.1824 0.0550 0.0341 0.0240 0.0233

range 0.0287 0.0220 0.0278 0.0454 0.0816 0.0667 0.0528 0.1068 0.0822

min 0.9184 0.8290 0.7132 0.5282 0.3064 0.2072 0.1359 0.0991 0.0381

range 0.0259 0.0124 0.0319 0.0317 0.0570 0.0580 0.0505 0.0804 0.0860

min 0.9322 0.8323 0.7386 0.6224 0.4696 0.2580 0.2222 0.1631 0.0751

horizontal max 0.9761 0.8877 0.7411 0.5536 0.3316 0.2133 0.1430 0.0995 0.1065 horizontal max 0.9901 0.9583 0.8423 0.5400 0.2876 0.1542 0.0957 0.1019 0.1034 horizontal max 0.9411 0.8715 0.7908 0.5824 0.3492 0.1875 0.1270 0.1158 0.1023 horizontal max 0.9369 0.8631 0.7892 0.6060 0.4085 0.2611 0.2104 0.1882 0.1489 horizontal max 0.9432 0.8632 0.7863 0.6496 0.4844 0.3724 0.3014 0.2491 0.1740 horizontal max 0.9472 0.8720 0.7985 0.7157 0.5942 0.3878 0.3722 0.3029 0.1957

range 0.0289 0.0641 0.1292 0.1868 0.2339 0.1827 0.1310 0.0940 0.0953 range 0.0151 0.0580 0.1935 0.2669 0.2521 0.1479 0.0938 0.1008 0.0967 range 0.0229 0.0557 0.1314 0.2101 0.2747 0.1798 0.1239 0.1141 0.0946 range 0.0199 0.0457 0.1003 0.1553 0.2262 0.2061 0.1764 0.1642 0.1256 range 0.0247 0.0342 0.0731 0.1214 0.1781 0.1651 0.1656 0.1500 0.1359 range 0.0150 0.0398 0.0599 0.0933 0.1245 0.1298 0.1501 0.1398 0.1206

11.1. BIPPCCA

x 10

ï3

403

Levien’s ErrorïDiffusion with h=2.0

x 10

0.5

6

5

2

0.3

0.7 0.8

0

0.25

x 10

ï3

0.5

0.9

0.75

1

0.3 0.2

0

0.8 0.9

0

0.25

x 10

ï3

0.5

0.75

1

BIPPCCA with ASOC=16 pixels

6

0.5

5

5

4

4

0.6

3

0.2 0.3 0.7 0.8 0.25

x 10

ï3

0.5

0.9

0.75

1

0

BIPPCCA with ASOC=25 pixels

5

5

4

4

0.2 0.3 0.1

0

0.25 ï3

0.5

0.8 0.9

0.75

1

BIPPCCA with ASOC=36 pixels

3

0.4

0.5

2

0.6

1

0.7 0

0.7

x 10 6

2

0.6

1

6

3

0.5

2 0.1

0.3 1 0.2 0.1 0

0.4

3

2 0.4

0

0.4

1 0.1

BIPPCCA with ASOC=9 pixels

6

0

0.7

3

0.6

1 0.2 0.1 0

0.6

0.5

4

0.4

3 2

BIPPCCA with ASOC=4 pixels

6

5 4

ï3

0.25

0.5

0.75

0.1

0.2 0.3 0.4

1

0.8 0.9 1

0

0.5 0.6 0.7

0

0.25

0.5

0.75

0.8 0.9 1

Figure 11.29: The resulting variations versus the average gray-level for green-noise halftoning. In each plot, the upper dashed lines indicate the variations using blue-noise, while the lower dashed lines indicate the variations using the printer’s default AM screen. The number appearing above each bar indicates the fraction of pixels that were white in the original halftone pattern.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

404

A special result that is also included in Fig. 11.29 is the resulting variance for Levien’s EDODF algorithm with h = 2.0. This particular result shows that the hysteresis constant is far too low for the amount of distortion in the Optra T610’s printing mechanism, but as shown in Fig. 10.47 for this arrangement of two hysteresis and two error filter weights, h > 2.0 does little to increase robustness, only creating strong horizontal artifacts. BIPPCCA’s advantage over EDODF is, therefore, that it can create clusters of virtually unlimited size, allowing BIPPCCA to create the 36-pixel cluster capable of maintaining a total variance across the page below 0.002 at any gray-level.

11.1.2

Constructing the Green-Noise Mask

The basic premise for constructing a green-noise mask (GNM) is to first construct a set of binary dither patterns, {Ig : 0 ≤ g ≤ 1}, using BIP˜ g (r) : 0 ≤ g ≤ 1}. PCA from a set of user defined shaping functions, {R In order for {Ig : 0 ≤ g ≤ 1} to satisfy the stacking constraint, BIPPCCA must, like BIPPSMA, be constrained in several ways. Initially, a dither pattern Ik can be constructed given Ig such that Ik ⊂ Ig by constraining step 2 of BIPPCCA to only consider those pixels of Ik for which Ig [n] = 1, and finally, a dither pattern Ig can be constructed given Ik such that Ik ⊂ Ig , if, in step 1, Ig is initialized to Ik and each value U [n] is scaled according to the location of minority pixels in Ig ˜ θ). In addition to the above modifications to BIPas defined by K(r, ˜ θ), must also observe the stacking PCCA, the shaping function, K(r, constraint as minority pixel clusters in pattern Ig cannot decrease in size from those in Ik . ˜ g (r) : 0 ≤ g ≤ 1} are of the Assuming all shaping functions of {R ˜ g (r) : 0 ≤ g ≤ 1} for the construction of a form of Fig. 11.1, defining {R given mask is done by specifying, for each gray-level, a desired cluster ˜ g (r) to observe size to form the set {Mg : 0 ≤ g ≤ 1}. In order for R the stacking constraint, {Mg : 0 ≤ g ≤ 1} should be monotonically increasing with increasing g for g < 12 and monotonically decreasing with increasing g for g > 12 . The shaping functions are then defined by the desired principal wavelength such that ⎧ ⎨



¯g , for 0 < g ≤ 1/2 D/ (g)/M  λg = ⎩ D/ (1 − g)/M ¯g , for 1/2 < g ≤ 1.

(11.4)

˜ g (r) : 0 ≤ g ≤ 1}, the steps for constructing the greenGiven {R

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

405

noise masks are described as follows 1. Given g1 = 0 and gN = 1, define Ig1 as an all-zero matrix and IgN as an all-one matrix. 2. Define the sequence {gki : i = 1, 2, . . . , N } as a rearrangement or re-ordering of the sequence {gi : i = 1, 2, . . . , N } such that every gray-level gi appears once and only once in the sequence {gki : i = 1, 2, . . . , N } with gk1 = 0 and gk2 = 1 (k1 = 1 and k2 = N ). 3. Set j = 3. 4. Construct the dither pattern Igkj under the stacking constraint for all patterns Igki for i = 1, 2, . . . , j − 1 (for all patterns that have been generated up to the current iteration). 5. Set j = j + 1. If j = N + 1, the process is complete; otherwise, continue at step 4. Note that, like the blue-noise mask, the patterns forming the set {Ig : 0 ≤ g ≤ 1} need not be generated in any particular order and that generating patterns in a random order may offer better results than by constructing Ig by consecutive gray-levels [68]. Based on the success of the sequence {0, 1, 14 , 34 , 12 , 18 , 38 , 58 , 78 , . . .} for constructing blue-noise masks, this same sequence will be used here. Shown in Fig. 11.30 are the four sets of desired cluster sizes where in sets (a–c), the average cluster size remains constant for intensities I ¯ constant for varying such that 0 ≤ I ≤ 14 . Referring back to (10.4), M g is the behavior of an FM halftoning technique. These regions will, therefore, be referred to as the FM regions in the corresponding masks (Fig. 11.31). The intensity region 14 < I ≤ 12 is the AM region where the average size of clusters increases linearly with I such that the total number of clusters remains constant. The behavior that the sets (a–c) lead to in their corresponding masks is that as I increases from 0 to 1 , the number of clusters (not the size) per unit area increases with 4 I, but at I = 14 , clusters begin to grow without the introduction of new clusters. The principal wavelength remains constant in these AM regions, leading to a clearly visible ring in the spectral domain, as seen in Fig. 11.32, where the magnitudes of the Fourier transform of each mask are illustrated.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

406

Mg

16

Mg

16

4

8

4

2 0

0

0.25

0.5 g

0.75

0

1

0

0.25

(a)

0.5 g

0.75

1

0.75

1

(b)

16

16

12

Mg

Mg

10

6 5

0

0

0.25

0.5 g

0.75

(c)

1

1 0

0

0.25

0.5 g

(d)

Figure 11.30: Four sets, {Mg : 0 ≤ g ≤ 1}, that define the shaping functions for constructing green-noise masks with sets (a–c) used to construct symmetric masks with increasing coarseness and set (d) to construct a non-symmetric mask where the white minority pixel clusters are larger than the corresponding black clusters.

© 2008 by Taylor & Francis Group, LLC

11.1. BIPPCCA

407

(a)

(b)

(c)

(d)

Figure 11.31: Four constructed green-noise masks.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 11. GREEN-NOISE MASKS

408

(a)

(b)

(c)

(d)

Figure 11.32: The magnitudes of the Fourier transforms of the masks of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

409

In mask (d), the minority pixel clusters formed by white pixels for gray-level g < 12 are larger in size than those formed by black pixels for gray-level 1 − g. Such a mask would be employed in a printer that prints black minority pixels with greater reliability than white minority pixels. The difference in principal wavelengths for g < 12 with those for which g > 12 accounts for the appearance of two rings in the Fourier domain. For further comparison of masks (a–d), Table 11.2 lists the ¯ , λg , and fg for each mask desired and the resulting parameters M for several gray-levels, and Figs. 11.33–11.40 show the corresponding gray-scale ramps and images. Figures 11.41–11.44 show the resulting gray scale images where Ulichney’s edge sharpening (β = 3.0) has been applied prior to halftoning.

11.2

Optimal Green-Noise Masks

Since the green-noise model allows the choice of various parameters, how one determines which parameter values are optimal depends upon the specific printing mechanism as well as the subjective evaluation of the subsequent print quality [22]. One important aspect to designing an optimal green-noise mask is the cluster size required to reduce the spatial variation in tone below a pre-specified tolerance. Using Table 11.1 as a reference, reducing the total variation in tone to that of the default AM screen requires that the green-noise mask for this printer must have a cluster size of at least 5 × 5 for lighter shades and 6 × 6 for darker shades.

© 2008 by Taylor & Francis Group, LLC

410

CHAPTER 11. GREEN-NOISE MASKS

Table 11.2: Table showing the average number of pixels per cluster, ¯ ; the principal wavelength, λg ; and the principal frequency, fg , for M masks (a–d). Shown in parantheses are the ideal parameters used to construct Rg (r).

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

411

Figure 11.33: Gray-scale ramp halftoned using mask (a) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

412

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.34: Gray-scale image halftoned using mask (a) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

413

Figure 11.35: Gray-scale ramp halftoned using mask (b) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

414

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.36: Gray-scale image halftoned using mask (b) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

415

Figure 11.37: Gray-scale ramp halftoned using mask (c) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

416

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.38: Gray-scale image halftoned using mask (c) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

417

Figure 11.39: Gray-scale ramp halftoned using mask (d) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

418

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.40: Gray-scale image halftoned using mask (d) of Fig. 11.31.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

419

Figure 11.41: Gray-scale image halftoned using mask (a) of Fig. 11.31, where Ulichney’s edge sharpening (β = 3.0) has been applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

420

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.42: Gray-scale image halftoned using mask (b) of Fig. 11.31, where Ulichney’s edge sharpening (β = 3.0) has been applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

11.2. OPTIMAL GREEN-NOISE MASKS

421

Figure 11.43: Gray-scale image halftoned using mask (c) of Fig. 11.31, where Ulichney’s edge sharpening (β = 3.0) has been applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

422

CHAPTER 11. GREEN-NOISE MASKS

Figure 11.44: Gray-scale image halftoned using mask (d) of Fig. 11.31, where Ulichney’s edge sharpening (β = 3.0) has been applied prior to halftoning.

© 2008 by Taylor & Francis Group, LLC

Chapter 12 Color Printing A major advantage of using stochastic halftoning techniques such as blue-noise over AM halftoning is the absence of moir´e when superimposing the halftone patterns of cyan, magenta, yellow, and black – eliminating the need for screen angles and screen rulings [74]. FM halftoning has even been shown to increase the tolerances for screen registration in commercial presses [104], but even with these properties, the increased susceptibility of isolated dots to distortion from process variation, as seen in monochrome halftoning (Chapter 9) which also occurs for color halftoning, makes AM halftoning the preferred technique in many devices. Just as green-noise has been tuned to the constraints of the printed dot in monochrome halftoning, green-noise can also be tuned to the constraints of the printed dots in color halftoning. In this chapter, we establish a general framework for color halftoning with green-noise that not only regulates the clustering of dots within a specific color or channel, but also regulates the amount of clustering across channels. Printers can, therefore, be tuned to either increase or decrease the amount of dot overlap between dots of different colors.

12.1

Generalized Error-diffusion

In Chapter 5, we reviewed the process error-diffusion and showed how it could be used to create monochrome blue-noise patterns. In Chapter 10, we extended the capabilities of error-diffusion by using Levien’s error-diffusion with output-dependent feedback to create monochrome green-noise patterns. We now consider the C-channel (colors) case, 423 © 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

424

where an output pixel is not the binary pixel y[n] but the C-dimensional vector y [n] such that ⎡

y [n] =

⎢ ⎢ ⎢ ⎢ ⎣

⎤

y1 [n] y2 [n] .. .

⎥ ⎥ ⎥, ⎥ ⎦

(12.1)

yC [n] where yi [n] is the binary output pixel of color i. Assuming all C channels are halftoned independently, the binary output pixel yi [n] is determined as  1, if (xi [n] + xei [n] + xhi [n]) ≥ 0 (12.2) yi [n] = 0, else, where xei [n] and xhi [n] are the error and hysteresis terms, respectively, for the ith color. The error term, being a vector, is calculated as ⎡

bT 0 1 ⎢ T ⎢ 0 b2 ⎢ x e [n] = ⎢ .. .. ⎣ . . 0 0 e x e [n] = BY [n],

··· ··· ...

0 0 .. .

· · · bT C

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

y1e [n] y2e [n] .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

yCe [n] (12.3)

where bi are the filter weights regulating the diffusion of error in the ith channel and yei [n] is the vector [yei [n−1], yei [n−2], . . . , yei [n−N ]]T composed exclusively from errors in channel i such that yei [n] = yi [n] − (xi [n] + xei [n]). The hysteresis term, xh [n], also a vector, is calculated as ⎡ ⎤⎡ T ⎤⎡ ⎤ h1 0 · · · 0 a1 0 · · · 0 y1 [n] ⎢ ⎥⎢ ⎥⎢ ⎥ T ⎢ 0 h2 · · · 0 ⎥ ⎢ 0 a2 · · · 0 ⎥ ⎢ y2 [n] ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x h [n] = ⎢ .. .. . . . .. . . . .. ⎥ ⎢ . ⎢ ⎥ . .. ⎥ . .. ⎥ ⎣ . ⎦ ⎣ .. ⎦⎣ . . . ⎦ 0 0 · · · hC x h [n] = HAY[n],

0

0

· · · aT C

yC [n] (12.4)

where ai are the filter weights and hi is the hysteresis constant that regulates the diffusion of feedback in the ith channel. Generalized even further, (12.3) becomes ⎡

x e [n] =

⎢ ⎢ ⎢ ⎢ ⎣

bT 1,1 bT 2,1 .. . bT C,1

© 2008 by Taylor & Francis Group, LLC

T bT 1,2 · · · b1,C T b2,2 · · · bT 2,C .. .. ... . . T T bC,2 · · · bC,C

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

y1e [n] y2e [n] .. . yCe [n]

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

(12.5)

12.1. GENERALIZED ERROR-DIFFUSION

425

where quantization errors can now be diffused between channels through bi,j , the error filter weights that regulate the diffusion of error from channel j to channel i; furthermore, (12.4) becomes ⎡

x h [n] =

⎢ ⎢ ⎢ ⎢ ⎣

h1,1 h2,1 .. . hC,1

h1,2 · · · h1,C h2,2 · · · h2,C .. .. .. . . . hC,2 · · · hC,C

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

aT 1,1 aT 2,1 .. . aT C,1

T aT 1,2 · · · a1,C T aT 2,2 · · · a2,C .. .. .. . . . T T aC,2 · · · aC,C

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

y1 [n] y2 [n] .. .

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

yC [n]

where the previous outputs of channel j can impact all other channels where hi,j = 0 for i = j as ai,j and hi,j regulates the diffusion of feedback from channel j to i. To include Eschbach’s and Knox’s [34] threshold modulation into color halftoning, we redefine the accumulated input pixel as xa [n] = x[n] + xf [n] + xe [n] + xh [n],

(12.6)

where xf [n] is the feed-through term defined as xf [n] = K x[n] such that ⎡

K =

⎢ ⎢ ⎢ ⎢ ⎣

k1 0 0 k2 .. .. . . 0 0

··· ··· ...

0 0 .. .

· · · kC

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(12.7)

We can also allow inputs of different channels to impact other channels by setting ⎡

K =

⎢ ⎢ ⎢ ⎢ ⎣

k1,1 k2,1 .. . kC,1

k1,2 · · · k1,C k2,2 · · · k2,C .. .. ... . . kC,2 · · · kC,C

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(12.8)

Before concluding this section, we make one last, but significant, modification to error-diffusion (Fig. 12.1) by first defining the thresholding function T(·) as y [n] = T( x[n] + x e [n] + x h [n]) = T(x a [n]),

(12.9)

where x a [n] = x[n] + x e [n] + x h [n] is the accumulated input pixel. The C × C interference matrix S is added to (12.9) as y [n] = T(Sx a [n])

© 2008 by Taylor & Francis Group, LLC

(12.10)

CHAPTER 12. COLOR PRINTING

426

x h [n] x [n] +

Y

+

+ Y

x a [n]

H S

A y [n]

T( ) -

+ x e [n]

+ Y

B

y e [n]

Figure 12.1: The generalized error-diffusion algorithm. such that Si,j is the influence on the thresholding function of color i by the accumulated input of color j. The effect of Si,j is to increase (Si,j > 0) or decrease (Si,j < 0) the likelihood of a minority pixel at yi [n] based on the likelihood of a minority pixel at yj [n]. Finally, we summarize error-diffusion by the generalized error-diffusion equation y [n] = T (S( x[n] + BYe [n] + HAY[n])) .

(12.11)

Shown in Fig. 12.2 is the resulting CMYK dither pattern created by halftoning a 72 × 216 pixel color image of constant color value x[n] = [ 18 , 18 , 18 , 18 ]T using Floyd’s and Steinberg’s error filter weights with no hysteresis and no dependencies between colors. Shown in Fig. 12.3 are the resulting CMYK dither patterns created using generalized errordiffusion with balanced weights and (top) a small hysteresis constant (h = 0.5), (middle) a medium hysteresis constant (h = 1.0), and (bottom) a large hysteresis constant (h = 2.0). Before halftoning, low-level white-noise (V AR = 0.01) was added to the first scan line of the original color image in order to minimize edge effects and also to unsynchronize the resulting dither patterns. Since, in this configuration, where all colors are halftoned exactly the same way and with each color of the original image identical to the other colors, the resulting pattern of each color will also be identical (synchronized) to the other patterns. So adding a single or even a few lines of low-level noise eliminates this synchronization between colors; furthermore, adding several columns of white-noise also minimizes edge effects. In this book, CMYK dither patterns created by generalized error-diffusion are the result of using a serpentine raster scan on a continuous-tone image where low-level white-noise has been added to both the edge rows and columns. The halftoned images are then cropped to exclude those same rows and columns.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

427

Figure 12.2: (See color insert following page 480.) CMYK dither pattern created via generalized error-diffusion using Floyd’s and Steinberg’s error filter on a serpentine raster with no diffusion between colors. For a statistical analysis of the resulting CMYK dither patterns, Fig. 12.4 shows four plots labeled cyan, magenta, yellow, and black corresponding to the CMYK dither pattern of Fig. 12.2. Shown in the first plot (labeled cyan) is the pair correlation between colors cyan versus cyan (Rc,c (r)), cyan versus magenta (Rc,m (r)), cyan versus yellow (Rc,y (r)), and cyan versus black (Rc,k (r)). The small diamonds placed along the horizontal axis indicate the principal wavelengths and cluster radii for Rc,c (r). As would be expected for a monochrome image, the pair correlation Rc,c (r) exhibits blue-noise characteristics as the pair correlation shows: (i) Rc,c (r) < 1 for r near zero, (ii) a frequent occurrence of the inter-point distance λg , and (iii) a decreasing influence with increasing r. Having no diffusion between colors and zero interference (S = I, the identity matrix), the pair correlations between channels are predominantly flat as minority pixels of color i have no influence on minority pixels of color j. The remaining plots show similar relationships for colors magenta, yellow, and black. Shown in Fig. 12.5 are the resulting pair correlations to Fig. 12.3 (top) using balanced weights with a low hysteresis constant h = 0.5, no diffusion between colors and zero interference. With a low hysteresis constant, this scheme generates blue-noise patterns very similar to those generated using the Floyd-Steinberg filter weights. Figure 12.6 (Fig. 12.3 (middle)) shows balanced weights with a medium hysteresis constant h = 1.0, where the patterns begin to exhibit clustering as the average size of a minority pixel cluster is 1.95 pixels. In each color, the pair correlation exhibits strong green-noise characteristics as each plot

© 2008 by Taylor & Francis Group, LLC

428

CHAPTER 12. COLOR PRINTING

Figure 12.3: (See color insert following page 480.) CMYK dither patterns created via generalized error-diffusion with no diffusion between colors and (top) a small hysteresis constant h = 0.5, (middle) a medium hysteresis constant h = 1.0, and (bottom) a large hysteresis constant h = 2.0 using balanced weights and a serpentine raster scan.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

429

2 CYAN

Rc,c(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.4: The pair correlations for CMYK dither patterns created via generalized error-diffusion using Floyd’s and Steinberg’s error filter on a serpentine raster with no diffusion between colors.

© 2008 by Taylor & Francis Group, LLC

430

CHAPTER 12. COLOR PRINTING

shows: (i) clustering as indicated by Ri,i (r) ≥ 1 for r ≤ rc (rc = 0.79), (ii) a frequent occurrence of the inter-cluster distance λg = 3.95, and (iii) a decreasing influence with increasing r. As before with zero influence and no diffusion between colors, the pair correlations between colors remain predominantly flat for all r. Figure 12.7 (Fig. 12.3 (bottom)) shows balanced weights with a large hysteresis constant h = 2.0, where the patterns exhibit a great degree of clustering as the average size of a minority pixel cluster is 5.38 pixels. Shown in Figs. 12.8–12.11 are the CMYK images corresponding to the patterns of Figs. 12.2 and 12.3. The dither patterns of Fig. 12.12 illustrate the effects of S, the interference matrix, on generalized error-diffusion using Floyd’s and Steinberg’s error filter with Fig. 12.12 (top) showing the case where S is the matrix defined by Si,j = 1 for i = j and Si,j = −0.2 for i = j. In this specific instance, S has the effect of minimizing the superposition of minority pixels of different colors. That is, given that a cyan pixel is very likely to be printed, minority pixels for magenta, yellow, and black are less likely to be printed at that same pixel location. This behavior is illustrated in the pair correlations of Fig. 12.13, where Ri,j < 1 for Ri,i > 1. For comparison, Fig. 12.12 (middle) shows the case where S is the identity matrix (no interference) with a flat pair correlation (Fig. 12.4) between minority pixels of different colors. Figure 12.12 (bottom) shows the case where S is the matrix defined by Si,j = 1 for i = j and Si,j = +0.2 for i = j. Here, the effect of S is to increase the superposition of minority pixels such that a minority pixel of color i with a high likelihood of being printed makes a minority pixel of any color j more likely (Fig. 12.14). The dither patterns of Fig. 12.15 illustrate the effects of S, the interference matrix, on generalized error-diffusion using balanced weights and a small hysteresis constant (h = 0.5). With a small hysteresis, these patterns follow the same behavior as seen with Floyd’s and Steinberg’s error filter. Figures 12.16 and 12.17 show the corresponding pair correlations that follow closely with those of Figs.12.13 and 12.14. Shown in Figs. 12.18 and 12.19 are the CMYK images corresponding to the patterns of Fig. 12.15. With a medium (Figs. 12.20–12.24) or large (Figs. 12.25–12.29) hysteresis constant, Si,j has a greater impact on the amount of overlap. As seen in the pair correlations, Si,j = −0.2 results is a greater peak in Ri,j for Ri,i < 1 while Si,j = +0.2 results is a greater peak in Ri,j for Ri,i > 1.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

431

CYAN

2 Rc,c(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.5: The pair correlations for CMYK halftone patterns using balanced weights on a serpentine raster scan; with no diffusion between colors; and a small hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

432

CYAN

2 Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

2 BLACK

R

(r)

k,k

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.6: The pair correlations for CMYK halftone patterns using balanced weights on a serpentine raster scan; with no diffusion between colors; and a medium hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

433

2 R

(r)

CYAN

c,c

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 R

(r)

m,m

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

2 BLACK

R

(r)

k,k

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.7: The pair correlations for CMYK halftone patterns using balanced weights on a serpentine raster scan; with no diffusion between colors; and a large hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

434

CHAPTER 12. COLOR PRINTING

Figure 12.8: (See color insert following page 480.) CMYK image created via generalized error-diffusion using Floyd’s and Steinberg’s error filter on a serpentine raster with no diffusion between colors.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

435

Figure 12.9: (See color insert following page 480.) CMYK image created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; and a small hysteresis constant h = 0.5.

© 2008 by Taylor & Francis Group, LLC

436

CHAPTER 12. COLOR PRINTING

Figure 12.10: (See color insert following page 480.) CMYK image created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; and a medium hysteresis constant h = 1.0.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

437

Figure 12.11: (See color insert following page 480.) CMYK image created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; and a large hysteresis constant h = 2.0.

© 2008 by Taylor & Francis Group, LLC

438

CHAPTER 12. COLOR PRINTING

Figure 12.12: (See color insert following page 480.) CMYK dither patterns created using Floyd’s and Steinberg’s error filter on a serpentine raster scan; with no diffusion between colors; and with (top) Si,j = −0.2 for i = j, (middle) Si,j = 0 for i = j, and (bottom) Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

439

2 CYAN

R

(r)

c,c

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.13: The pair correlations for CMYK dither patterns created using Floyd’s and Steinberg’s error filter on a serpentine raster scan; with no diffusion between colors; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

440

2 CYAN

R

(r)

c,c

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 R

(r)

m,m

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 R (r) y,y

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.14: The pair correlations for CMYK dither patterns created using Floyd’s and Steinberg’s error filter on a serpentine raster scan; with no diffusion between colors; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

441

Figure 12.15: (See color insert following page 480.) CMYK dither patterns created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; a small hysteresis constant h = 0.5; and (top) Si,j = −0.2 for i = j, (middle) Si,j = 0 for i = j, and (bottom) Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

442

CYAN

2 Rc,c(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 R

(r)

m,m

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.16: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a small hysteresis constant h = 0.5; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

443

2 CYAN

Rc,c(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 R

(r)

m,m

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

hg

0

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.17: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a small hysteresis constant h = 0.5; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

444

CHAPTER 12. COLOR PRINTING

Figure 12.18: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a small hysteresis constant h = 0.5; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

445

Figure 12.19: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a small hysteresis constant h = 0.5; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

446

CHAPTER 12. COLOR PRINTING

Figure 12.20: (See color insert following page 480.) CMYK dither patterns created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; a medium hysteresis constant h = 1.0; and (top) Si,j = −0.2 for i = j, (middle) Si,j = 0 for i = j, and (bottom) Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

447

CYAN

2 Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

2 BLACK

R

(r)

k,k

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.21: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a medium hysteresis constant h = 1.0; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

448

2 CYAN

Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.22: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a medium hysteresis constant h = 1.0; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

449

Figure 12.23: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a medium hysteresis constant h = 1.0; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

450

CHAPTER 12. COLOR PRINTING

Figure 12.24: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a medium hysteresis constant h = 1.0; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

451

Figure 12.25: (See color insert following page 480.) CMYK dither patterns created via generalized error-diffusion using balanced weights on a serpentine raster; with no diffusion between colors; a large hysteresis constant h = 2.0; and (top) Si,j = −0.2 for i = j, (middle) Si,j = 0 for i = j, and (bottom) Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

452

CYAN

2 Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.26: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a high hysteresis constant h = 2.0; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

453

CYAN

2 Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.27: The pair correlations for CMYK halftone patterns created using balanced weights on a serpentine raster scan; with no diffusion between colors; a high hysteresis constant h = 2.0; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

454

CHAPTER 12. COLOR PRINTING

Figure 12.28: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a large hysteresis constant h = 2.0; and Si,j = −0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

12.1. GENERALIZED ERROR-DIFFUSION

455

Figure 12.29: (See color insert following page 480.) CMYK image with no diffusion between colors created using balanced weights on a serpentine raster scan; with no diffusion between colors; a large hysteresis constant h = 2.0; and Si,j = +0.2 for i = j.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

456

12.2

Multi-Channel Green-Noise Masks

The green-noise mask is a novel approach to dither array screening where a continuous-tone image is converted to a binary halftone image by performing a pixel-wise comparison between the original and the dither array or mask. Previously, halftoning with green-noise has implied error-diffusion-based methods that, although are tunable (capable of creating halftone patterns with large clusters for printers with high dot-gain characteristics and small clusters for printers with low dot-gain characteristics), carry a high computational cost. Now, through the use of a green-noise mask, halftoning can create a stochastic patterning of dots with adjustable coarseness but with the same computational freedom as ordered-dither halftoning schemes. Introduced in Chapter 10 for monochrome images, the greennoise mask is defined by the set {Ig : 0 ≤ g ≤ 1} of M × N binary green-noise dither patterns with one pattern, Ig , corresponding to each possible discrete gray-level g (256 patterns for 8 bit gray-scale images). This set satisfies the stacking constraint that for any two gray-levels k and g with k < g, Ik ⊂ Ig (if Ik [m, n] = 1, then Ig [m, n] = 1). As a consequence, a pixel of the M × N dither array or mask DA[m, n] is defined simply as the minimum g for which Ig [m, n] = 1. The size parameters M and N are arbitrary integers with larger masks constructed by tiling edge-to-edge the original M × N mask such that the output pixel, y[m, n], after halftoning the input pixel, x[m, n], is defined as y[m, n] = T (x[m, n] − DA[modM (m), modN (n)]) , (12.12) where T is the thresholding function of (12.9). For color halftoning, the multi-channel green-noise mask is defined by the set {Ic,g : c = 1, 2, . . . , C and 0 ≤ g ≤ 1}, where Ic,g is the binary green-noise dither pattern for color c and intensity level g (for 24-bit RGB color this corresponds to 256 patterns per channel or 3 × 256 total patterns). Like the monochrome set, this set must also satisfy the stacking constraint, but only within a given color such that Ic,g [m, n] = 1 if Ic,k [m, n] = 1 for color c and intensity levels k and g with k < g. A pixel, DA[m, n], of the multi-channel green-noise mask is therefore defined as ⎡ ⎤ min{g : I1,g [m, n] = 1} ⎢ ⎥ ⎢ min{g : I2,g [m, n] = 1} ⎥ ⎢ ⎥, DA[m, n] = ⎢ (12.13) .. ⎥ ⎣ ⎦ . min{g : IC,g [m, n] = 1}

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

457

where the output after halftoning is defined by 



y [m, n] = T x[m, n] − DA[mod M (m), modN (n)] . (12.14)

12.2.1

Color BIPPCCA

The physical construction of binary dither patterns for the multi-channel green-noise mask is done through VBIPPCCA, the vector binary pair correlation construction algorithm. In color, a binary dither pattern representing the color g = [g1 , g2 , . . . , gC ]T is defined by the set of monochrome images {I1,g1 , I2,g2 , . . . , IC,gC }, where Ii,gi is the binary dither pattern corresponding to the ith color with intensity gi . VBIPPCCA constructs these monochrome images according to the previous algorithm, but unlike BIPPCCA, when a minority pixel is added to the ith color, the probabilities corresponding to majority pixels of color j are adjusted according to Rj,i (r), the desired pair correlation between minority pixels of colors j and i. So, for a CMYK dither pattern, for each minority cyan pixel added, VBIPPCCA will make use of the ˜ c,c (r), R ˜ m,c (r), R ˜ y,c (r), and R ˜ k,c (r) to user-defined shaping functions R adjust the probabilities of majority pixels in the cyan, magenta, yellow, and black colors, respectively. Because stationarity is also a desired property for digital color halftoning, VBIPPCCA will apply CM just as in the monochrome case with each color filtered independently of the others. Returning to the CMYK case, this implies that the maximum likely majority pixel of the cyan color is the majority pixel of the cyan color with the highest product Uc [m, n] × CMc [m, n], where Uc [m, n] is the probability array for cyan pixels and CMc [m, n] is the concentration matrix formed by applying a user-defined mapping to the concentration of minority cyan pixels. In summary, the steps for VBIPPCCA are performed as follows, where {Ii,gi : i = 1, 2, . . . , C} is the initial set of empty M × N arrays 1. Create a set of M × N arrays, {U1 , U2 , . . . , UC }, of uniformly distributed random numbers such that Ui [m, n] ∈ (0, 1] is the probability that Ii,gi [m, n] will become a minority pixel. 2. For each pattern Ii,gi , where the ratio of the total number of minority pixels to the total number of pixels is less than gi :

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

458

(a) Construct the concentration matrix CMi using a user-defined mapping of {HLP ⊗ Ii,gi }, the output after filtering Ii,gi with the low-pass filter HLP using circular convolution. (b) Locate the majority pixel in Ii,gi with the highest modified probability (the majority pixel Ii,gi [m, n] such that Ui [m, n]× CMi [m, n] > Ui [o, p] × CMi [o, p] for all 1 ≤ o ≤ M and 1 ≤ p ≤ N , where Ii,gi [o, p] is also a majority pixel), and replace that pixel, Ii,gi [m, n], with a minority pixel. (c) Given the new minority pixel, Ii,gi [m, n], adjust the probability of each and every majority pixel, Il,gl [o, p] for l = 1, 2, . . . , C, such that ˜ l,i (r), (Ul [o, p])new = (Ul [o, p])old × R

(12.15)

where r is the minimum wrap-around distance from the majority pixel Il,gl [o, p] to the new minority pixel Ii,gi [m, n]. 3. If, for all colors i, the ratio of the total number of minority pixels to the total number of pixels in Ii,gi is equal to gi , the desired intensity of color i, then quit with the desired color dither pattern given by the set {Ii,gi : i = 1, 2, . . . , C}; otherwise, continue at step 2b. Like BIPPCCA, the above algorithm is not suited to the design of multi-channel green-noise masks as the stacking constraint will not be satisfied for all patterns. VBIPPCCA must, therefore, satisfy the same constraints as BIPPCCA in order to be used for mask construction. The first of these two constraints, Ii,gi ⊂ Ii,ki , is satisfied by first initializing Ii,gi to Ii,ki and applying step 2c for each and every minority pixel in Ii,ki . VBIPPCCA can then continue at step 3. The second constraint is satisfied in step 2b of VBIPPCCA when locating the maximum likely majority pixel in color i by considering only those majority pixels in Ii,gi that correspond to minority pixels in the constructed patterns, Ii,li , for which l > g. Note that, with these constraints, patterns of the multichannel green-noise mask can be constructed in any order, and that the order, to which patterns of any color i are constructed, need not be the same as any other color j. Before constructing masks, Fig. 12.30 shows a set of pair cor˜ i,i (r) shapes the pair relation shaping functions where the function R ˜ i,j (r) correlation between pixels of the same color and the function R

© 2008 by Taylor & Francis Group, LLC

SHAPING FUNCTION

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

459

G 1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

RADIAL DISTANCE (h ) g

˜ i,i (r) (solid line) Figure 12.30: The pair correlation shaping functions, R ˜ i,j (r) (dashed line), used to construct color green-noise with prinand R cipal wavelength λg and decreased overlap of minority pixels of different colors. shapes the pair correlation between pixels of different colors. This pair of shaping functions is used to reduce the amount of overlap between pixels of different colors (Fig. 12.31 (top)). Using this same pair of ˜ i,j (r) = 1 for all r, patterns with no corshaping functions, but with R relation between channels (Fig. 12.31 (middle)) can be constructed. To create a pattern where the overlapping of pixels of different colors is ˜ i,j (r) is set to have the same shape as R ˜ i,i (r) increased, the function R (Fig. 12.31 (bottom)). These patterns of Fig. 12.31, generated by VBIPPCCA, were constructed to represent a 72 × 216 pixel input image of constant color x[n] = [ 18 , 18 , 18 , 18 ]T with an average of 5 pixels per cluster (λg = 6.32 pixels). The statistical measures of the spatial relationships between pixels for these three patterns are shown in Figs. 12.32–12.34. The results, shown here, demonstrate VBIPPCCA’s ability to capture the same spatial relationships between pixels as those created in Fig. 12.25 via generalized error-diffusion. The key is in the shaping functions, and through these shaping functions, the same relationships between minority pixels can be encouraged in the design of green-noise masks. Mask design can be seen in Fig. 12.35, where the three design criteria: (1) decreased, (2) uncorrelated, and (3) increased pixel overlap are employed in masks (a), (b) and (c), respectively. Mask (d) is a special mask designed more to demonstrate the range of possibilities for dither array generation. In this instance, the colors cyan and yellow

© 2008 by Taylor & Francis Group, LLC

460

CHAPTER 12. COLOR PRINTING

Figure 12.31: (See color insert following page 480.) CMYK dither patterns constructed by VBIPPCCA having (top) decreased overlap, (middle) uncorrelated overlap, and (bottom) increased overlap.

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

461

2 CYAN

Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.32: The pair correlations for CMYK halftone patterns created using VBIPPCCA with decreased overlap.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

462

2 R

(r)

CYAN

c,c

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 Rm,m(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 R (r) y,y

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

BLACK

2 Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.33: The pair correlations for CMYK halftone patterns created using VBIPPCCA with uncorrelated overlap.

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

463

2 CYAN

Rc,c(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

MAGENTA

2 R

(r)

m,m

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

YELLOW

2 Ry,y(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

2 BLACK

Rk,k(r)

1 rc

0

0

hg

5 10 15 RADIAL DISTANCE (pixels)

20

Figure 12.34: The pair correlations for CMYK halftone patterns created using VBIPPCCA with increased overlap.

© 2008 by Taylor & Francis Group, LLC

CHAPTER 12. COLOR PRINTING

464

(a)

(b)

(c)

(d)

Figure 12.35: (See color insert following page 480.) CMYK green-noise masks constructed from VBIPPCCA to (a) minimize dot overlap, (b) have uncorrelated overlap, (c) maximize overlap, and (d) maximize overlap between cyan and yellow. are designed to overlap while not overlapping with black or magenta. These colors, black and magenta, are uncorrelated with respect to each other. The CMYK color scales shown in Fig. 12.36 are given to further illustrate the clustering behavior of each mask. By design, each mask has an average cluster size of 2 pixels at extreme gray-levels (g = 0, 1) and an average cluster size of 12 pixels at g = 12 . Shown in Figs. 12.37– 12.40 are the CMYK images halftoned via the masks of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

(a)

(b)

(c)

465

(d)

Figure 12.36: (See color insert following page 480.) CMYK gradient ramps halftoned using the masks of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

466

CHAPTER 12. COLOR PRINTING

Figure 12.37: (See color insert following page 480.) CMYK image halftoned using mask (a) of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

467

Figure 12.38: (See color insert following page 480.) CMYK image halftoned using mask (b) of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

468

CHAPTER 12. COLOR PRINTING

Figure 12.39: (See color insert following page 480.) CMYK image halftoned using mask (c) of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

12.2. MULTI-CHANNEL GREEN-NOISE MASKS

469

Figure 12.40: (See color insert following page 480.) CMYK image halftoned using mask (d) of Fig. 12.35.

© 2008 by Taylor & Francis Group, LLC

Chapter 13 Stochastic Moir´ e As described in Chapter 2, amplitude modulated (AM) halftoning refers to algorithms that create the illusion of continuous-tone in bi-level imaging devices by producing a regular pattern of round dots that vary in size according to tone such that dark shades of gray are represented by large printed dots and light shades of gray are represented by small. In color printers, continuous shades of color are produced by superimposing the halftone patterns of cyan, magenta, yellow, and black inks. While aligning each AM grid, such that the round dots of each color are superimposed directly on top of one another (a process referred to as dot-on-dot), produces a full spectrum of colors [53], even slight misregistration (misalignment of the grids) can drastically degrade the visual quality of the printed image. Instead of overlapping dots, each AM screen is typically given its own orientation or screen angle. The problem now is not the distortion created by misregistration, but rather that caused by the introduction of a visual interference pattern created by superimposing two or more regular patterns, a phenomenon known as moir´e. Looking at just two patterns, Fig. 13.1 shows the moir´e patterns created by superimposing a regular grid at a screen angle of 45 degrees with grids at angles of (a) 30 degrees and (b) 15 degrees. From visual inspection, the appearance or visibility of the artificial texture created by moir´e is minimized by increasing the screen angle offset. For three patterns, moir´e is minimized at 30 degrees ( 90 degrees) offsets, and for four patterns, the maximum 3 offset is 22.5 degrees ( 90 degrees). Because the luminance value of yel4 low is so much closer to white than any of the luminance values for cyan, magenta, or black, offsets of 30 degrees between cyan, magenta, and black and 15 degrees between yellow and magenta and yellow and 471 © 2008 by Taylor & Francis Group, LLC

472

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.1: The periodic moir´e created by superimposing two AM halftone patterns of screen angles (left) 45 degrees and 30 degrees and (right) 45 degrees and 15 degrees. black are typically employed (CMYK screen angles of 15 degrees, 75 degrees, 0 degrees, and 45 degrees), producing better results than 22.5 degrees offsets. Periodic moir´e created by the superposition of AM patterns has been studied in great detail by Amidror et al. [4], who model AM screens as cosinusoidal gratings and their superposition in the spatial domain as convolution in the spectral domain. Modeling binary dither patterns as continuous-tone cosinusoidal gratings leads to a Fourier domain representation of a pattern that is composed exclusively of three purely real impulses. Because the superposition of black with any other color leads to black, Amidror et al. adopt a multiplicative model where the pattern created by superposition is determined by a pixel-wise multiplication operation between the two patterns being superimposed. In the Fourier domain, this multiplicative relationship leads to a convolution of the impulses of each grating, as in Fig. 13.2. Approximating the human visual system (HVS) in the Fourier domain as a round-hat function, the optimal screen offset angle is the angle that keeps all but the DC impulse outside the cut-off frequency of the HVS. In frequency modulated (FM) halftoning, the illusion of continuous-tone is produced by a random arrangement of same-sized dots such that, in theory, superimposing two or more stochastic dither patterns does not lead to the appearance of moir´e, but, as seen in

© 2008 by Taylor & Francis Group, LLC

473

θ λ-1

HVS

Figure 13.2: Amidror et al.’s periodic moir´e analysis using cosinusoidal gratings in place of binary halftone patterns.

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

474

Fig. 13.3, superimposing two FM screens does lead to a halftone with low-frequency artifacts – creating a noisy (uncorrelated) appearance. This phenomenon of low-frequency graininess was referred to by Lau et al. [70] as stochastic moir´e. Like moir´e in AM halftoning, the visibility of stochastic moir´e cannot be eliminated from FM halftones, only minimized. For analysis, Amidror et al.’s spectral technique does not work because of its reliance on the simple characterization of cosinusoidal gratings in the spectral domain. As was demonstrated in Chapter 5, the spectral representations of stochastic halftones are very complex, containing a full spectrum of real and imaginary components. Only through the computational convenience of digital computers can the spectrums of stochastic halftones be managed, let alone convolved. Due to the void left by Amidror et al.’s analysis to characterize stochastic halftones, Lau et al. [70] propose a method of characterizing aperiodic moir´e by a relationship between in- and out-of-phase interactions between two or more dither patterns with the optimal halftoning algorithms being the ones that space the in- and out-of-phase points either as close together or as far apart as possible. Prior to Lau et al., the only significant research describing instances of the stochastic moir´e phenomenon was started by Glass [42, 43], who studied the visual perception of order created by superimposing a random pattern of dots onto a geometrically distorted version of itself (Fig. 13.4) to create what is now generally referred to as a glass pattern. From the visual perception of order created when a binary dot pattern is superimposed onto a rotated version of itself, Glass speculated that early steps of the human visual process may include the computation of local autocorrelations by excitation of line detectors and the subsequent averaging of local autocorrelations by collective excitation of the columns in the visual cortex [42].

13.1

Spatial Analysis of Periodic Moir´ e

The basic premise of Amidror et al.’s work on periodic moir´e is that the superposition of the two AM screens is a multiplicative relationship in the luminance image. That is, since the luminance values of cyan, magenta, yellow, and black inks are all less than one, a pixel-wise multiplication between the screens is also less than one, with black ink leading to a value of zero. In the Fourier domain, this multiplicative relationship leads to a convolution of the Fourier spectrums. Amidror

© 2008 by Taylor & Francis Group, LLC

´ 13.1. SPATIAL ANALYSIS OF PERIODIC MOIRE

475

Figure 13.3: The (right) low-frequency graininess produced by superimposing (left and center) two uncorrelated FM halftone patterns.

© 2008 by Taylor & Francis Group, LLC

476

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.4: A (left) binarized white-noise pattern and (right) Glass pattern formed by a superimposing rotated version (left) onto itself. et al. go further to simplify the problem by replacing binary screens with cosinusoidal gratings, a surface whose Fourier spectrum is characterized by a handful of impulses orderly arranged on the frequency plane. This model has effectively simplified periodic moir´e to a framework that can be implemented by hand, not computation. The optimal alignment of screens is now a simple problem of moving impulses as far away as possible from the origin of the spectral plane. Two features of Amidror et al.’s framework deserve repeating, as they will play a fundamental role in our stochastic moir´e work. The first is that Amidror et al. ignore color by dealing exclusively with luminance. This is in no way a shortcoming of Amidror et al.’s framework because they are trying to address the appearance of an unnatural texture and not differences in hue; furthermore, any deviations in color can be addressed by increasing/decreasing the total coverage of a given ink. The benefit of working with just luminance values is that moir´e is a feature of a two-dimensional function (coordinates X, the horizontal, and Y, the vertical) as opposed to X, Y, and Z (color). Another benefit of working in the luminance channel is that our understanding of the human visual system is well understood for monochrome black and white images and has successfully been characterized by low-pass FIR filters [119]. The same cannot be said for full-color RGB. The second feature of Amidror et al.’s work is their use of the Fourier domain, which it can be argued, lies primarily in the fact

© 2008 by Taylor & Francis Group, LLC

´ 13.1. SPATIAL ANALYSIS OF PERIODIC MOIRE

477

Figure 13.5: The peaks (solid lines) and valleys (dashed lines) of two cosinusoidal gratings. that it is easy to do the multiplicative relationship in the spatial domain. While not as simple a characterization in the Fourier domain as Amidror et al.’s, Lau et al.’s analysis of stochastic moir´e presents intuitive interpretations in the spectral domain, and like Amidror et al., their understanding of the HVS in the Fourier domain plays a key role in identifying good versus bad instances of moir´e. Before developing this spectral framework, though, Lau et al. build upon an understanding of stochastic moir´e from the spatial domain, beginning with defining moir´e as a periodic pattern of overlapping and non-overlapping dots [53]. Making Amidror et al.’s substitution of cosinusoidal gratings in place of binary screens, our basic premise for characterizing moir´e (both periodic and stochastic) is that moir´e is, in fact, a pattern of in-phase and out-of-phase alignments between two gratings. Illustrated in Fig. 13.5, the locations where the two gratings are in-phase are the points where the two peaks (solid lines) intersect or where two valleys (dashed lines) intersect. Out-of-phase points are the locations where a peak of one pattern intersects a valley of the other. Shown in Fig. 13.6 are the in-phase (X’s) and out-of-phase (O’s) points between two cosinusoidal gratings. The parameter r represents what we refer to as the principle wavelength of moir´e, and it is the average distance from an in-phase point to its nearest out-of-phase point. For the periodic gratings of Fig. 13.5, this distance r has a deterministic solution λ r= , (13.1) 2 cos(θ) where θ represents the relative screen angle between the two screens and λ is the inverse of the screen frequency. From our understanding of the human visual system presented in Chapter 4, minimizing the visibility

© 2008 by Taylor & Francis Group, LLC

478

´ CHAPTER 13. STOCHASTIC MOIRE

λ

Figure 13.6: The in-phase (X’s) and out-of-phase (O’s) points between two-cosinusoidal gratings. of moir´e becomes a problem of either maximizing (θ = 0 degrees) or minimizing r (θ = 90 degrees) such that the moir´e pattern fluctuates too slowly per unit length to be noticed by the HVS or too rapidly. Using two-dimensional gratings, Fig. 13.7 shows the locations of the inand out-of-phase points. Now, considering the periodic AM patterns for a given color composed of a given proportion of A and B, if all the dots of A are inphase with the dots of B, the halftone creates a certain color and texture for the viewer. If dots of A are all 180 degrees out-of-phase with B, the halftone creates a different color and/or texture for the viewer. So, in Fig. 13.7, ‘X’s and ‘O’s mark the extreme colors/textures, with all other areas falling somewhere between the two. If we choose θ such that r is minimized, the screens of A and B create the moir´e that minimizes the visibility of artificial textures created by superimposing A and B. There are two ways to interpret this result. The first is that, given the low-pass nature of the human visual system (HVS) [17], minimizing r creates a pattern that maximizes the frequency of spatial fluctuations in color/texture. Maximizing these fluctuations moves the spectral content of the halftone pattern to where the HVS is least sensitive, creating an apparent image composed exclusively of a DC color/texture. Another interpretation of minimizing r is to first suppose there is a disc of constant radius placed over the halftone pattern. Assuming this disc is sufficiently large, the area covered by the disc has proportions of colors/textures equal to the proportions of the entire halftone. These proportions are also independent of and do not change with a change in the location of the disc in the pattern. Looking at the proportions as the radius of the disc moves towards zero, these

© 2008 by Taylor & Francis Group, LLC

´ 13.2. SPATIAL ANALYSIS OF APERIODIC MOIRE

479

r

r

Figure 13.7: The in-phase (X’s) and out-of-phase (O’s) points for twodimensional gratings and the corresponding AM halftone patterns. proportions of colors/textures within the disc should remain constant and independent of location as long as the radius is greater than the distance r. The smaller the disc can be without changing proportions, the better it will be with respect to moir´e visibility. So minimizing r is the same as reducing the minimum radius of the disc that maintains proportions equal to the proportions of the entire halftone pattern.

13.2

Spatial Analysis of Aperiodic Moir´ e

While this framework of in-phase and out-of-phase points offers no advantage over Amidror et al.’s with regard to periodic moir´e, it can be extended to stochastic halftoning. Before doing so, Lau et al. [70] derive a concise definition of what stochastic moir´e is and what it is not using an illustration similar to Fig. 13.8, where several stochastic halftone patterns are shown each composed of 9% cyan coverage and 9% ma-

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

480

(a) perfectly out-of-phase

(b) uncorrelated

(c) perfectly in-phase

(d) maximized fluctuations

Figure 13.8: Four instances of two-color FM halftones. genta coverage. Pattern (a) shows the instance where minority pixels of cyan and magenta are homogeneously distributed such that magenta pixels are placed midway between cyan pixels. Shown in pattern (c) is the case where magenta pixels are placed directly on top of all cyan pixels. While it maybe argued by a given observer that pattern (a) creates a hue different from that of pattern (c), Lau et al. ignore those differences and focus on the fact that pattern (a) certainly portrays a different texture from that of (c). Any differences in hue are assumed to be corrected by manipulating the percentage of coverage of the various inks. Now while (a) and (c) have different textures, we make no claims as to whether (a) or (c) is a better texture. A more revealing look at stochastic moir´e occurs in pattern (b), where the magenta and cyan dither patterns are completely uncorre-

© 2008 by Taylor & Francis Group, LLC

´ 13.2. SPATIAL ANALYSIS OF APERIODIC MOIRE

481

lated with some minority pixels (of both patterns) overlapping and some falling directly in between those of the other pattern. All other minority pixels fall somewhere in between these two extremes. From pattern (b), stochastic moir´e is defined as the change in texture that occurs from one point to another within a given halftone. Like periodic moir´e, the optimal stochastic halftone is the one that either minimizes or maximizes the amount of fluctuation in this texture per unit area. In the case of pattern (a) and (c), the amount of fluctuation is minimized with patterns (a) and (c) offering equivalent optimality regarding moir´e. Pattern (d) offers the other moir´e extreme by maximizing the amount of fluctuation per unit area. Restricting the analysis to the spatial domain, one way to characterize fluctuations in texture is to identify the in-phase and out-of-phase points. In-phase points are the locations where minority pixels of both patterns overlap, while out-of-phase points are the locations where a minority pixel of one pattern falls directly between minority pixels of the other. A minority pixel is directly between minority pixels of the other pattern when it coincides with a vertex point of the voronoi mesh defined by the minority pixels of the other pattern. Lau et al. choose these vertices as the locations of valleys because they are mid-way between minority pixels, with any displacement moving the valley closer to one minority pixel and farther from the others. The principle wavelength of stochastic moir´e will, again, be the average distance between an in-phase point and its nearest out-of-phase point. The term moir´e distance is used by Lau et al. to describe how quickly the color halftone transitions from overlapping (in-phase) to non-overlapping (out-of-phase) textures. In AM halftoning, this distance of moir´e is the distance r from (13.1). In FM halftoning, the moir´e distance does not have a deterministic solution and must, therefore, rely on expectations and moments. To this end, we define S(φA , φB ) as the stochastic moir´e distance such that ¯ E{XO} S(φA , φB ) = , (13.2) max(λA , λB ) ¯ is the minimum distance from an in-phase (X) to the nearest where XO out-of-phase (O) point between the dither patterns of φA and φB . The terms λA and λB represent the average distance between minority pixels in φA and φB . Their presence in (13.2) normalizes S(φA , φB ) such that fluctuations in S are not just the result of the closer proximity of minority pixels in φA or φB as either of their intensities approaches 12 .

© 2008 by Taylor & Francis Group, LLC

482

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.9: The binary image created (top) before and (bottom) after applying a one-pixel vertical shift to one of the two channels forming a perfectly out-of-phase stochastic moir´e surface.

The max(·) function is used to normalize the result with respect to the least frequent minority pixels of φA or φB . To minimize the appearance of stochastic moir´e, maximizing S(φA , φB ) corresponds to creating φA and φB in such a way that their superposition results in a texture that does not change noticeably across a given area. This is equivalent to dot-on-dot and dot-off-dot alignments in AM halftoning. But, as previously noted, even slight a misregistration between the screens could result in drastic shifts in color/ texture as demonstrated in Fig. 13.9, where one of two correlated patterns is shifted by a single pixel down. We can, instead, try to minimize S(φA , φB ) such that the texture of the resulting superposition of φA and φB results in moir´e that fluctuates too rapidly to be perceived by the HVS. The advantage of this scheme is that it does not necessarily imply any correlation between the patterns, although correlation may improve results. Returning to Fig. 13.8, patterns (a) and (c) are composed of all out-of-phase points and all in-phase points, respectively. In either case, we say that the moir´e √ wavelength is infinitely long. In pattern (d), the moir´e wavelength is 1/ 2 and is the average distance between minority pixels in a blue-noise dither pattern with two times the gray-level of pattern φA . Figure 13.10 illustrates an example of how minimizing the primary frequency of stochastic moir´e improves the color halftone. In both dither patterns of Fig. 13.10, halftoning is done independently, resulting in two dither patterns whose number of minority pixels of cyan overlapping magenta are approximately equal. Looking at the differences

© 2008 by Taylor & Francis Group, LLC

´ 13.2. SPATIAL ANALYSIS OF APERIODIC MOIRE

483

Floyd and Steinberg

Ulichney

Figure 13.10: (See color insert following page 480.) Two-color halftone patterns generated using error-diffusion with the black and white diagrams indicating regions where the minority pixels of cyan and magenta do (black) and do not (white) overlap. These black and white diagrams were created by hand. in texture across each dither pattern (better indicated in the binary black and white diagrams of Fig. 13.10, where black indicates the regions where the minority pixels of cyan and magenta overlap and white indicates regions where minority pixels do not), Ulichney’s halftoning scheme of Fig. 13.10 (bottom) breaks up the large patches of texture, resulting in a more homogeneous arrangement of cyan and magenta pixels. For a given viewing distance and print resolution, the smaller patches that form Fig. 13.10 (bottoms) are much less visible than those of Fig. 13.10 (top). For experimental results, the plot of S(φA , φB ) versus gray-level

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

484

moire distance

4

1 0

0

0.25

0.33

0.5

gray level

Floyd and Steinberg moire distance

4

1 0

0

0.25

0.33

0.5

gray level

Ulichney

Figure 13.11: Plot of the stochastic moir´e distance versus gray-level g for two-color halftones of color (g, g). g, for two-color halftones of color (g, g), in Fig. 13.11 (top), for Floyd’s and Steinberg’s error-diffusion shows a near 1:1 relationship between ¯ and λA with peculiar spikes at g ≈ 1 , 1 , and 1 , where Knox [58] E{XO} 4 3 2 notes that Floyd’s and Steinberg’s error-diffusion locks into a regular pattern. The exponential growth of S(φA , φB ) as g approaches 0 is a consequence of the worm patterns that form at extreme gray-levels where φA and φB are correlated in such a fashion as to make both inphase and out-of-phase pixels especially rare and S(φA , φB ) especially high. For g ≈ 14 , 13 , and 12 , the regular patterns of Floyd’s and Steinberg’s error-diffusion lead to large patches of clusters of in- and out-ofphase pixels. This peaking of S(φA , φB ) is a consistent feature whenever both of the corresponding dither patterns have a strong correlation or ordered arrangement between minority pixels of like color. That is, whenever a strong correlation exists between minority pixels and that correlation exists in both patterns, then the two patterns are even more likely to be in-phase if the nearest neighboring minority pixels are also in-phase. This is clearly the case in Fig. 13.12 (top), where overlap-

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

485

Floyd and Steinberg

Ulichney

0.25

0.33

0.5

Figure 13.12: Two-color halftone ramps generated using (top) Floyd’s and Steinberg’s and (bottom) Ulichney’s error-diffusion. ping and non-overlapping pixels occur almost exclusively in clusters. Shown in Fig. 13.11 (bottom) and Fig. 13.12 (bottom) are the results when halftoning is done using Ulichney’s perturbed filter weight scheme, which randomizes the filter coefficients of error-diffusion to break up the regular patterns that occur at g ≈ 14 , 13 , and 12 . Ulichney’s scheme clearly performs its task, as S(φA , φB ) is now void of the spikes that plague Floyd’s and Steinberg’s error-diffusion.

13.3

Spectral Analysis of Aperiodic Moir´ e

A problem incurred by (13.2), measuring the distance between the inphase and out-of-phase points, is that it relies entirely on pixel overlap. This creates problems at lower gray-levels where the minority pixel population is so much less that the probability of overlapping of minority pixels from two independently generated processes is very low for a reasonable sample size. The effects of this shortcoming can be relatively minimized by increasing the sample size or by taking an average over a large ensemble. A different problem that cannot necessarily be alleviated by increasing the sample size is shown in Fig. 13.13, where the distance from an X to an O is not equal to the distance from an O to an X, a consequence of (13.2)’s ad hoc definition. Ultimately what is shown in Fig. 13.13 is that the distance in (13.2) is not an accurate measure of how quickly a two-color dither pattern changes from an area

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

TH E

THE DISTANCES WE GET

DI ST AN C

E

W

E

W AN T

486

Figure 13.13: Illustration showing a problem with measuring the stochastic moir´e distance as the average minimum distance between an X and its nearest O. From visual inspection, it should be obvious that the average minimum distance from an X to an O is not equal to an O to its nearest X. of perfectly in-phase pixels to an area of perfectly out-of-phase pixels, which is what the stochastic moir´e distance is intended to represent. The problems with (13.2) can be summarized by observing that it relies solely on the extreme phase points, of which there may be very few. The minority pixels that fall between extremes tell us something about how well aligned the two dither patterns are, and these pixels need to be made use of. To do so, Lau et al. return to their definition of moir´e as being a pattern of overlapping and non-overlapping dots. In-phase minority pixels are instances of overlapping pixels while outof-phase pixels are clearly non-overlapping. One can even say that these out-of-phase pixels are as non-overlapping as minority pixels can be for a given halftone. Minority pixels that are neither of these extremes are non-overlapping, but Lau et al. say that they are not as non-overlapping as out-of-phase pixels. How non-overlapping a minority pixel is can be measured as the distance from that minority pixel to its nearest neighboring minority pixel in the other pattern. In formal terms, the dither patterns φA = {ai : i = 1, 2, . . . , NA } and φB = {bj : j = 1, 2, . . . , NB } define a discrete-space, two-dimensional

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

487

Figure 13.14: A (left) two-color FM halftone and (right) its corresponding stochastic moir´e surface. The superimposed circles illustrate the minimum distance between the minority pixels of the cyan pattern located at the circles’ centers and their nearest neighboring minority pixels of the cyan pattern. function D[n] such that NA 1  D[n] = di δ[n − ai ], where λb i=1 di = min ||ai − bj ||2 . j

(13.3) (13.4)

In its (13.3) form, D[n] represents a continuous-space signal D(x), an image referred to as the stochastic moir´e surface, which has been sampled on a stochastic sampling grid defined by φA . If the sampling grid has a blue-noise distribution, as would be the case of a visually pleasing dither pattern with principle wavelength λa , D(x) can be obtained from D[n] using an ideal low-pass filter with cut-off frequency 0.5λ−1 a and magnitude λ2a , as in Fig. 13.14. It is worth emphasizing that φA must be a blue-noise process, and the reason lies in stochastic sampling theory whereby the point process is composed exclusively of a DC frequency component combined with spatial frequencies above 1/λa , where λa is the minimum distance between minority pixels in φA (Fig. 13.15). Perfect reconstruction of the continuous-space signal is guaranteed if the signal being sampled only contains spatial frequencies at or below 0.5/λa . Based on our definition of D[n], the stochastic moir´e surface is such that the maximum rate at which the surface can change is to go from a maximum value or peak at one sample to a minimum value at a nearest neighboring sam-

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

ALI A

488

DC 1 2

NG SI

λ-1A

λ-1A

λ-1A

(a) blue-noise point process

1 λ-1 2 A

(b) stochastic moire surface

low-pass filter

(c) sampled surface without aliasing

Figure 13.15: Diagram of the spectral domain representations of (a) the blue-noise point process defining the sampling grid, (b) the circularbandlimited stochastic moir´e surface with bandwidth less than half the principle frequency, and (c) the power spectrum of the sampled surface with no aliasing.

ple. Half the wavelength of this maximum rate component is, therefore, equal to λa , just as the distance from a peak to valley in a cosine is equal to half the cosine’s wavelength. The frequency of this maximum rate component is, therefore, equal to 0.5λ−1 a . We conclude that the maximum spatial frequency contained within the stochastic moir´e surface is 0.5λ−1 a , with the sampling rate being sufficiently high as to avoid aliasing. If, for instance, the reference pattern defining the sampling grid were from a white-noise pattern, then the stochastic moir´e surface would be impossible to extract from the spectral components of the sampling grid. Now suppose that both dither patterns are blue-noise patterns, but with different minority pixel intensities. In such an instance, the sampling grid, φA , is defined by the pattern with the higher intensity. This follows from the earlier comment about how fast the surface can change from a peak to a valley. While it is not possible for the surface to move from a peak to a valley between any two neighboring minority pixels if sampling is defined by the higher intensity pattern where the stochastic moir´e surface is being over-sampled, it is possible to go from

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

489

a peak to a valley in less distance than between any two neighboring samples when sampling is defined by the lower intensity pattern where the stochastic moir´e surface is being under-sampled. Given our stochastic moir´e surface, characterizing stochastic moir´e is now reduced to the problem of characterizing the spatial fluctuations in the continuous-space, two-dimensional surface D(x). How we measure spatial fluctuations in this new monochrome image is arbitrary but should take into account our understanding of the human visual system to put a higher cost on low- and mid-frequency fluctuations as opposed to high. Perhaps the easiest way to measure the visibility of stochastic moir´e is to measure the visual cost of the stochastic moir´e surface as V C(φA , φB ) = Average AC Power (D(f ) × HV S(f )),

(13.5)

where HV S(f ) is the spectral, low-pass filter model of the human visual system used by Sullivan et al. [119]. For the remainder of this chapter, we will make this substitution, and to see an illustration of (13.5) as our stochastic moir´e metric for stochastic halftones, Fig. 13.16 shows two instances of stochastic moir´es. These particular surfaces have visual cost measures of (left) 0.0484 and (right) 0.0311. Here, we are assuming a 300 dpi printer and a viewing distance of 20 inches. Comparison of the two visual costs suggests an improvement in color rendition from the top pattern of Fig. 13.16 to the bottom. Visual inspection validates these results. To help further appreciate the relationship between the moir´e surface, D(x), and the overlapping component dither patterns, φA and φB , Fig. 13.17 shows a two-color stochastic dither pattern and its moir´e surface with the pixels corresponding to locations where φA [n] = φB [n] = 1 indicated by white circles. What this particular figure demonstrates is that direct correlation of the Fourier transforms of φA and φB only reveals information regarding these white circles and that our new stochastic sampling technique provides much more information by presenting the interaction between φA and φB in the non-black regions where D(x) > 0. What Fig. 13.17 also helps to illustrate is the intended relationship between (13.2) and the stochastic moir´e surface. What Lau et al. were trying to measure in (13.2), although crudely, was the dominant wavelength in the moir´e surface such that, perhaps, by measuring the average distance between the perfectly inphase pixels and the perfectly out-of-phase pixels, (13.2) was somehow

© 2008 by Taylor & Francis Group, LLC

490

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.16: (See color insert following page 480.) Two examples of stochastic moir´e and the corresponding moir´e surfaces produced by superimposing two FM halftone patterns.

identifying the frequency that all the spectral power in the moir´e surface would be centered around. If it could determine that the average distance from in-phase clusters to out-of-phase clusters was large, then the moir´e surface would be fluctuating slowly, while short distances would mean the surface was fluctuating rapidly. With this sampling approach, we see now that directly measuring the average distance from in-phase clusters to out-of-phase clusters is not necessary. The stochastic moir´e surface completely characterizes the phase relationship between two overlapping dither patterns at all pixels. For a further demonstration of the new sampling/reconstruction approach to moir´e analysis, we can apply it to the periodic moir´e found in AM halftoning where every cluster is replaced with a single minority pixel at the cluster’s centroid. We will assume that the reference pattern that defines the sampling grid will have a screen angle of 45 de-

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

φA

φA && φB

491

φB

moire surface

Figure 13.17: Illustration of the relationship between the stochastic moir´e surface and the dither produced by a pixel-wise logical AND operation between dither patterns φA and φB . grees and a periodicity of 8 pixels on a 1,200 dpi halftone (150 lines per inch). For this reference pattern, the low-pass filter has a cut-off frequency of π/8 radians (75 Hz), and shown in Fig. 13.18 are the surfaces corresponding to the periodic moir´e created by screen angle offsets of 5 degrees, 15 degrees, and 30 degrees. For a measure of the visual cost of each surface, Fig. 13.18 also shows the same surfaces but after applying our chosen HVS model. The average AC powers of these surfaces are

© 2008 by Taylor & Francis Group, LLC

492

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.18: The moir´e surfaces and their corresponding modeled visual responses for 1,200 dpi, 150 lpi AM halftoning with screens angles 5 degrees, 15 degrees, and 30 degrees.

0.0220, 0.0134, and 0.0063, respectively. An interesting point that should be noticed from Fig. 13.18 is that, for any reasonable print resolution and viewing distance, the HVS model will have a cut-off frequency significantly lower than that of the low-pass filter used to obtain the stochastic moir´e surface from D[n]. So, as the power spectrum of the reference dither pattern differs from the ideal blue-noise spectrum causing the round-hat low-pass filter to seem invalid, the difference in power spectrums will have little impact on the visual cost. We could, in fact, skip the low-pass filter and apply

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

493

our HVS directly to the Fourier transform of D[n]. Now, having just showed that this new moir´e framework can be applied to periodic moir´es, perhaps we should compare it to Amidror et al.’s in terms of the one-dimensional cosinusoidal gratings of Fig. 13.2, where sampling is said to occur at the peaks of the reference gradient. Shown in Fig. 13.19 are the spatial patterns and their corresponding Fourier domain representations for our new stochastic moir´e analysis. The spectral domain representations for Amidror et al.’s are repeated from Fig. 13.2 for convenience. Unlike in Amidror et al.’s analysis, the sampling pattern of the reference pattern leads to an infinite impulse train along the horizontal, instead of just the three impulses that make up the Fourier transform of the grating, and convolution of this infinite impulse train with the three impulses of the superimposed grating leads to three infinite impulse trains. And if we apply a stripe filter such that all horizontal frequencies greater in magnitude than 12 λ−1 are removed, then only the three impulses that form the moir´e grating remain. These are the same moir´e components that Amidror et al. identify as the most noticeable, with the optimal screen angle being the one that maximizes the distance between the two AC impulses and the DC impulse. So we see that, in fact, minimizing the visual cost of moir´e is done by maximizing the spatial frequencies of the moir´e surface. As a demonstration of (13.5)’s feasibility at quantitatively measuring stochastic moir´e, Lau et al. compare the resulting visibility measures for two-color halftoning using Floyd’s and Steinberg’s [40], Jarvis et al.’s [50], Stucki’s [117], and Ulichney’s [123] error-diffusion. As was demonstrated in Chapter 5, Floyd’s and Steinberg’s error-diffusion creates strong periodic textures at gray-levels 1/4, 1/3, and 1/2 while Jarvis et al.’s do so at gray-level 1/3 and Stucki’s at 1/3. Through a perturbation of error filter coefficients, Ulichney’s error-diffusion breaks up periodic textures in the dither pattern – creating a far more pleasant blue-noise halftone. In all cases, strongly correlated textures result in distinct features in the visual cost plots of Figs. 13.20–13.23. In Fig. 13.24, we show several of the corresponding stochastic moir´e surfaces, indicated in Figs. 13.20–13.23 by the dotted grid lines. In Figs. 13.20–13.23, the most noticeable feature that exists in all four plots is the high visual cost along the main diagonal where ga = gb . Surprisingly, the increased visibility of stochastic moir´e along the cyan equal to magenta line is a natural property of blue-noise, and this can be seen if we note that blue-noise dither patterns maximize spatial fluctuations in texture by optimally breaking up clusters of in-phase or

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

494

reference pattern

pattern A

λ-1 superimposed pattern

λ-1 pattern B

θ

θ

convolution

convolution

low-pass filtering

HVS

visual cost

HVS spatial domain

spectral domain

spectral domain

New Moire Framework

Amidror et al's

Figure 13.19: A step-by-step comparison of the new stochastic moir´e analysis and Amidror et al.’s for two superimposed, one-dimensional cosinusoidal gradients.

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

495

0.25

0.33

0.5

0.25

0.33

0.5

0.25

0.33

0.5

Figure 13.20: The (top) dither pattern and (bottom) corresponding two-dimensional plot of stochastic moir´e visibility (black=0, white=0.035) for Floyd’s and Steinberg’s error-diffusion.

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

496

0.25

0.33

0.5

0.25

0.33

0.5

0.25

0.33

0.5

Figure 13.21: The (top) dither pattern and (bottom) corresponding two-dimensional plot of stochastic moir´e visibility (black=0, white=0.035) for Jarvis et al.’s error-diffusion.

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

497

0.25

0.33

0.5

0.25

0.33

0.5

0.25

0.33

0.5

Figure 13.22: The (top) dither pattern and (bottom) corresponding two-dimensional plot of stochastic moir´e visibility (black=0, white=0.035) for Stucki’s error-diffusion.

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

498

0.25

0.33

0.5

0.25

0.33

0.5

0.25

0.33

0.5

Figure 13.23: The (top) dither pattern and (bottom) corresponding two-dimensional plot of stochastic moir´e visibility (black=0, white=0.035) for Ulichney’s error-diffusion.

© 2008 by Taylor & Francis Group, LLC

´ 13.3. SPECTRAL ANALYSIS OF APERIODIC MOIRE

(ga,gb)=(0.25,0.25)

(ga,gb)=(0.33,0.33)

499

(g a,gb)=(0.50,0.50)

(a) Floyd and Steinberg

(b) Jarvis et al

(c) Stucki

(d) Ulichney

Figure 13.24: Selected stochastic moir´e surfaces for several errordiffusion techniques.

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

500

clusters of out-of-phase pixels. Lau et al. note this in one-dimensional given that φA = {am : m = 1, 2, . . .} and φB = {bn : n = 1, 2, . . .} represent two blue-noise point processes with an average separation between consecutive points of λA and λB , respectively. Now suppose that ai ∈ φA and bj ∈ φB are such that ai = bj , indicating that φA and φB are in-phase at ai . In order to minimize the fluctuation in texture between φA and φB around ai , the point process needs to maximize the probability that ai+1 overlaps bj+1 such that the texture remains constant in the region [ai , ai+1 ], where Pr(ai+1 = bj+1 ) =

+∞ x=ai

Pr(ai+1 = bj+1 |bj+1 = x) · Pr(bj+1 = x) dx.

(13.6) From (13.6), the probability that ai+1 overlaps bj+1 is maximized when λA = λB for uncorrelated φA and φB . It is therefore stated that since φA and φB are functions of their corresponding minority pixel intensities, fluctuations in texture are minimized when the intensities of the twocomponent dither patterns are equal. Returning to specific instances of moir´e, Floyd’s and Steinberg’s strong periodic textures for gray-levels (ga , g + b) = (0.25, 0.25) and (ga , gb ) = (0.50, 0.50) create large patches of fixed phase between A and B (the component dither patterns) – leading to large near-DC components in the stochastic moir´e surface and, therefore, large visibility measures. A similar phenomenon occurs at gray-level (ga , g + b) = (0.33, 0.33), but to a much lesser degree. Jarvis et al.’s suffers similar correlation to Floyd’s and Steinberg’s, at gray-level (ga , g + b) = (0.33, 0.33) but eliminates the high visual cost everywhere else. Stucki’s follows the behavior of Jarvis et al.’s closely but with added correlation at (ga , gb ) = (0.50, 0.50), where the visual cost is very high. What we see in Ulichney’s error-diffusion is a halftoning algorithm with consistent behavior across all gray-levels and without the outliers or spikes that exist in the other three techniques. While Ulichney’s error-diffusion does not have the lowest mean or median values, the lack of any wildly varying or abrupt changes in visual cost means that smooth color gradients will show a more consistent behavior in texture. This is clearly visible in Fig. 13.14 for two-color gradients where a binary halftone is produced by combining the component halftones using a pixel-wise logical or function (1=black). While all four patterns have their problems maintaining a smooth appearance, the large visibility measure of Floyd’s and Steinberg’s along the lines

© 2008 by Taylor & Francis Group, LLC

´ 13.4. MINIMIZING STOCHASTIC MOIRE

501

ga = 0.25 and gb = 0.25 is clearly apparent by observation in Fig. 13.14. Jarvis et al.’s and Stucki’s have the same problem along the ga = 0.33 line and the gb = 0.33 line. With the especially high visibility measure along these lines, the fluctuations in texture in these regions are very uncharacteristic of the rest of the halftone. What we see in Ulichney’s is a halftone with no dramatic shifts in texture with the same visible impression of noise across the entire pattern. What you may notice, if anything, is a greater variation in texture along the diagonal where ga = gb .

13.4

Minimizing Stochastic Moir´ e

As previously discussed, minimizing the visibility of stochastic moir´e is accomplished by either maximizing or minimizing the spatial fluctuations in texture. Examples of minimizing fluctuations were shown in Fig. 13.8 for patterns (a) and (c), with pattern (c) arguably the preferred dither pattern of the two because it simultaneously minimizes fluctuations in texture and maximizes the frequencies of the spectral components that form the luminance image. An example of a specific instance where this type of stochastic moir´e reduction occurs is in [138], where Yao and Parker generate strongly correlated blue-noise masks, and in [68], where Lau et al., generate inter-locking green-noise masks. The problem with this type of moir´e reduction, where fluctuations in texture are minimized in FM halftones, is that it requires perfect alignment of screens by the printing device, a problem shared by dot-on-dot and dot-off-dot AM halftoning. Figure 13.9 demonstrated the effect that misalignment can have on such halftones where one of the two channels is shifted down by a single pixel. In terms of Amidror et al.’s moir´e framework, minimizing spatial fluctuations in texture generates unstable halftone patterns. Because of the detrimental impact that only one pixel of misalignment could have for minimized textures, maximizing spatial fluctuations would seem to be a far more attractive solution for minimizing stochastic moir´e visibility even at the cost of introducing low-frequency graininess into the halftone. But maximizing spatial fluctuations in texture does not necessarily alleviate constraints on screen alignment. It will, in general, reduce the impact of misalignment with the amount of impact that misalignment has on moir´e visibility, depending on how strong a correlation exists between patterns. To truly alleviate con-

© 2008 by Taylor & Francis Group, LLC

502

´ CHAPTER 13. STOCHASTIC MOIRE

straints on screen alignment requires that overlapping patterns be completely uncorrelated such that the probability that a cyan dot overlaps a given magenta dot be equal to the intensity of cyan dots. Using uncorrelated screens makes minimizing fluctuations in texture across any appreciably sized area statistically impossible, and therefore, only maximizing fluctuations minimizes the visibility of stochastic moir´e and the overall impact it has on the aesthetic quality of the printed image. In practice, identifying the optimal uncorrelated halftoning algorithm reduces to experimenting with various algorithms and then selecting the one that produces the lower visual cost. We did this in Sec. 5.2 when we identified Ulichney’s error-diffusion as the preferred technique relative to several other variations of error-diffusion, but given a particular halftoning algorithm, can we make improvements that further minimize moir´e visibility without introducing correlation between component colors? Looking at the stochastic moir´e visibility measure of Fig. 13.13 for Ulichney’s error-diffusion, a noticeable increase in visibility occurs along the cyan equal to magenta line. So one way to minimize stochastic moir´e visibility, and not a good way, would be to manipulate the amounts of cyan and magenta in regions where the intensities of cyan and magenta are equal in the original continuous-tone image. This technique has the very undesirable affect of changing hues in the output relative to the original. A similar phenomenon occurs in the direct binary search [78] algorithm for monochrome images where the dither pattern associated with gray-level 0.5 offers enough improvement in visual cost that the error in tone produced by halftoning near 0.5 graylevels as 0.5 is an acceptable sacrifice overall. Given the natural tendency of blue-noise to maximize stochastic moir´e visibility when two superimposed dither patterns have the same intensity, minimizing stochastic moir´e visibility in these regions requires either that the intensity of one pattern be manipulated or that the statistical character of one of the patterns be varied. The second, and preferred approach, can be achieved by adding noise to the image prior to halftoning in these regions of equal intensity such that the average intensity, and hence the average hue, does not change [122]. This can also be achieved by perturbing the quantization threshold used in error-diffusion with low-variance white-noise in these same regions. The overall effect is to “whiten” the spectral content of the corresponding pattern [65], and thereby increase the variability in the pattern with a minority pixel less likely to be found an exact distance λA from its

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

503

nearest neighbors.

13.5

Stochastic Moir´ e and Green-Noise

In green-noise halftoning, stochastic moir´e has the same impact as in FM halftoning expect that the lower frequency content of green-noise halftones results in lower frequency fluctuations in texture for stochastic moir´e [67]. The HVS being more sensitive to low-frequency textures makes the stochastic moir´e of green-noise more noticeable and, therefore, of greater impact on the resulting print [70]. This was visible in the green-noise mask color plates most noticeable in regions in the area of the violin surface. Like FM halftoning, stochastic moir´e visibility is minimized by varying the statistical character of overlapping patterns, but unlike FM halftones, the statistical nature of green-noise halftones can be manipulated by varying the coarseness of overlapping patterns such that the average size of clusters and the related spacing between clusters change without modifying the pattern’s intensity. So in green-noise halftoning, stochastic moir´e visibility is minimized by varying the coarseness between overlapping dither patterns [70]. Shown in Fig. 13.25 is proof of this concept, where the patterns on top show the overlapping of two uncorrelated halftone gradients produced using blue-noise masks and patterns on the bottom show the overlapping of uncorrelated gradients using a blue-noise mask for one pattern and a green-noise mask for the other. The drawback of using increased pattern coarseness to minimize stochastic moir´e is that in areas of the digital print where the halftone is composed of dots from a single color, the high coarseness this pattern may have, in the hopes of minimizing stochastic moir´e in other areas, has the undesirable effect of creating a visually apparent texture due to increased visibility of printed dots that now occur in clusters [66]. So in general, minimizing stochastic moir´e by means of green-noise is a balancing act of reducing moir´e visibility at the cost of low-frequency graininess introduced by coarse halftones with the printer manufacturer choosing the optimal ratio for their device [62]. But what would clearly help is a halftoning scheme that adaptively modifies coarseness according to the color composition of the original image at each pixel. Such a technique is proposed by employing Levien’s [75] error-diffusion with output-dependent feedback for two-color halftones where coarseness is determined according to the difference in tone at each pixel such that

© 2008 by Taylor & Francis Group, LLC

504

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.25: The composite halftones created by (top) two uncorrelated blue-noise masks and (bottom) an uncorrelated combination of a blue-noise and a green-noise mask.

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

505

as the difference in tone gets smaller, the difference in coarseness gets larger. For deriving the stochastic moir´e surface from two overlapping green-noise dither patterns, Lau et al. [70] rely on the parent process of a green-noise pattern where each minority pixel cluster is replaced with a single minority pixel at the cluster’s centroid. This new pattern has statistical characteristics identical to a blue-noise dither pattern. So, ¯ in addition to g, it is therefore possible to given λg ’s dependence on M modify the average spacing between parent process points (the underlying blue-noise process) without modifying g. In terms of stochastic ¯ allows green-noise patterns to move off moir´e visibility, modifying M the 1:1 line in a 2-color halftone without changing hue. So combining a blue-noise and a green-noise dither pattern allows us to avoid the worst case scenario of stochastic moir´e that occurs when overlapping patterns have equal spacing between their parent points. As was demonstrated previously in Fig. 13.3, blue- and green-noise combinations will show a dramatic improvement in pattern homogeneity in the distribution of black pixels. The drawback to increasing pattern coarseness, for the purpose of minimizing stochastic moir´e visibility, is the increased low-frequency graininess of the component dither patterns caused by the increased visibility of the clusters. So given the detrimental impact of employing larger average cluster sizes, a printer manufacturer has the task of choosing the optimal trade-off between halftone visibility and stochastic moir´e visibility, keeping in mind the additional impact of printer distortion [70]. In the case of halftoning by means of blue- or green-noise masks, halftone coarseness is fixed at the time of mask construction and is independent of the image content. But using an adaptive scheme like Levien’s error-diffusion with output-dependent feedback which we described in Chapter 10, halftone coarseness could be a function of the local color content of the image. For a demonstration of stochastic moir´e using error-diffusion with output-dependent feedback, Fig. 13.26 (top) is the composite halftone pattern created by superimposing two halftone gradients both generated with h = 0.0. In Fig. 13.26 (bottom), h = 1.0 for the second gradient. Clearly, setting h = 1.0 for pattern B has greatly improved the homogeneity of black pixels across the image, but at the same time, it has greatly increased the coarseness of the pattern in the lower-left corner where ga = 0 and gb = 0.5. Following Lau’s [64] suggestion of an image dependent hysteresis

© 2008 by Taylor & Francis Group, LLC

506

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.26: The composite halftones generated using error-diffusion with output-dependent feedback with (top) hb [n] = 0 and (bottom) hb [n] = 1.0.

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

507

term, we can define h[n] for the second of two, A and B, patterns as hB [n] = hA [n] +

C1 1 + C2 · |ga − gb |2

(13.7)

where the scalar constants C1 and C2 now regulating coarseness with C1 hB [n] ranging from hA + 1+C when |ga − gb | = 1 and hA + C1 when 2 |ga − gb | = 0. Using the two error and two hysteresis weight filter originally specified by Levien and setting C1 = 1.0, C2 = 10.0, and hA [n] = 0, Fig. 13.27 (top) shows the hysteresis value for the combined gradient pattern of Fig. 13.27 (bottom). Shown in Fig. 13.28 are further demonstrations of varying hb [n] with color composition where (C1 , C2 ) = (0.6, 10.0) in Fig. 13.28 (top) and (C1 , C2 ) = (1.0, 4.0) in Fig. 13.28 (bottom). Visual inspection shows the dramatic improvement in homogeneity along the ga = gb line, where the difference in coarseness is maximized, as well as the reduction in coarseness near (ga , gb ) = (0.5, 0) in the lower-left corner of the pattern relative to the patterns of Fig. 13.26. Moving now to more than two-color images, it is believed by Lau et al. [70] that minimizing stochastic moir´e visibility between any twocolors minimizes moir´e for all colors with coarseness between patterns ranked according to luminance. In the case of CMYK screens, this would imply a blue-noise screen for black, with a coarser pattern for magenta, with a coarser pattern for cyan, and with the coarsest pattern for yellow. For a demonstration, Fig. 13.17 compares the use of uncorrelated blue-noise dithering versus green-noise with varying coarseness between colors where, for a pixel in channel K, hK [n] is set constant at 0.3 due to 0.3’s visually desirable attributes using two hysteresis and two error filter weights in monochrome halftones. In order to minimize stochastic moir´e visibility between magenta and black, hM [n] is set according to (13.7) for channel M such that hM [n] = hK [n] +

C1 , 1 + C2 · |gM − gK |2

(13.8)

where C1 = 0.6 and C2 = 4.0. For cyan, hC [n] is set to minimize moir´e visibility between cyan and black as well as cyan and magenta; therefore, hC [n] is set such that hC [n] = hM [n] + max(hCK [n], hCM [n]), where C1 hCK [n] = and 1 + C2 · |gC − gK |2

© 2008 by Taylor & Francis Group, LLC

(13.9)

508

´ CHAPTER 13. STOCHASTIC MOIRE

Figure 13.27: The (bottom) composite halftone generated using errordiffusion with output-dependent feedback, where hA [n] was set to 0 for all n while hB [n] was set equal to the image shown on the (top) derived from (13.7) with C1 = 1.0 and C2 = 10.0.

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

509

Figure 13.28: The composite halftones generated using error-diffusion with output-dependent feedback, where (top) (C1 , C2 ) = (0.6, 10.0) and (bottom) (C1 , C2 ) = (1.0, 4.0).

© 2008 by Taylor & Francis Group, LLC

´ CHAPTER 13. STOCHASTIC MOIRE

510 hCM [n] =

C1 . 1 + C2 · |gC − gM |2

The same approach is applied to selecting the hysteresis parameter for channel Y such that hY [n] = hC [n] + max(hY K [n], hY M [n], hY C [n]), where(13.10) C1 hY K [n] = , 1 + C2 · |gY − gK |2 C1 hY M [n] = , and 1 + C2 · |gY − gM |2 C1 hY C [n] = . 1 + C2 · |gY − gC |2 Looking at the resulting CMYK halftones in Fig. 13.29, the difference in quality is most apparent in the fluctuations between red and yellow patches, assuming the print page is held at an appropriate distance from the observer’s eye. Please note that, if viewed from too close a distance, the graininess of the green-noise obscures the improvement in stochastic moir´e. From too far a distance, the two patterns blur to the point where fluctuations in color become indistinguishable. From a 20-inch viewing distance, the stochastic moir´e seems to become most apparent, overtaking the graininess of green-noise as the predominant halftone artifact. The viewing distance and the visibility of moir´e varying with this distance bring up a very important issue regarding the use of greennoise as a means of minimizing moir´e visibility, and that issue is how much of a difference in coarseness between colors should one have? Ultimately, coarseness should be a function of several factors, including the printer manufacturer’s personal interests to balance halftone visibility with stochastic moir´e visibility and the device’s ability to print dots reliably. A third factor is the HVS’s sensitivity to stochastic moir´e visibility between certain colors, such as when it occurs between cyan and magenta as opposed to black and magenta or black and cyan. As described, the existing stochastic moir´e framework ignores these issues relating to color. But it should be known by the reader that the stochastic moir´e framework of Lau et al. was specifically designed to avoid such issues simply because of the many difficult questions that color assumes, aside from those relating to moir´e. On this same issue, Amidror et al. [4] simplified their analysis by restricting the problem of periodic moir´e for color halftones to the luminance channel such that

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

511

Figure 13.29: (See color insert following page 480.) CMYK halftones showing, from left to right, the improvement in stochastic moir´e using varying levels of coarseness when patterns are uncorrelated. Holding these images 20 to 30 inches from the eye maximizes the visual differences between halftones. the strength of the spectral impulses proportional to the luminance of their corresponding color. What Lau et al. [70] say is that it is assumed that by minimizing moir´e between any two channels, moir´e will be minimized across any three channels and across the entire four-channel image. And while this assumption seems short-sighted, studies of visual perception using glass patterns suggest very similar conclusions. In particular, Earle [31]

© 2008 by Taylor & Francis Group, LLC

512

´ CHAPTER 13. STOCHASTIC MOIRE

shows that, given a glass pattern composed of dots of three different luminance values, the HVS is more sensitive to correlation between the two most similar luminance values (cyan and magenta) and not necessarily the two lowest luminance values (black and magenta). So, for CMYK screens, Earle’s work implies that the difference in coarseness between magenta and cyan should be larger than the difference between black and magenta, and that given a pattern composed of the three, the HVS is unlikely, or at least less likely, to notice the stochastic moir´e formed between cyan and black. In the above derivation of the hysteresis values for the four channels, (13.8-13.10) do not guarantee, with the exception of black and magenta, that two channels will not have the same principle wavelength. It is possible for cyan to be so coarse that even though it has a different intensity from black, it has a principle wavelength equal to that of black. Consider a situation where the intensity of black is low while both cyan and magenta have similar and high intensity values. In this situation, the cyan would be made coarse due to its similarity with magenta, but this coarseness comes as no result of the low intensity of black such that cyan and black have the same principal wavelength, and similarity in principal wavelengths is the one indicator of a visually apparent stochastic moir´e. To modify (13.8-13.10) in order to prevent any two-colors from having the same principal wavelength, we would need to measure the principal wavelength versus intensity versus hysteresis value, h, for a monochrome halftone and then use these data to identify the optimal hysteresis value for magenta as being the h that adequately separates magenta’s principal wavelength from that of black. For cyan, this would imply finding the h that adequately separates its wavelength from that of both black and magenta, and lastly, repeat for yellow constrained by the three prior colors. But this may be unnecessary given studies by Prazdny [99], who shows that the perception of order in glass patterns is also affected by the similarity in dot sizes, where differences in dot size erode the HVS’s ability to detect correlations between points. What this means for stochastic moir´e is that the difference in cluster size between cyan and black acts as an additional deterrent to stochastic moir´e visibility. What (13.8-13.10) do guarantee is that colors with neighboring luminance values will have different principal wavelengths with magenta always more coarse than black, cyan always more coarse than magenta, and yellow always more coarse than cyan. And this dissimilarity is, as

© 2008 by Taylor & Francis Group, LLC

´ AND GREEN-NOISE 13.5. STOCHASTIC MOIRE

513

Earle’s and Prazdny’s glass pattern studies would suggest, adequate for minimizing stochastic moir´e in four-color prints. But clearly, psychovisual evaluations of stochastic moir´e are a necessary and immediate concern for future stochastic moir´e research. In concluding this chapter, the reader should note that stochastic moir´e is a geometric phenomenon resulting from the superposition of two or more stochastic patterns, and it should be noted that Lau et al. [69] theorize that it is this stochastic moir´e phenomenon that leads to low-frequency graininess in color FM halftones. The framework that followed the work of Lau et al. for measuring the visibility of stochastic moir´e should not be confused with stochastic moir´e itself, and it should be noted that there may exist other methods by which moir´e visibility can be analyzed. If nothing else, the framework described here supports Lau et al.’s assertion of the relationship between low-frequency graininess and moir´e visibility. Furthermore, it justifies the combination of using blue- and green-noise halftoning as a means of minimizing low-frequency graininess between colors. It also adds more justification to the use of jointly blue-noise masks [119] and other halftoning algorithms [79] that closely correlate the component dither patterns of CMYK halftones. With regard to the impact of stochastic moir´e on green-noise, the results presented by Lau et al. and repeated here are surprising when you consider that green-noise had only been considered to be the halftoning technique of choice if and only if the printing device could not reproduce blue-noise halftones consistently. So the work of Lau et al. represents a major milestone in the study of green-noise halftoning because it is the first to claim that green-noise may be the preferred halftoning technique in ideal printing devices that, traditionally, would have only considered employing blue-noise.

© 2008 by Taylor & Francis Group, LLC

Chapter 14 Multi-Tone Dithering While the majority of this book has focused on the generation of dither patterns composed purely of black and white pixels, many instances of printing devices capable of multi-tone output exist that print dots of different intensities either through the manipulation of dye/pigment content or of droplet/laser spot size. For halftoning, the ability to print multiple shades/dot sizes of a given ink color creates a new range of problems in what is generally referred to as multi-toning. Specifically, how does one optimize the distribution of dots from an available set of ink intensities and droplet sizes for minimizing halftoning artifacts? Conventional error-diffusion for binary devices can easily be extended to the multilevel case by replacing the thresholding stage by a multilevel quantizer [53]. Katoh et al. [109] first applied error-diffusion to multilevel printing devices. The images halftoned using this conventional multilevel halftoning algorithm exhibit much lower quantization error as compared to the bilevel case, however, they suffer with banding artifacts in the regions of intermediate printable gray-levels. The reason is that, at the intermediate gray-level, there is no quantization error or contouring as we have the ink for that particular tonal level available. In the neighborhood of this zero error region, there is a sparse distribution of the black and white pixels in an otherwise constant gray region. This appears as a banding artifact to the human observer. A number of algorithms have been proposed to eliminate the problem of banding, focusing on equalizing the distribution of mean square error (MSE) between the halftoned and the original images by distributing the MSE uniformly over all the gray-levels. Ochi [93] has addressed the same problem by iterating the error-diffusion with a layered structure to remove the contouring at midtones. This improvement 515 © 2008 by Taylor & Francis Group, LLC

516

CHAPTER 14. MULTI-TONE DITHERING

is at the expense of increasing the deviation in the uniform density areas. Sugiura and Makita [118] developed a new multi-toning algorithm, based on error-diffusion, by adding noise to increase the deviation around the density, printed by the ink of intermediate gray-level. Spaulding et al. [113] generalized blue-noise dither arrays to the multi-tone problem by scaling the subject dither array to a certain intermediate range before thresholding. Miller and Smith [85] described another implementation of multilevel halftoning where a modularly addressed matrix is used to store pointers to a series of dither LUTs instead of actual dither values. In this method, the results of the screening process for each of the possible input levels are precalculated and stored in these LUTs. A major advantage of this approach is that any conceivable dot growth pattern can be specified, hence smoother visual transition can be achieved at intermediate tones thereby minimizing banding artifacts. Yu et al. [102, 101] introduced an overmodulation method, to achieve a smoother transition at the intermediate output levels. A preprocessing step is added before the screening, where the input pixel value is checked to see whether it is inside a predetermined range of any intermediate output levels. If not, this pixel is passed to the screening stage; otherwise, an overmodulation function is called to modify the input pixel value before passing it to the screen. While it has been determined that by mixing ink levels above and below the nearest level minimizes banding [35], no means has yet been devised for finding the optimal distribution of a given set of inks for any intermediate gray-level, nor have any means been derived for defining the optimal distribution of said dots for that gray-level. So while the optimal distribution of dots for producing aperiodic, disperseddot halftones can be described in terms of the blue-noise model [123], and aperiodic, clustered-dot halftones can be described in terms of the green-noise model [66], no such model exists for multi-tones satisfying similar constraints regarding the human visual system. In this chapter, we review such a multi-toning blue-noise model proposed by Bacca [103] intended to serve as a standard by which multi-tone conversion algorithms are optimized, with the better of two algorithms being the one whose spectral features more closely match those of the proposed model. In order to introduce such a model, a series of new tools for the analysis of multi-tone dither patterns are introduced. In particular, Bacca applies threshold decomposition [7], which allows for the representation of multi-tones as the superposition of N spatially correlated halftone patterns, where N is the number of

© 2008 by Taylor & Francis Group, LLC

14.1. SPECTRAL STATISTICS OF MULTI-TONES

517

available inks. Each one of these patterns can then be characterized itself as a blue-noise pattern. The spectral profile of blue-noise multitones is thus defined as the aggregation of the profiles of these halftones plus the cross-spectra generated by the interaction between dots of different intensities. Having a means by which we can evaluate the graininess of multi-tone textures for a given concentration of gray-level ink dot intensities, Bacca then derives the optimal concentration of inks to produce a particular shade of gray from the component ink set can. This chapter concludes be reviewing a set of algorithms, introduced by Bacca, that combine threshold decomposition with wellknown blue-noise halftoning algorithms to generate multi-tone dither patters that show the optimal spectral characteristics. In particular, we will look at the extension of error-diffusion and of DBS to multitoning. A spectral and spatial analysis of dither patterns generated with these methods is also developed based on well-known measures defined in Chapter 5. This analysis will show how the most pleasing patterns present the characteristics indicated in the development of the model.

14.1

Spectral Statistics of Multi-tones

multi-tone dither patterns representing a constant gray-level g can be modeled as stochastic processes, just as it was done with halftone patterns in Chapter 3. Assume that a multi-tone dither pattern is created using N different inks of intensities (g1 , g2 , ..., gN ) sorted according to intensity, starting with the lightest. A white pixel (where nothing is printed) is said to have intensity g0 = 0 while a black pixel is printed using intensity gN = 1. The dither pattern, therefore, contains pixels of N + 1 different intensities. Each multi-tone pixel M [n] is thus considered a realization of a discrete random variable obeying a probability density function P (M [n] = gi ) = pi |N (14.1) i=0 , where the probabilities pi |N of the i=0 indicate the proportion of pixels N corresponding inks included in the multi-tone, such that i=0 pi = 1. Furthermore, the probabilities are such that the mean or expected  value of M [n], E(M [N ]) = N i=0 pi gi , is equal to the gray-level g to be reproduced, while the variance is given by Var(M [n]) = E(M 2 [n]) − (E(M [n]))2

© 2008 by Taylor & Francis Group, LLC

518

CHAPTER 14. MULTI-TONE DITHERING =

N 



pi gi2

− g2 = σ2.

(14.2)

i=0

Looking now at the higher order moments for M [n] and hence obeying an optimal dot distribution as Ulichney did for binary bluenoise, we note that some of the fundamental principles of halftoning naturally extend to multi-toning since both aim to produce visually pleasant images. But, the analysis and synthesis of multi-tones presents new challenges when compared to halftones. First, effects in the spatial domain should be evaluated since the average intensity or the textures of the dither pattern can be affected by the possible superposition of dots of different intensities or by clustering of different kinds of pixels. Second, the spectral domain analysis of multi-tones becomes more complex as the number of inks increases. Patterns formed with dots of the same ink will have their own spectral profile and their superposition generates spectral cross-terms. The characterization of the spectral profile of blue-noise multi-tones needs to take into account all these components and evaluate their interaction. Treating a multi-tone as the superposition of a series of halftone patterns printed on top of each other with different inks, it can be related to color halftoning where three or more halftones, one for each of the primaries used by the printing system, are superimposed in order to generate the appearance of a continuous tone color image. One of the phenomena observed when overlapping halftones is the appearance of stochastic moir´e described in Chapter 11. In the multi-tone case the superposition of blue-noise patterns does not necessarily result in a good-quality pattern and, just as it happens in color, some correlation must be introduced between the patterns printed with different inks in order to obtain a visually pleasant result. The observations above provide a strong motivation to incorporate the correlation between different inks into the analysis and synthesis of multi-tones. To this end, a simple yet elegant method is proposed based on the threshold decomposition representation of signals [7, 39]. Threshold decomposition states that a discrete signal taken on one of k possible values can be represented as the weighted sum of k − 1 binary signals. For the case of multi-toning, define M as the multi-tone dither pattern, such that M [n] is its nth pixel. We further define the series of (binary) halftones Hi |N i=1 as 

Hi [n] =

© 2008 by Taylor & Francis Group, LLC

1, if M [n] ≥ gi 0, else.

(14.3)

14.1. SPECTRAL STATISTICS OF MULTI-TONES

519

M

d 3 H3

d 2 H2

d1 H1

Figure 14.1: Decomposition of a three-ink multi-tone M in a series of halftones Hi |3i=1 satisfying the stacking constraint.

The halftone Hi represents the level i threshold decomposition of the multi-tone M . According to this definition, a printed pixel in Hi indicates that a printed pixel of intensity gi or darker appears in the multitone in the same position. This also implies that there is a printed pixel in the same position in Hj for all j ≤ i. That is, the halftones in the threshold decomposition of M are constrained to stack. The multi-tone can be described in terms of its threshold decomposition representation as M [n] =

N 

di Hi [n],

(14.4)

i=1

where di = gi − gi−1 |N i=1 are the relative differences between the intensities of the printable inks. An example of how this decomposition is performed is shown in Fig. 14.1. The multi-tone Mis a 3 × 3 image printed with three inks with intensities (g1 , g2 , g3 ) = 31 , 23 , 1 . According to (14.3), the halftone H2 , for example, will contain a printed pixel in all positions where M has dots of intensity greater that or equal to 23 . Looking at Fig. 14.1, this results in all dots printed with inks g2 and g3 in M being represented by printed pixels in H2 . The other two halftones exhibit similar characteristics. The set of halftones Hi |N i=1 can, thus, be described as a set of correlated stochastic processes whose marginal densities are P (Hi [n]) =

© 2008 by Taylor & Francis Group, LLC

  N pj , j=i i−1

for Hi [n] = 1 j=0 pj , for Hi [n] = 0,

(14.5)

CHAPTER 14. MULTI-TONE DITHERING

520

with means and variances given by N 

μi =

pj

(14.6)

j=i

σi2 = μi (1 − μi ).

(14.7)

The mean of the multi-tone can be expressed as a function of the characteristics of the halftones Hi as E[M ] = E

N 



di Hi =

i=1

N 

di μi = g,

(14.8)

i=1

and since the random processes Hi are correlated, the variance of its linear combination is Var(I) = Var

N 



di Hi

i=1

=

N 

d2i Var(Hi ) + 2

i=1

N  N 

di dj Cov(Hi , Hj ), (14.9)

i=1 j=i+1

where Cov(Hi , Hj ) = E(Hi Hj ) − E(Hi )E(Hj ) is the covariance of the random processes Hi and Hj . The product Hi Hj with j ≥ i is equal to Hj , thus the covariance reduces to Cov(Hi , Hj ) = μj (1 − μi ), for j ≥ i.

(14.10)

Replacing (14.10) in (14.9) yields Var(M ) =

=

N 

d2i σi2

+2

N  N 

i=1

i=1 j=i+1

N 

N  N 

d2i σi2 + 2

i=1

i=1 j=i+1

di dj μj (1 − μi ) di dj

(14.11)

   μj (1 − μi )  σ

μi (1 − μj )

i σj .

The variance of the multi-tone thus results as the weighted sum of the variances of each one of the halftones in the threshold decomposition plus a weighted sum of cross-terms that indicate the interactions between dots of different intensities.

© 2008 by Taylor & Francis Group, LLC

RAPSD

14.2. MULTI-TONE BLUE-NOISE MODEL

521

σB2 σA2

fA

0

fB

Radial Frequency

2 __ 2

Figure 14.2: Optimal RAPSD for a two-ink multi-tone dither pattern with principal frequencies fA and fB and pattern variances of σA2 and σB2 , respectively.

14.2

Multi-Tone Blue-Noise Model

In deriving the optimal blue-noise multi-tone spectra, Bacca assumes a multi-tone, M [n], is represented as the superposition of a series of halftones as indicated in (14.4), and supposing that the Hi s are bluenoise binary dither patterns, then their spectra will have the shape indicated in Fig. 5.4, with amplitudes σi and principal frequencies fi given by ⎧ √ 1 ⎪ ⎨ μi , for μi < 4 1 , for 14 ≤ μi ≤ 34 (14.12) fi = ⎪ √ ⎩ 2 1 − μi , for μi > 34 . Note that the RAPSD of the patterns is not normalized by σ 2 , as in the case of binary blue-noise halftoning, and if the multi-tone dither pattern was generated as white-noise, its spectrum would be flat and have an amplitude σ 2 as indicated in (14.2). Since the multi-tone is a linear combination of stacking binary blue-noise patterns, the spectrum of the aggregate should preserve some of the spectral characteristics of the individual patterns. For example, these patterns have a low-frequency response close to zero and a flat high-frequency response originated by the elimination of clustering and the preservation of the high-frequency components of white-noise. These characteristics are also required for a multi-tone dither pattern. The mid-frequency range, however, should exhibit accumulations (peaks) of energy around the principal frequencies of each one of the halftones in the threshold decomposition representation. An example of two inks is shown in Fig. 14.2, where the characteristics just described can be appreciated. It has been stated earlier that there must be a correlation be-

© 2008 by Taylor & Francis Group, LLC

522

CHAPTER 14. MULTI-TONE DITHERING

tween the patterns Hi |N i=1 in order for the multi-tone to be visually pleasant. And in order to characterize this correlation, the theory of statistical signal analysis provides a series of measures that allow the analysis and quantification of the relationship between two or more signals in the spectral domain. Since the covariance allows us to study the similarities or differences between two signals in the time/spatial domain, it is natural for its Fourier transform, the cross-spectral density function (CSD), to be the first choice when analyzing the correlation of two signals in the spectral domain. This function is complex-valued and, in consequence, we should resort to analyze its magnitude and phase. The magnitude of the CSD is known as the cross-amplitude spectrum, and it represents the average value of the product of the components for each frequency. The phase of the CSD, the phase spectrum, is the average phaseshift between the components of the two signals at each frequency [100]. Since the CSD is the Fourier transform of the cross-correlation of the two signals, it can be calculated directly in the frequency domain by multiplying their PSDs. In consequence, spectral peaks corresponding to only one of the two signals can appear in the CSD even when there is no real relationship between the signals at that point. In order to avoid these kind of effects, another measure of the correlation of two signals in the frequency domain is required. The magnitude-squared-coherence function (MSC) is defined as the normalized modulus of the cross-power spectrum [112] 2 Kxy =

|Pxy (f )|2 , Px (f )Py (f )

(14.13)

where Pxy (f ) is the cross-spectral density function (CSD) of the patterns, and Py (f ) and Px (f ) are the PSDs of the signals y and x, respectively. Equation (14.13) is the frequency domain equivalent of the correlation coefficient and it can be interpreted as a measure of the correlation of two signals at each frequency. The correlation coefficient is defined as 2 rxy

(Cov(x, y))2 = . σx2 σy2

(14.14)

Applying this definition to a pair of sub-halftones Hi and Hj (j ≥ i) and replacing the variances and covariances with the values calculated

© 2008 by Taylor & Francis Group, LLC

14.2. MULTI-TONE BLUE-NOISE MODEL

523

previously leads to 2 rij =

(Cov(Hi , Hj ))2 μj (1 − μi ) , = 2 2 σi σj μi (1 − μj )

(14.15)

so the the terms in the last sum in (14.23) can be calculated as di dj rij σi σj . The MSC has several interesting properties; it is bounded between 0 and 1, being 0 for independent processes and 1 for signals that are linearly related (one is the result of filtering the other with a linear filter). In general, the MSC represents the portion of power of a signal at a given frequency that can be accounted for by its linear regression on the other. The coherence is invariant to linear filtration and is 2 2 symmetric (Kxy = Kyx ). To apply the MSC to the analysis of multi-tones, one must calculate it for the different pairs of halftones Hi |N i=1 defined in (14.3). The method of averaged periodograms used to estimate the PSDs can also be used to estimate the CSDs. Since the patterns analyzed are isotropic, the radial average of the MSC is representative of the behavior of the function in the two-dimensional plane and can be analyzed instead. Assuming that K periodograms are used to estimate the MSC, a value of K1 or −KdB is considered noise and indicates that the two patterns are independent. To study the behavior of the MSC, a series of multi-tones were generated using different mechanisms. A representative example is shown in Fig. 14.3, which shows the multi-tones of gray-level 150 out of 255, generated with the same inks and ink concentrations but with different methods. It also shows the RAPSD of the corresponding subhalftones and their radial MSCs. The top-left plot is a multi-tone generated using independent white-noise. The RASPDs of the sub-halftones are flat, as expected, and so are their MSCs, whose low values (-10dB corresponding to K = 10 periodograms) reflect no correlation between the patterns for any frequency. The top-right plot corresponds to the superposition of two blue-noise patterns generated independently. The pattern looks noisy and the lack of correlation between the patterns shows as an almost constant low level for the coherence, similar to the one observed for white-noise. At the same time, the RASPD plots of Fig. 14.3 show that each of the patterns used to create this multi-tone are blue-noise, proving that some correlation needs to be introduced between the blue-noise patterns used to generate the multi-tone. The bottom-left plots show a

© 2008 by Taylor & Francis Group, LLC

CHAPTER 14. MULTI-TONE DITHERING

524

f1 0

f2

0.28 0.5 RADIAL FREQUENCY

0.7071

0 ï5 ï10 0

0.5 RADIAL FREQUENCY

f1 0

RADIAL MSC

RADIAL MSC

0

RAPSD

0.5

RAPSD

0.5

0.7071

0

0.28 0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

0 ï5 ï10

(a)

(b)

f1

f2

0.28 0.5 RADIAL FREQUENCY

0.7071

0 ï5 ï10 0

0.5 RADIAL FREQUENCY

(c)

0.7071

f1 0

RADIAL MSC

RADIAL MSC

0

RAPSD

0.5

RAPSD

0.5

0

f2

f2

0

0.28 0.5 RADIAL FREQUENCY

0.7071

0

0.5 RADIAL FREQUENCY

0.7071

0 ï5 ï10

(d)

Figure 14.3: Radial MSCs, along with RAPSDs and the radial MSCs of the patterns used for their generation, of multi-tones of gray 150 generated as: (a) the superposition of two independent white-noise patterns, (b) two independent blue-noise patterns, (c) a suboptimal multi-tone generated with DBS, and (d) an optimal blue-noise multitone.

© 2008 by Taylor & Francis Group, LLC

Radial MSC

14.3. BLUE-NOISE MULTI-TONING

525

rij fi

0

fj

Radial Frequency

2 __ 2

Figure 14.4: Radial MSC of an ideal blue-noise multi-tone. multi-tone generated with DBS for multi-toning as in [79]. The pattern looks more uniform but presents clustering that reflects in high values for the coherence for the lower frequencies. Again, the RASPD of the sub-halftones shows they are blue-noise but the coherence shows an inadequate correlation between them that results in the artifacts mentioned before. The bottom-right pattern was generated following the blue-noise theory developed in this chapter. Looking at Fig. 14.3 (bottom-right), the pattern is the most visually pleasant of the four, where its radial MSC plot shows low coherence values for the lower frequencies and an almost constant value for all frequencies above the lowest cut-off frequency of the multi-tone. The conclusion obtained from these results is that the spectral coherence of the sub-halftones in a blue-noise multi-tone should be low for the low-frequency band and rise to the value of the correlation coefficient rij for all frequencies above the lowest principal frequency of the patterns being evaluated. An ideal plot of this function is shown in Fig. 14.4.

14.3

Blue-Noise Multi-Toning

Several algorithms for multi-toning have been proposed in the literature, mostly as extensions of previously developed halftoning algorithms. For example, the error-diffusion algorithm was modified by replacing the binary thresholding by a multi-level quantizer. Gentile [41] et al. used this approach with image-dependant and image-independent quantizers. The first problem these approaches faced was the appearance of banding artifacts in intensity levels close to those of the available inks. In order to avoid banding artifacts, new algorithms attempted to redistribute the error over the gray-scale. Faheem et al. [35] proposed

© 2008 by Taylor & Francis Group, LLC

CHAPTER 14. MULTI-TONE DITHERING

Y

SEPARATE

B

G

ERROR DIFFUSION

ERROR DIFFUSION

Bh

Gh

SUPERIMPOSE

526

M

Figure 14.5: Error-diffusion using gray-level separation [35]. to apply correlated error-diffusion in different channels that correspond to each of the available inks. An extension of DBS for the case of multi-toning has also been presented [79] and applied to the design of a multi-toning dither array. Faheem et al. divide the input image in two channels (or more, depending on the number of inks). One of these channels represents the proportion of gray ink to be used in the final halftone and the other the amount of black ink. The two channels are halftoned using errordiffusion in a correlated fashion, making sure that a black pixel does not overlap a gray pixel. The two halftones obtained are combined to generate the final multi-tone. The process is summarized in the block diagram in Fig. 14.5. The DBS algorithm presented in Chapter 7 was extended to multi-toning just by changing the definition of the coefficient a0 to 

a0 =

g[m0 ] − g[m1 ] for a swap g[m0 ] − gi for a toggle

(14.16)

where g[m0 ] and g[m1 ] represent the intensity of pixels printed at locations m0 and m1 , and gi is the intensity of the ith available ink in the printing system. These algorithms were designed before the development of the blue-noise theory for multi-toning. Examples of the results obtained with them can be seen in Figs. 14.6 to 14.9. In order for a multi-tone to be optimal according to the bluenoise theory just derived, the dots of different inks should be located in a correlated fashion in order to attain the spectral profile required. In Sec. 14.4, threshold decomposition was used to break down multi-tones into halftones in order to facilitate their analysis. A similar scheme can be applied to the synthesis of multi-tones of continuous-tone pictures

© 2008 by Taylor & Francis Group, LLC

14.3. BLUE-NOISE MULTI-TONING

527

Figure 14.6: Multi-toning of an 8 bit grayscale ramp with inks g1 = 127 and g2 = 255 using Faheem et al.

© 2008 by Taylor & Francis Group, LLC

528

CHAPTER 14. MULTI-TONE DITHERING

Figure 14.7: Multi-toning of an 8 bit grayscale image with inks g1 = 127 and g2 = 255 using Faheem et al.

© 2008 by Taylor & Francis Group, LLC

14.3. BLUE-NOISE MULTI-TONING

529

Figure 14.8: Multi-toning of an 8 bit grayscale ramp with inks g1 = 127 and g2 = 255 using multi-toning DBS

© 2008 by Taylor & Francis Group, LLC

530

CHAPTER 14. MULTI-TONE DITHERING

Figure 14.9: Multi-toning of an 8 bit grayscale image with inks g1 = 127 and g2 = 255 using multi-toning DBS

© 2008 by Taylor & Francis Group, LLC

14.3. BLUE-NOISE MULTI-TONING

531

N

di Hi 3 i=1

M i (Y)| N

i=1

Figure 14.10: Blue-noise multi-toning, where the continuous-tone image Y is divided into N components that are halftoned with any algorithm in a correlated fashion to generate a set of halftones Hi |N i=1 , which are the threshold decomposition representation of the final multi-tone.

to ensure optimality. This process is summarized in the block diagram in Fig. 14.10. Here, an image Y is divided into N sub-images Yi |N i=1 , one for each of the available inks. The sub-images are halftoned in a correlated fashion using any halftoning algorithm. Finally, the resulting halftones Hi |N i=1 are weighted and added together to obtain the final multi-tone M . This process can be justified as follows: Assume a constant patch of intensity g is to be reproduced using the inks gi |N i=1 , in proportions pi |N . The intensity of the patch can be represented as i=1 g=

N  i=1



gi pi (g) =

N 

di μi (g),

(14.17)

i=1

where μi (g) = N j=i pi (g) and di = gi − gi−1 . If a patch of intensity μ1 (g) is halftoned using blue-noise, the resulting dither pattern will have the same statistics as H1 , the level one threshold decomposition representation of an ideal multi-tone as defined in (14.3). The process is repeated for a patch of intensity μ2 (g) with the constraint that the resulting halftone should stack on the first one. The resulting dither pattern holds the same properties required by H2 . If the procedure is repeated for the remaining μi (g)|N i=3 , ensuring that the ith halftone stacks on the i − 1st, the result is a series of N halftones that meet

© 2008 by Taylor & Francis Group, LLC

CHAPTER 14. MULTI-TONE DITHERING

532

all the requirements indicated in (14.3) to (14.7). In consequence, a linear combination of these halftones results in an optimal blue-noise multi-tone. If instead of a patch of a constant value, the process is to be applied to a continuous-tone picture Y , a similar procedure should be carried out for each pixel in the picture, resulting in a set of sub-images Yi |N i=1 , defined by Yi [n] = μi (Y [n]) . (14.18) Each of the images Yi can be halftoned using any known blue noise halftoning algorithm to generate the halftones Hi , taking into account that the halftone Hj should stack on the halftone Hj−1 for all j. The final multi-tone is obtained as (see Sec. 14.1) M [n] =

N 

di Hi [n].

(14.19)

i=1

14.3.1

Blue-Noise Multi-Toning with Error-Diffusion

In order to generate blue-noise multi-tones by means of error diffusion, the stacking constraint should be involved in the quantization of the pixels such that 

Hi [n] =

1 if Yi [n] + ew i [n] ≥ 0 else

1 2

and Hi−1 [n] = 1

(14.20)

for i = 1, · · · , N . If i = 1, it is assumed that H0 [n] = 1∀n. Where, in order to set the pixel Hi [n] to 1, the original pixel should have a value larger than the threshold and also, the corresponding pixel in the previous sub-halftone Hi−1 [n] should be set to 1. In summary, to generate blue-noise multi-tones using error-diffusion, the original image should be divided into N sub-images as indicated in (14.18). These images should be halftoned in ascending order, taking into account the stacking constraint as indicated in (14.20). Examples of the results obtained with this algorithm are shown in Figs. 14.11 and 14.12.

14.3.2

Multi-Toning with DBS

In order to incorporate DBS as the halftoning algorithm to use in the multi-toning structure in Fig. 14.10, a few considerations need to be made. Assume DBS is applied to the sub-halftone Hi . When a toggle or a swap is performed, it is mandatory to ensure that the stacking

© 2008 by Taylor & Francis Group, LLC

14.3. BLUE-NOISE MULTI-TONING

533

Figure 14.11: Multi-toning of an 8 bit ramp with inks g1 = 127 and g2 = 255 using blue-noise error-diffusion.

© 2008 by Taylor & Francis Group, LLC

534

CHAPTER 14. MULTI-TONE DITHERING

Figure 14.12: Multi-toning of an 8 bit image with inks g1 = 127 and g2 = 255 using blue-noise error-diffusion

© 2008 by Taylor & Francis Group, LLC

14.4. OPTIMIZATION

535

constraint is maintained. In consequence, a change of the pixel Hi [n] from 1 to 0 will require that all the pixels Hj [n]|N j=i+1 are changed to zero. If the change is the opposite (from 0 to 1), all the pixels Hj [n]|i−1 j=1 have to be changed to 1. Since a change in a pixel in one of the subhalftones implies a change in several of them, the quality metric used to determine if a change is accepted needs to include all sub-halftones. Such a metric could be defined as E=

N 



Ei , where Ei =

ˆ i (x) − Yˆi (x)|2 dx. |H

(14.21)

i=1

The efficient implementation of the algorithm described by Analoui and Allebach should be applied to each sub-halftone independently, taking into account the previous considerations. Examples of the results obtained with this algorithm can be appreciated in Figs. 14.13 and 14.14.

14.4

Optimization

In the previous sections, the spectral characteristics of a multi-tone dither pattern were described based on its threshold decomposition, but the model presented is still incomplete since it depends on the proportions of each ink used to generate the pattern, and these proportions have a great impact in the appearance of the final halftone. As such, we will now determine the optimal amounts of each ink for a given gray-level assuming ideal printing. Once these values are found, the blue-noise model for multi-tones will be uniquely determined by the inks used and the tone to be reproduced. First, an optimization criteria must be chosen, and, since the final goal is to obtain a uniform pattern, one approach would be to optimize the proportions pi |N i=0 so that the variance of the pattern calculated in (14.2) is minimized. Suppose that it is required to reproduce a pattern of intensity g = gk , where gk is the intensity of one of the inks available for printing; the pattern with the lowest variance will be the one where every pixel takes the value gk . If such a scheme is applied to a picture, the regions of intensity gk will be clearly differentiable from their surroundings to generate an effect known as banding, which undesirable in multi-toning (See Figs. 14.18 and 14.19). Thus, as described next, the optimization cost function must rely on the human visual system model.

© 2008 by Taylor & Francis Group, LLC

536

CHAPTER 14. MULTI-TONE DITHERING

Figure 14.13: Gray-scale ramp multi-toned with blue-noise DBS multitoning.

© 2008 by Taylor & Francis Group, LLC

14.4. OPTIMIZATION

537

Figure 14.14: Gray-scale image multi-toned with blue-noise DBS multitoning.

© 2008 by Taylor & Francis Group, LLC

538

CHAPTER 14. MULTI-TONE DITHERING

Noting that the optimal multi-tone dither pattern is the one closer to a constant value patch of the corresponding intensity when viewed by a human observer, an approximation of the perceived multitone can be obtained by filtering it with an appropriate HVS model. The output of this process is an image that can be thought of as a constant signal plus some quantization noise. The optimal multi-tone dither pattern will, therefore, be the one that, after filtered with an HVS model filter, contains the least amount of remanent quantization noise. The metric used for this characterization is the integral of the power of the filtered multi-tone over the whole baseband, after removing the DC component. For this purpose we will use N¨as¨anen’s HVS model introduced in Chapter 4, since the simplicity of its mathematical formulation makes it useful for spectral analysis and optimization in the frequency domain. The model will be calculated with a value of S = 5 in × 300 dpi. On the other hand, a mathematical description of the spectra of the multi-tone is required in order to develop a process to optimize its characteristics. The model to be created is based on the optimal spectral profile (PSD) of a blue-noise halftone depicted in Fig. 5.4 and the threshold decomposition representation developed in Sec. 14.1. While the PSD is well defined, it does not have a closed mathematical form; however, it can be approximated by a properly weighted step function of the radial frequency as follows PSD(Hi ) = σi2 U (f − fi )

(14.22)

where U (x) = 1 for x ≥ 0 and U (x) = 0 otherwise. Assume the spectrum of the superposition of the first two halftones H1 and H2 needs to be found and suppose f1 ≤ f2 . The amplitude of the frequency response for f ≥ f2 is equal to the variance of the pattern obtained by the superposition of the two halftones and it can be calculated using (14.11). According to the model, the frequency components of the second halftone are zero for f < f2 . In consequence, the response of the superposition of these two patterns should be equal to the frequency response of H1 in this interval. The interaction between dots of different patterns is represented by the cross-term in (14.11). If a third halftone with a principal frequency f3 is superimposed to these two patterns, the frequency components for f < f3 remain unaltered and the amplitude for f ≥ f3 can be calculated again using (14.11). Under this model,

© 2008 by Taylor & Francis Group, LLC

14.4. OPTIMIZATION

539

the spectrum of a linear combination of halftones can be expressed as PSD(M ) =

N 

d2i σi2 U (f − fi ) + 2

i=1

N  N 

cij σi σj U (f − fM ij ), (14.23)

i=1 j=i+1

where fM ij = max(fi , fj ) and cij = di dj rij . In summary, the optimal blue-noise spectra of Eq. (14.23) will show a staircase-like pattern, starting with a low-frequency response close to zero, then ascending with peaks in the principal frequencies of each one of the patterns Hi , and ending in a flat high-frequency region. The noise power is defined as the integral over the baseband of the product of (14.23) and the frequency response of the HVS filter indicated in

PN (g, gi , pi ) =

1 2

− 12



1 2

− 12



−kf

e

 N 



d2i σi2 U (f − fi )

(14.24)

i=i

+2

⎞

N N  

cij σi σj U (f − fM ij )⎠ dxdy,

i=1 j=i+1

where f =

√

x2 + y 2 . Exchanging the integrals and sums results in

PN (g, gi , pi ) =

N 

d2i σi2

i=1

+2





1 2

− 12

N  N 

1 2

− 12

e−kf U (f − fi )dxdy

cij σi σj

i=1 j=i+1

1 2

− 12



1 2

− 12

(14.25)

e−kf U (f − fM ij )dxdy.

In (14.25), all integral terms are similar and represent the product of a decaying exponential by a step function over a square area. The result of this product is the value of the exponential in the area where the step function equals one and zero everywhere else. This integral is, therefore, equivalent to the volume under the exponential in the entire baseband after subtracting the undesired region around the origin, which results in

1 2

− 12



1 2

e1 −kf U (f − fi )dxdy =

−2



1 2



− 12

= VS −

© 2008 by Taylor & Francis Group, LLC

1 2

− 12

e−kf dxdy −

2π fi 0

0

f e−kf df dθ

 2π  −kfi 1 − e (kf + 1) , i k2

(14.26)

CHAPTER 14. MULTI-TONE DITHERING

540

where VS represents the value of the first integral that can be calculated numerically. Substituting (14.26) into (14.25) and after some manipulation results in ⎛

PN (g, gi , pi ) = VS ⎝

N 

d2i σi2 + 2

i=1



N N  

⎞

cij σi σj ⎠

(14.27)

i=1 j=i+1

N   2π  d2i σi2 1 − e−kfi (kfi + 1) − 2 k i=1

+2

N N  



⎞



cij σi σj 1 − e−kfM (kfM + 1) ⎠ .

i=1 j=i+1

Replacing the characteristics of the halftones (di , σi , cij , fi ) with the values calculated in Sec. 14.1 leads to a cost PN that is a function of the N intensities of the inks gi |N i=1 , the proportions of each ink pi |i=1 , and the gray-level g to be reproduced. The minimization of this cost function must yield the values of the pi s minimizing PN for each gray-level g. For simplicity, a solution to the case of multi-toning with two inks is presented first. This solution is then generalized to multi-toning with multiple inks. The optimization problem can be formulated for each value of the intensity g, in the interval 0 ≤ g ≤ 1, and a constant set of inks (0, g1 , 1) to obtain the optimum distribution of inks such that the noise power PN is minimized. All variables required for the calculation of the cost function can be written as functions of the graylevel g and the amount of black ink p2 since g = g1 p1 + p2 . The cost function is then a piece-wise function of only two parameters, as depicted in Fig. 14.15. For a better visualization, Fig. 14.15 depicts the cost as a function of the concentrations of the inks (p1 and p2 ). The analysis is performed with the cost as a function of the gray level to be reproduced g and the concentration of black ink p2 . According to previous definitions in Sec. 14.1, the constants in (14.27) can be rewritten in terms of g and p2 as 

σ12 σ22 f1

g − p2 + g1 p2 g − p2 + g1 p2 1− = g1 g1 = p2 (1 − p2 ) ⎧  g−p2 +g1 p2 ⎪ , ⎪ g1 ⎨



for g < g41 + p2 (1 − g1 ) 1 , for g41 + p2 (1 − g1 ) ≤ g ≤ = ⎪  2 ⎪ ⎩ 1 − g−p2 +g1 p2 , for g < 3g1 + p (1 − g ) 2 1 g1 4

© 2008 by Taylor & Francis Group, LLC

3g1 4

+ p2 (1 − g1 )

14.4. OPTIMIZATION

541

PN

1 __ 3 4

1

p2

__ 1 4

0

__ 1 4

0

p1

__ 3 4

Figure 14.15: Cost function in (14.27) for the case of two inks (127 and 255) as a function of the concentrations of the inks. ⎧ ⎪ ⎨

f2

√

for p2 < 14 1 , for 14 ≤ p2 ≤ = ⎪ √ ⎩ 2 1 − p2 , for p2 > 34 p2 ,

c1,2 = g1 (1 − g1

3 4

   p2 (g1 )

− g + p2 − g1 p2 ) . (g − p2 + g1 p2 )(1 − p2 )

It can be seen from the definitions above that the piece-wise nature of the cost function is due to the piece-wise definition of the principal frequencies of the halftones. The domain of the cost function is a section of the two-dimen1 sional plane limited by the lines p2 = g, p2 = g−g , and the positive 1−g1 g axis. The boundaries between the regions where the principal frequencies (and the cost function) change definitions are given by the lines p2 = 14 , p2 = 34 , g = g41 + p2 (1 − g1 ), and g = 3g41 + p2 (1 − g1 ). These lines divide the domain of the function in six sections, as shown in Fig. 14.16. The cost function takes on different values on each of the sections shown, but in all cases, the function can be shown to

© 2008 by Taylor & Francis Group, LLC

CHAPTER 14. MULTI-TONE DITHERING

542 1

f1= 1−p1

F

__ 3 4

f2= 1−p2

1 f1= __ 2

p2

E

1 f2= __ 2

D 1 __ 4 f1= p1

B 0

A 0

g 4

1 ____

g 4

3g + g 4

2 2 _________ ____ 1

f2= p2

C

2g1+ g2 _________ g1+ 2g2 g1 _________ 4 4

3g 4

_____ 2

g + 3g 4

_________ 2 1

1

g Figure 14.16: Detail of the domain of (14.27) in the plane g − p2 where the domain is divided into the six regions indicated A to F due to the discontinuities in the principal frequencies of the sub-halftones H1 and H2 . The values of the principal frequencies corresponding to each region are also indicated.

be concave. In consequence, its minima are located along the edges. Comparing the values on the edges, the minima for all values of g are found to be along the dotted trajectory in Fig. 14.17(a). Examples of multi-tones generated using this gray-level concentration are shown in Figs. 14.18 and 14.19. If each tone is considered individually, the textures and quality of reproduction are pleasant, but, when analyzing the pictures globally, banding artifacts are clearly appreciated in the fourth and seventh lines of the grayscale and some regions of the picture. The sharp transitions in the trajectory translate into sharp transitions in

© 2008 by Taylor & Francis Group, LLC

14.4. OPTIMIZATION

543

the multi-tone as well. To avoid banding, we can constrain the optimization process to find a curve where a small change in the gray-level Δg results in a small change in the concentration of black pixels Δp2 . The resulting trajectory replaces the sharp transitions with a smoother path following the local minima of the cost function. This is indicated by a dashed line in Fig. 14.17(a) with the obtained results in Figs. 14.20 and 14.21. Banding is eliminated, and the pleasant textures observed in the previous plot are maintained. Figure 14.17(b) shows the cost for each one of the trajectories, where the total cost incurred in each multi-tone is 68 for Fig. 14.18 and 83.9 for Fig. 14.20. These results can be compared with the ones obtained with blue-noise halftoning in Chapters 3 and 6. Now, in order to complete our description of the optimal distributions of the two inks versus the gray-level g, general form, for the concentrations of the inks and the principal frequencies of the model, as functions of the inks intensities, given as ⎧ g ⎪ g1 ⎪ ⎪ ⎪ ⎨ 3g2 −4g

p1 = ⎪ ⎪ ⎪ ⎪ ⎩

4(g2 −g1 ) 4g−4g1 +g2 4(g2 −g1 ) g2 −g g2 −g1

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎨ 4g−3g1

p2 = ⎪ ⎪ ⎪

f1 =

4(g2 −g1 ) 1 4 g−g1 g2 −g1

⎪ ⎩ ⎧  g ⎪ ⎪ ⎪ g1 ⎪ ⎪ ⎨ 1 2  4g−5g1 +2g2 ⎪ ⎪ ⎪ 4(g2 −g1 ) ⎪ ⎪ ⎩ 0 ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 4g−3g1 ⎨

f2 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 2 

4(g2 −g1 ) g2 −g g2 −g1

g ≤ 3g41 3g1 < g ≤ 2g14+g2 4 2g1 +g2 < g ≤ 3g14+g2 4 3g1 +g2