2,109 433 61MB
Pages 362 Page size 595 x 842 pts (A4) Year 2008
The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance
The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance
Paul S Addison Napier University, Edinburgh, UK
loP
Institute of Physics Publishing Bristol and Philadelphia
© lOP Publishing Ltd 2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library CataloguinginPublication Data
A catalogue record of this book is available from the British Library. ISBN 0 7503 06920 Library of Congress CataloginginPublication Data are available
Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Frederique Swist Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS 1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Academic + Technical, Bristol Printed in the UK by MPG Books Ltd, Bodmin, Cornwall
For Hannah, Stephen, Anthony and Michael
Contents
Preface 1
2
3
Getting started 1.1 Introduction 1.2 The wavelet transform 1.3 Reading the book
xi
1 1 2 3
The continuous wavelet transform 2.1 Introduction 2.2 The wavelet 2.3 Requirements for the wavelet 2.4 The energy spectrum of the wavelet 2.5 The wavelet transform 2.6 Identification of coherent structures 2.7 Edge detection 2.8 The inverse wavelet transform 2.9 The signal energy: waveletbased energy and power spectra 2.10 The wavelet transform in terms of the Fourier transform 2.11 Complex wavelets: the Morlet wavelet 2.12 The wavelet transform, short time Fourier transform and Heisenberg boxes 2.13 Adaptive transforms: matching pursuits 2.14 Wavelets in two or more dimensions 2.15 The CWT: computation, boundary effects and viewing 2.16 Endnotes 2.16.1 Chapter keywords and phrases 2.16.2 Further resources
6 6 6 9 9 11 14 21 25 28 33 35 45 51 55 56 63 63 63
The discrete wavelet transform 3.1 Introduction 3.2 Frames and orthogonal wavelet bases 3.2.1 Frames
65 65 65 65 Vll
Contents
Vlll
Dyadic grid scaling and orthonormal wavelet transforms The scaling function and the multiresolution representation The scaling equation, scaling coefficients and associated wavelet equation 3.2.5 The Haar wavelet 3.2.6 Coefficients from coefficients: the fast wavelet transform Discrete input signals of finite length 3.3.1 Approximations and details 3.3.2 The multiresolution algorithman example 3.3.3 Wavelet energy 3.3.4 Alternative indexing of dyadic grid coefficients 3.3.5 A simple worked example: the Haar wavelet transform Everything discrete 3.4.1 Discrete experimental input signals 3.4.2 Smoothing, thresholding and denoising Daubechies wavelets 3.5.1 Filtering 3.5.2 Symmlets and coiflets Translation invariance Biorthogonal wavelets Twodimensional wavelet transforms Adaptive transforms: wavelet packets Endnotes 3.10.1 Chapter keywords and phrases 3.10.2 Further resources 3.2.2 3.2.3 3.2.4
3.3
3.4
3.5
3.6 3.7 3.8 3.9 3.10
4
Fluids
4.1 4.2
4.3
4.4
4.5
Introduction Statistical measures 4.2.1 Moments, energy and power spectra 4.2.2 Intermittency and correlation 4.2.3 Wavelet thresholding 4.2.4 Wavelet selection using entropy measures Engineering flows 4.3.1 Jets, wakes, turbulence and coherent structures 4.3.2 Fluidstructure interaction 4.3.3 Twodimensional flow fields Geophysical flows 4.4.1 Atmospheric processes 4.4.2 Ocean processes Other applications in fluids and further resources
5 Engineering testing, monitoring and characterization 5.1 5.2 5.3
Introduction Machining processes: control, chatter, wear and breakage Rotating machinery
67 69 72 73 75 77 77 81 83 85 87 91 91 96 104 112 115 117 119 121 133 141 141 141 144 144 145 145 152 153 159 160 160 171 174 178 178 186 187 189 189 189 195
Contents
5.4 5.5 5.6 5.7 5.8
6
7
5.3.1 Gears 5.3.2 Shafts, bearings and blades Dynamics Chaos Nondestructive testing Surface characterization Other applications in engineering and further resources 5.8.1 Impacting 5.8.2 Data compression 5.8.3 Engines 5.8.4 Miscellaneous
IX
195 199 202 208 211 221 224 224 225 228 229
Medicine 6.1 Introduction 6.2 The electrocardiogram 6.2.1 ECG timing, distortions and noise 6.2.2 Detection of abnormalities 6.2.3 Heart rate variability 6.2.4 Cardiac arrhythmias 6.2.5 ECG data compression 6.3 Neuroe1ectric waveforms 6.3.1 Evoked potentials and eventrelated potentials 6.3.2 Epileptic seizures and epileptogenic foci 6.3.3 Classification of the EEG using artificial neural networks 6.4 Pathological sounds, ultrasounds and vibrations 6.4.1 Blood flow sounds 6.4.2 Heart sounds and heart rates 6.4.3 Lung sounds 6.4.4 Acoustic response 6.5 Blood flow and blood pressure 6.6 Medical imaging 6.6.1 Ultrasonic images 6.6.2 Magnetic resonance imaging, computed tomography and other radiographic images 6.6.3 Optical imaging 6.7 Other applications in medicine 6.7.1 Electromyographic signals 6.7.2 Sleep apnoea 6.7.3 DNA 6.7.4 Miscellaneous 6.7.5 Further resources
230 230 230 231 234 236 239 248 248 249 252 255 258 259 260 263 264 267 270 270
Fractals, finance, geophysics and other areas 7.1 Introduction 7.2 Fractals 7.2.1 Exactly selfsimilar fractals
278 278 278 279
270 273 275 275 275 276 276 277
Contents
x
7.2.2 Stochastic fractals 7.2.3 Multifractals 7.3 Finance 7.4 Geophysics 7.4.1 Properties of subsurface media 7.4.2 Surface feature analysis 7.4.3 Climate, clouds, rainfall and river levels 7.5 Other areas 7.5.1 Astronomy 7.5.2 Chemistry and chemical engineering 7.5.3 Plasmas 7.5.4 Electrical systems 7.5.5 Sound and speech 7.5.6 Miscellaneous Appendix Useful books, papers and websites 1 Useful books and papers 2 Useful websites
282 292 294 298 299 305 307 309 309 310 311 311
312 313
314 314 315
References
317
Index
351
Preface
Over the past decade or so wavelet transform analysis has emerged as a major new timefrequency decomposition tool for data analysis. This book is intended to provide the reader with an overview of the theory and practical application of wavelet transform methods. It is designed specifically for the 'applied' reader, whether he or she be a scientist, engineer, medic, financier or other. The book is split into two parts: theory and application. After a brief first chapter which introduces the main text, the book tackles the theory of the continuous wavelet transform in chapter 2 and the discrete wavelet transform in chapter 3. The rest of the book provides an overview of a variety of applications. Chapter 4 covers fluid flows. Chapter 5 tackles engineering testing, monitoring and characterization. Chapter 6 deals with a wide variety of medical research topics. The final chapter, chapter 7, covers a number of subject areas. In this chapter, three main topics are considered firstfractals, finance and geophysicsand these are followed by a general discussion which includes many of other areas not covered in the rest of the book. The theory chapters (2 and 3) are written at an advanced undergraduate level. In these chapters I have used italics for both mathematical symbols and key words and phrases. The key words and phrases are listed at the end of each chapter and the reader new to the subject might find it useful to jot down the meaning of each key word or phrase to test his or her understanding of them. The applications chapters (4 to 7) are at the same level, although a considerable amount of useful information can be gained without an indepth knowledge of the theory in chapters 2 and 3, especially in providing an overview of the application of the theory. It is envisaged that the book will be of use both to those new to the subject, who want somewhere to begin learning about the topic, and also those who have been working in a particular area for some time and would like to broaden their perspective. It can be used as a handbook, or 'handy book', which can be referred to when appropriate for information. The book is very much 'figure driven' as I believe that figures are extremely useful for illustrating the mathematics and conveying the concepts. The application chapters of the book aim to make the reader aware of the similarities that exist in the usage of wavelet transform analysis across disciplines. In addition, and perhaps more importantly, it is intended to make the reader aware of waveletbased methods in use in unfamiliar disciplines which may be transferred to Xl
xu
Preface
his or her own areathus promoting an interchange of ideas across discipline boundanes. The application chapters are essentially a whistlestop tour of work by a large number of researchers around the globe. Some examples of this work are discussed in more detail than others and, in addition, a large number of illustrations have been used which have been taken (with permission) from a variety of published material. The examples and illustrations used have been chosen to provide an appropriate range to best illustrate the waveletbased work being carried out in each subject area. It is not intended to delve deeply into each subject but rather provide a brief overview. It is then left to the reader to follow up the relevant references cited in the text for themselves in order to delve more deeply into each particular topic as he or she requires. I refer to over seven hundred scientific papers in this book which I have collected and read over the past three or so years. I have made every effort to describe the work of others as concisely and accurately as possible. However, if I have misquoted, misrepresented, misinterpreted, or simply missed out something I apologise in advance. Of course, all comments are welcomeemail address below. The book stems from my own interest in wavelet transform analysis over the past few years. This interest has led to a number of research projects concerning the waveletbased analysis of both engineering and medical signals including: nondestructive testing signals, vortex shedding signals in turbulent fluid flows, digitized spatial profiles of structural cracks, river bed sediment surface data sets, phonocardiographic signals, pulse oximetry traces (photoplethysmograms) and the electrocardiogram (ECG), the latter leading to patent applications and a university spinoff company, Cardiodigital Ltd. Quite a mixed bag, at first appearance, but with a common thread of wavelet analysis running throughout. I have featured some of this work in the appropriate chapters. However, I have tried not to swamp the application chapters with my own workalthough the temptation was high for a number of reasons including knowledge of the work, ease of reproduction etc. I hope I have struck the correct balance. All books reflect, to some extent, the interests and opinions of the author and, although I have tried to cover as broad a range of examples as possible, this one is no exception. Coverage is weighted to those areas in which I have more interest: fluids, engineering, medicine and fractal geometry. Geophysics and finance are given less space and other areas (e.g. astronomy, chemistry, physics, nonmedical biology, power systems analysis) are detailed briefly in the final chapter. There are some idiosyncrasies in the text which are worth pointing out. I am anf person not an w person: I prefer Hertz to radians per second. I can tap my fingers at approximately 5 Hz, or 1 Hz, I know what 50 Hz means (mains hum in the UK) and so on: w, I have to convert. Hence the frequencies in the text are in the form of l/time either in Hertz or nondimensionalized. The small downside is that the mathematics, in general, contain a few more termsmostly 2s and 'iTs. I have devoted a whole chapter to the continuous wavelet transform. It is noticeable that many current wavelet texts on the market deal only with the discrete wavelet transform, or give the continuous wavelet transform a brief mention en route to the theory of the discrete
Preface
Xlll
wavelet transform. I believe that the continuous wavelet transform has much to offer a wide variety of data analysis tasks and I attempt, through this text, to redress the balance somewhat. (Actually, the proportion of published papers which concern the continuous wavelet transform as opposed to the discrete wavelet transform is much higher than that represented by the currently available wavelet text books.) The book is focused on the wavelet transform and makes only passing reference in the application chapters to some of the other timefrequency methods now available. However, I have added sections on the short time Fourier transform and matching pursuits towards the end of chapter 2 and on wavelet packets at the end of chapter 3 respectively. Finally, note that I have developed the discrete wavelet transform theory in chapter 3 in terms of scale rather than resolution, although the relationship between the alternative notations is explained. I would like to thank the following people for taking the time to comment on various drafts of the manuscript: Andrew Chan of Birmingham University, Gareth Clegg of Edinburgh University (formerly at The Royal Infirmary of Edinburgh), Maria Haase of Stuttgart University and Alexander Droujinine of HeriotWatt University. I would like to thank Jamie Watson of CardioDigital Ltd for his comments on the draft manuscript and for his close collaboration over the years (and various bits of computer code!). I would also like to thank all those authors and publishers who gave their consent to reproduce their figures within this text. I am grateful to those funding bodies who have supported my research in wavelet analysis and other areas over the years, including the Engineering and Physical Science Research Council (EPSRC), the Medical Research Council (MRC) and the Leverhulme Trust. And to those other colleagues and collaborators with whom my wavelet research is conducted and who make it so interesting, thanks. Special thanks to my wife, Stephanie, who has supported and encouraged me during the writing of this book. Special thanks also to my parents for their support and great interest in what I do. Although it has been a long hard task, I have enjoyed putting this book together. I have certainly got a lot out of it. I hope you find it useful. Paul S Addison January 2002 [email protected]
Chapter 1
Getting started
1.1
Introduction
The wavelet transform (WT) has been found to be particularly useful for analysing signals which can best be described as aperiodic, noisy, intermittent, transient and so on. Its ability to examine the signal simultaneously in both time and frequency in a distinctly different way from the traditional short time Fourier transform (STFT) has spawned a number of sophisticated waveletbased methods for signal manipulation and interrogation. Wavelet transform analysis has now been applied in the investigation of a multitude of diverse physical phenomena, from climate analysis to the analysis of financial indices, from heart monitoring to the condition monitoring of rotating machinery, from seismic signal denoising to the denoising of astronomical images, from crack surface characterization to the characterization of turbulent intermittency, from video image compression to the compression of medical signal records, and so on. Many of the ideas behind wavelet transforms have been in existence for a long time. However, wavelet transform analysis as we now know it really began in the mid1980s where they were developed to interrogate seismic signals. Interest in wavelet analysis remained within a small, mainly mathematical community during the rest of the 1980s with only a handful of scientific papers coming out each year. The application of wavelet transform analysis in science and engineering really began to take off at the beginning of the 1990s, with a rapid growth in the numbers of researchers turning their attention to wavelet analysis during that decade. The past few years have each seen the publication of over one thousand refereed journal papers concerning the wavelet transform, covering all numerate disciplines. Figure 1.1 shows this rapid increase in waveletbased scientific papers published in recent years. The wavelet transform is a mathematical tool which is now common in many data analysts' toolboxes. This book aims to provide the reader both with an introduction to the theory of wavelet transforms and an overview of its use in practice. The two remaining sections of this short chapter contain, respectively, a brief nonmathematical description of the wavelet transform and a guide to subsequent chapters of the book.
1
2
Getfing started 1400
~
1200
tOOO
~
400 200
o
/
/
./
./ ~
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 ye"
Figure 1.1. Yearly count of scientific papers concerning wavelets 19902001. The plot contains the number of papers with 'wavelet' or 'wavelets' in the title, keywords or abstract of refereed journal papers. Source: Web of Science, httpllwos.mimas.ac.uk/ (Nole that a handful of these papers do not concern wavelet transforms, but rather refer to physical phenomena where the term wavelet has been used to describe a small localized wave.)
1.2
The wavelet transform
Wavelet transform analysis uses little wavelike functions known as wavelets. Actually. 'local' wavelike function is a more accurate description of a wavelet. Figure 1.2(a) shows a few examples of wavelets commonly used in practice. Wavelets are used to transform the signal under investigation into another representation which presents the signal information in a more useful form. This transformation of the signal is known as the wavelel trallSform. Mathematically speaking, the wavelet transform is a convolution of the wavelet function with the signal and we will see exactly how this is done in chapters 2 and 3. Here we stick to schematics. The wavelet can be manipulated in two ways: it can be moved to various locations
on the signal (figure 1.2(b)) and it can be stretched or squeezed (figure 1.2(c)). Figure 1.3 shows a schematic of the wavelet transform which basically quantifies the local matching of the wavelet with the signal. ff the wavelet matches the shape of the signal well at a specific scale and location, as it happens to do in the top plot of figure 1.3, then a large transform value is obtained. If, however, the wavelet and the signal do not correlate well, a low value of the transform is obtained. The transform value is then located in the twodimensional transform plane shown at the bottom of figure 1.3 (indicated by the black dot). The transform is computed at various locations of the signal and for various scales of the wavelet, thus filling up the transform plane: this is done in a smooth continuous fashion for the cOllfinuol/s wal'eleI tram/arm (CWT) or in discrete steps for the discrete wavelet trans/arm (DWT).
Reading the book
3
(a)
(b)
(c)
/\
Figure 1.2. The little wave. (a) Some wavelets. (b) Location. (c) Scale.
Plotting the wavelet transform allows a picture to be built up of the correlation between the waveletat various scales and locationsand the signal. In subsequent chapters we will cover the wavelet transform in more mathematical detail.
1.3
Reading the book
The purpose of the book is both to introduce the wavelet transform and to convey its multidisciplinary nature. This is done in the subsequent chapters of the book by first providing an elementary introduction to wavelet transform theory and then presenting a wide range of examples of its application. It will quickly become apparent that
4
Getting started
local matching of wavelet and signal leads to a large transform value
Wavelet Transform
scale ,                          ,
current wavelet  . scale
current wavelet location
location
Figure 1.3. The wavelet, the signal and the transform.
very often the same wavelet methods are used to interrogate signals from very different subject areas, where quite unrelated phenomena are under investigation. The book is split into two distinct parts: the first part, chapters 2 and 3, deals respectively with continuous wavelet transform theory and discrete transform theory; while the second part, chapters 4, 5, 6 and 7, presents examples of their application in science, engineering, medicine and finance. There are a number of ways to read this book: from the linear (beginning to end) via the targeted (employing the index) to the random (flicking through) approach. The reader unfamiliar with wavelet theory should read chapters 2 and 3 before moving on to the sections of particular relevance to his or her area of interest. The reader is also advised to look outside his or her own area to see how wavelets are being employed elsewhere. Details of further resources concerning the theory and applications of wavelet analysis are provided at the end of each chapter. The appendix lists a selection of useful books, papers and websites. The book contents are outlined in more detail as follows: Chapter 2: This chapter presents the basic theory of the continuous wavelet transform. It outlines what constitutes a wavelet and how it is used in the transformation of a signal. In the latter part of the chapter the continuous wavelet transform is
Reading the book
5
compared both with the short time Fourier transform and the matching pursuit method. Chapter 3: The discrete wavelet transform is described in this chapter. Orthonormal discrete wavelet transforms are considered in detail, in particular those of Haar and Daubechies. These wavelets fit into a multiresolution analysis framework where a discrete input signal can be represented at successive approximations by a combination of a smoothed signal component plus a sum of detailed wavelet components. The chapter ends by looking briefly at wavelet packets, a generalization of the discrete wavelet transform which allows for adaptive partitioning of the timefrequency plane. Chapter 4: This chapter deals with fluid mechanics, a subject that is always hungry for new mathematical techniques. The timefrequency localization properties of the wavelet transform have been employed extensively in the study of a wide variety of fluid phenomena including the intermittent nature of fluid turbulence, the characteristics of turbulent jets, the nature of fluidstructure interactions and the behaviour of large scale geophysical flows. Chapter 4 also contains the mathematics for discrete wavelet statistics and power spectra following on from some of the basic theory given in chapter 3. Chapter 5: In this chapter a close look is taken at the application of wavelet transforms to a variety of pertinent problems in engineering. These applications include the assessment of machine processes behaviour, condition monitoring of rotating machinery, the analysis of nonlinear and transient oscillations, the characterization of repeated impacting on structural elements, the interrogation of nondestructive testing signals, and the characterization of rough surfaces. Chapter 6: Medical applications of wavelet transform analysis are covered in this chapter. Wavelet transform methods have been used to characterize a wide variety of medical signals. Many of these are reviewed in this chapter, including the ECG, EEG, EMG, pathological sounds (lung sounds, heart sounds and arterial sounds), blood flows, blood pressures, DNA sequences and medical images (optical, xray, NMR, ultrasound etc.). Chapter 7: This final chapter covers a variety of areas of application. Most of the chapter is devoted to three main subjectsfractal geometry, finance and geophysicswith a separate section devoted to each of them. The final part of the chapter provides a brief account of the role wavelet transform analysis has played in a number of other areas including astronomy, plasma physics, electrical power systems, chemical analysis and more. Appendix: The appendix contains a list of useful papers, books and websites concerning wavelet transform theory and application. These have been chosen by the author for their extensive content and/or clarity of presentation.
Chapter 2
The continuous wavelet transform
2.1
Introduction
This chapter covers the basic theory of the continuous wavelet transform (CWT). We will first determine what constitutes a wavelet, how it is used in the transformation of a signal and what it can tell us about the signal. We then consider the inverse wavelet transform and the reconstruction of the original signal. We will look at the energypreserving features of the wavelet transform and how it may be used to produce wavelet power spectra. Finally, we will compare the wavelet transform both with the short time Fourier transform (STFT) and matching pursuit (MP) method.
2.2
The wavelet
The wavelet transform is a method of converting a function (or signal) into another form which either makes certain features of the original signal more amenable to study or enables the original data set to be described more succinctly. To perform a wavelet transform we need a wavelet which, as the name suggests, is a localized waveform. In fact, a wavelet is a function 1jJ(t) which satisfies certain mathematical criteria. As we saw briefly in the previous chapter, these functions are manipulated through a process of translation (i.e. movements along the time axis) and dilation (i.e. spreading out of the wavelet) to transform the signal into another form which 'unfolds' it in time and scale. Note that in this chapter and the next we assume that the signal to be analysed is a temporal signal, i.e. some function of time such as a velocity trace from a fluid, vibration data from an engine casing, an ECG signal and so on. However, many of the applications discussed later in the book concern wavelet analysis of spatial signals, such as well logged geophysical data, crack surface profiles, etc. In these cases, the independent variable is space rather than time; however, the wavelet analysis is performed in exactly the same way. A selection of wavelets commonly used in practice is shown in figure 2.1. We will consider some of them in more detail as we proceed through the text. As we can see from the figure they have the form of a small wave, localized on the time axis. There 6
The wavelet ,/(1) ,         ,
7
,/(1) ,            ,
v (,) '            " t
(b) L 
_____='
,/(1) ,           ,
(e) ' 
J
(d) L 
'
t
Figure 2.1. Four wavelets. (a) Gaussian wave (first derivative of a Gaussian), (b) Mexican hat (second derivative of a Gaussian). (c) Haar. (d) Morlel (real part).
are, in fact, a large number of wavelets to choose from for use in the analysis of our data. The best one for a particular application depends on both the nature of the signal and what we require from the analysis (Le. what physical phenomena or process we are looking to interrogate, or how we are trying to manipulate the signal). We will begin this chapter by concentrating on a specific wavelet, the Mexican hat, which is very good at illustrating many of the properties of continuous wavelet transform analysis. The Mexican hat wavelet is shown in figure 2.I(b) and in more detail in figure 2.2(a). The Mexican hat wavelet is defined as
(2.1 ) The wavelet described by equation (2.1) is known as the mother wavelet or analysing wavelet. This is the basic form of the wavelet from which dilated and translated versions are derived and used in the wavelet transform. The Mexican hat is, in fact, t2j2 the second derivative of the Gaussian distribution function e : that is, with unit variance but without the usual 1/ J2n normalization factor. The Mexican hat normally used in practice, Le. that given by equation (2.1) and shown in figure 2.2(a), is actually the negative of the second derivative of the Gaussian function. Al! derivatives of the Gaussian function may be employed as a wavelet. Which is the most appropriate one to use depends on the application. Both the first and second derivatives of the Gaussian are shown in figures I(a) and I(b). These are the two that are most often used in practice. Higherorder derivatives are less commonplace.
The continuous wavelet transform
8
1,....""7"1',........, 0.5
....
~
''
7
0 0.5
1
4
3
1
2
1
0
4
t
I
a
I
(a)
3
2
~
I
a.
b
4
o 1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
I Ip Ie a
I I I
.... I
,I
(b)
Peak frequency  I p
= Vi 2:rt
Bandpass frequency  Ie =
V5i2 2:rt
V2
and also  I p = 2:rt
Figure 2.2. The Mexican hat mother wavelet and its associated energy spectrum. (a) The Mexican hat mother wavelet (named for an obvious reason!). Notice that, for the Mexican hat, the dilation parameter a is the distance from the centre of the wavelet to where it crosses the horizontal axis. (b) The energy spectrum of the Mexican hat shown in (a). Note that as it is a real wavelet its Fourier spectrum is mirrored around the zero axis. (a is the standard deviation of the spectrum around the vertical axis.)
The energy spectrum of the wavelet
2.3
9
Requirements for the wavelet
In order to be classified as a wavelet, a function must satisfy certain mathematical criteria. These are: 1. A wavelet must have finite energy: (2.2)
where E is the energy of a function equal to the integral of its squared magnitude and the vertical brackets II represent the modulus operator which gives the magnitude of 1jJ(t). If 1jJ(t) is a complex function the magnitude must be found using both its real and complex parts. 2. If ~(f) is the Fourier tran~form of 1jJ(t), i.e.
~(f)
=
f~oo
1jJ( t) e i(27rf)t dt
(2.3)
then the following condition must hold: (2.4)
This implies that the wavelet has no zero frequency component, ~(O) = 0 or, to put it another way, the wavelet 1jJ(t) must have a zero mean. Equation (2.4) is known as the admissibility condition and Cg is called the admissibility constant. The value of Cg depends on the chosen wavelet and is equal to Jr for the Mexican hat wavelet given in equation (2.1). 3. An additional criterion that must hold for complex wavelets is that the Fourier transform must both be real and vanish for negative frequencies. We shall consider complex wavelets towards the end of this chapter when we will take a close look at the Morlet wavelet.
2.4
The energy spectrum of the wavelet
Wavelets satisfying the admissibility condition (equation (2.4)) are in fact bandpass filters. This means in simple terms that they let through only those signal components within a finite range of frequencies (the passband) and in proportions characterized by the energy spectrum of the wavelet. A plot of the squared magnitude of the Fourier transform against frequency for the wavelet gives its energy spectrum. For example, the Fourier energy spectrum of the Mexican hat wavelet is given by
(2.5) where the subscript F is used to denote the Fourier spectrum as distinct from the waveletbased spectrum defined later in section 2.9. A plot of the energy spectrum
l a T h e continuous wavelet transform
of the Mexican hat wavelet is shown in figure 2(b). Note that, as the Mexican hat wavelet is a real function, its Fourier spectrum is symmetric about zero. We will see later that complex wavelets do not have negative frequency components (requirement 3 above). The peak of the energy spectrum occurs at a dominant frequency of f p = ±J2/27!". The second moment of area of the energy spectrum is used to define the passband centre of the energy spectrum, fo as follows:
50 f21~(f) 2 df 2 50 I~(f) 4f 00
1
(2.6)
00
1
where fe is simply the standard deviation of the energy spectrum about the vertical axis. For the Mexican hat mother wavelet, j~ is equal to V572/27!" or 0.251 Hz. In practice we require a characteristic frequency of the mother wavelet, such as ./'p, fe or some other, in order to relate the frequency spectra obtained using Fourier transforms to those obtained using wavelet transforms. Later we will see how these characteristic frequencies change as the mother wavelet is stretched and squeezed through its dilation parameter. When performing wavelet transform analysis it is important that the energy spectrum of the wavelet is considered, as it indicates the range and character of the frequencies making up the wavelet. From equations (2.1) and (2.2) we see that the total energy of the Mexican hat wavelet is finite and given by E
=
.f~oo 'IjJ(t)2 dt = .f~oo [(1 
2 t2 2 t ) e / ]2 dt
=
~ J7f
(2.7)
The energy of a function is also given by the area under its energy spectrum. For the Mexican hat wavelet this gives us
(2.8) Hence
(2.9) This is a result we would expect for any function from Parseval's theorem. Often, in practice, the wavelet function is normalized so that it has unit energy. To do this for the Mexican hat we modify its definition given in equation (2.1). From equation (2.7) we see that it is normalized to have unit energy by dividing it by (3J7f/4) 1/2. This gives (2.10) Both equation (2.1) and equation (2.10) are commonplace in the literature. The only alteration necessary when employing the normalized Mexican hat of equation (2.10) rather than that defined in equation (2.1) is in the value of the admissibility constant C g , which must be changed from 7r to 4J7f/3. In the rest of this chapter we will stick to our original definition of the Mexican hat given by equation (2.1).
The wavelet transform
2.5
11
The wavelet transform
Now we have chosen a mother wavelet, how do we put it to good use in a signal analysis capacity? First we require our wavelet to be more flexible than that defined earlier, i.e. 'IjJ(t). We can perform two basic manipulations to make our wavelet more flexible: we can stretch and squeeze it (dilation) or we can move it (translation). Figure 2.3(a) shows the Mexican hat wavelet stretched and squeezed to respectively 1.0
0.5 ,....
(:l
0
b
?
0.5
1.0
L..'''1I++1f'''.....
(a)
1.0,.....r··
Signal set to zero here
j
r (f)
Beginning of signal segment
End of signal segment
Figure 2.35 (continued). (d) Periodization involves the repetition of the signal along the time axis. (e) Reflection. (f) Smoothing window.
extent as the wavelet a scale increases and the wavelet extends further beyond the edge of the signal when close to it. Finally, when plotting the wavelet transform, we want to be able to discern features associated with the signal at specific a scales. It is often the case that features
The CWT: computation, boundary effects and viewing
61
x(t)
Polynomial fit
(g)
Beginning of signal segment
End of signal segment
x(t)
Signal outwith segment
(h)
Beginning of signal segment
r
End of signal segment
Figure 2.35 (continued). (g) Polynomial fitting. (h) Signal following.
at certain scales dominate the transform plot, obscuring the detail at other scales. In order to accentuate this hidden detail we can change the weighting parameter w(a) in value to a value which is more suitable at the transform from the usual 1/ highlighting features at a specific a scale of interest. This is shown in figure 2.36 for a sinusoidal signal containing a burst of noise. The transform plot with w( a) set to 1/ shows up the sinusoidal waveform well, but this dominates and the noise is not highlighted particularly well. By changing the weighting to w(a) = a1.5 we can enhance the noise within the plot at lower a scales. When w(a) is set to a 2.S we can see that the noise now dominates the transform plot. It is sometimes useful to vary w(a) in this way to accentuate features at different scales in the transform plot which might otherwise have been missed. Another way to weight the transform plot to show up small and large amplitude features is to use a logarithmic scale for T(a, b). However, two problems arise with this approach. The first is that we cannot take the logarithm of the negative values of T(a, b) and the second is that taking the logarithm of nearzero values of T(a, b) produces very large negative
va
va
62
The comiflliOUS wavelet transform
ty\/V\JI (aJ
(b)
II"
(0)
(d)
(eJ Figure 2.36. Accentuating features in the wavelet transform plot. (a) Original signal. (b) Transform plot with \I'(a) = lj.fii. (c) Transform plot with \I'(a) = al.5. (d) Transform plot with \I'(a) = a 2.5 . (e) Logarithmic plot of the modulus of the Mexican hat wavelet transform IT(a,b)l. Nearzero values of IT(a, b)1 produce large negative logarithmic values. These are omilled using a minimum cutofT value, where aIlIT(a,b)1 values less than the cutoff are set to this value before computing logarithms.
numbers. We can get around these two problems, respectively, by using the modulus of the transform IT(a , b)1 to avoid negatives and setting a cutoff, or floor, value of IT(a,b)1 in order to both avoid zero and nearzero values and limit the extent of the logarithmic scale. This is illustrated in figure 2.36(e). Logarithmic plotting is particularly good at highlighting simultaneously features in the signal occurring at very different orders of magnitude, see for example the logarithmic plots of the ECG signals in figures 6.12 to 6.15 of chapter 6.
Endnotes
2.16
Endnotes
2.16.1
Chapter keywords and phrases
63
(You may find it helpful to jot down your understanding of each of them.) wavelet mother/analysing wavelet Mexican hat wavelet Fourier transform admissibility condition admissibility constant energy spectrum bandpass filter
2.16.2
passband centre dilation parameter location parameter wavelet transform inverse wavelet transform wavelet transform modulus maxima scalogram spectrogram
wavelet variance M orlet wavelet wavelet transform ridges short time Fourier transform windowed Fourier atom timefrequency atom matching pursuit boundary effects
Further resources
There are a number of sources of information at an introductory level concerning the continuous wavelet transform. Gade and GramHansen (1997) compare wavelet transforms with short time Fourier transforms, presenting timefrequency plots of a variety of signals including speech and engine vibrations. Sarkar and Su (1998) detail the properties of the continuous wavelet transform from the perspective of electrical engineers. The relationship between Fourier and wavelet spectra is described by Perrier et al (1995). Wong and Chen (2001) provide a number of illustrative examples of phase and modulus plots in their Morlet waveletbased study of nonlinear structural oscillations. See also chapter 1 of the book by Holschneider (1995) which contains many informative modulus and phase plots for a variety of wavelets and various simple signal features. Holshneider gives details of many other wavelets including Bessel, chirp, Cauchy, Poisson and Marr. Many websites contain good introductory material on the continuous wavelet transform. The appendix lists some websites from which to begin a search. Concise treatments of the continuous wavelet transform which provide a little more of the mathematical background at a reasonable mathematical level are to be found in chapter 3 of the book by Kaiser (1994), chapter 3 of the book by Blatter (1998), chapter 5 of the book by Vetterli and Kovacevic (1995) and the paper by Koornwinder in the book edited by the same author (Koornwinder, 1993). Heisenberg's uncertainty principle is covered by many authors; see for example Mallat (1998) or Kaiser (1994) for some extra information. Ridges have been used by Staszewski (1997, 1998a) in a new procedure for nonlinear system identification. (More details are given in chapter 5, section 5.4.) See also the early paper by Delprat et al (1992) and the algorithms for ridge detection of noisy signals by Carmona et al (1997, 1999). On a related topic, the relationship between standard wavelet scalograms and Fourier wavelengths is discussed by Meyers et al (1993), who provide brief mathematical detail in the appendix. See also Torrence and Compo (1998) who provide the information on how to find the Fourier wavelengths for the Morlet, Paul and all the derivatives of Gaussian wavelets (including the Mexican
64
The continuous wavelet transform
hat). More information on modulus maxima and signal reconstruction can be found in the papers by Mallat and Hwang (1992) and Mallat and Zhong (1992). Examples of the use of modulus maxima and ridge methods are to be found in chapters 4 to 7 of this book (e.g. figures 5.27 and 5.28 in chapter 5 and figures 6.20 and 6.21 in chapter 6). Further information on the application of ridges, modulus maxima, the complete Morlet wavelet and complex Mexican hat can also be found in Addison et al (2002a). In this chapter we considered the twodimensional Mexican hat wavelet. More information concerning the twodimensional continuous wavelet transform can be found in the book by Antoine (1999). A twodimensional Morlet wavelet is given by Kumar and FoufoulaGeorgiou (1994, p 29) and Peyrin and Zaim (1996). Koornwinder (1993) provides a brief outline of the mathematics of multidimensional wavelet transforms. See also the discussion by Daubechies (1992, pp 3334) concerning the continuous wavelet in higher dimensions where rotations can be introduced into the definition of higherdimensional wavelets. Numerous, diverse applications of the continuous wavelet transform to real signals are given in chapters 4 to 7 of this book, all of which require a discretized version of the transform. We dealt briefly with this issue in section 2.15. For more information on the discretization of the continuous wavelet transform see for example Jones and Baraniuk (1991), Sadowsky (1994) and Jordan et al (1997). The latter authors discuss many of the implementation issues that arise from the discretization itself. In particular they show how to convert from the nondimensional times and frequencies of the mother wavelet to physical measures. They use the Morlet wavelet to illustrate their discussion, applying it to the velocity fluctuations measured in a subsonic wake undergoing transition to turbulence. Note that they use angular frequency and a Morlet wavelet which is not normalized to have unit energy. In this chapter we have not dealt directly with the issue of the physical meaning of the a scale parameter. It is nondimensional. However, in practice it has to be linked to the timescales under investigation. The easiest thing to do is relate a = 1 to an appropriate unit of time, e.g. 1 second, 1 day, 1 year, etc. This has been done in this chapter when necessary: see for example figure 2.18 containing the vortex shedding time series where a = 1 corresponded to 1 second. It can also be linked to the sampling interval ~t or some multiple of it. We do not even have to use the a scale parameter to link to a temporal scale but can employ another measure of spread such as the standard deviation of the wavelet energy density in time; this has been done for the Morlet wavelet by Jordan et al (1997). A good account of the matching pursuit method is given in the original paper by Mallat and Zhang (1993). The method has been applied in many areas of science, technology and finance. We will come across some of these applications in subsequent chapters of this book. However, the paper by Zygierewicz et al (1999), which investigates sleep patterns in the EEG, gives a concise account in its appendix of the timefrequency (Wigner) representation of energy distribution used for the matching pursuit (something we have not gone into detail with here). The matching pursuit is a member of a larger family of adaptive approximation techniques. Jaggi et al (1998) give a brief account of related methods and introduce a high resolution pursuit which overcomes some of the shortcomings of the matching pursuit. Goodwin and Vetterli (1999) used a matching pursuit method which incorporates damped sinusoids as the analysing functions.
Chapter 3
The discrete wavelet transform
3.1
Introduction
In this chapter we consider the discrete wavelet transform (DWT). We will see that when certain criteria are met it is possible to completely reconstruct the original signal using infinite summations of discrete wavelet coefficients rather than continuous integrals (as required for the CWT). This leads to a fast wavelet transform for the rapid computation of the discrete wavelet transform and its inverse. We will then see how to perform a discrete wavelet transform on discrete input signals of finite length: the kind of signal we might be presented with in practice. We will also consider briefly biorthogonal wavelets, which come in pairs, and some space is devoted to twodimensional discrete wavelet transforms. The chapter ends with wavelet packets: a generalization of the discrete wavelet transform which allows for adaptive partitioning of the timefrequency plane.
3.2 3.2.1
Frames and orthogonal wavelet bases Frames
In chapter 2, the wavelet function was defined at scale a and location b as
VJa,b(t)
=
va1 VJ (tb) a
(3.1 )
In this section the wavelet transform of a continuous time signal, x(t), is considered where discrete values of the dilation and translation parameters, a and b, are used. A natural way to sample the parameters a and b is to use a logarithmic discretization of the a scale and link this, in turn, to the size of steps taken between b locations. To link b to a we move in discrete steps to each location b which are proportional to the a scale. This kind of discretization of the wavelet has the form (3.2)
65
66
The discrete wavelet transform
where the integers m and n control the wavelet dilation and translation respectively; ao is a specified fixed dilation step parameter set at a value greater than 1, and bo is the location parameter which must be greater than zero. The control parameters m and n are contained in the set of all integers, both positive and negative. It can be seen from the above equation that the size of the translation steps, !:1b = bodf/, is directly proportional to the wavelet scale, a3 The wavelet transform of a continuous signal, x( t), using discrete wavelets of the form of equation (3.2) is then 1
•
Tm,11
=
foo00 x(t) a1~/2 'I/J(ao mt 
nb o) dt
(3.3a)
o
which can also be expressed as the inner product ~11,11 = (x, 'l/Jm,l1)
(3.3b)
where T ml1, are the discrete wavelet transform values given on a scalelocation grid of index m, n. For the discrete wavelet transform, the values Tm,11 are known as wavelet coefficients or detail coefficients. These two terms are used interchangeably in this chapter as they are in the general wavelet literature. To determine how 'good' the representation of the signal is in wavelet space using this decomposition, we can resort to the theory of wavelet frames which provides a general framework for studying the properties of discrete wavelets. Wavelet frames are constructed by discretely sampling the time and scale parameters of a continuous wavelet transform as we have done above. The family of wavelet functions that constitute a frame are such that the energy of the resulting wavelet coefficients lies within a certain bounded range of the energy of the original signal, i.e. 00
AE:::;
00
L 11 L T m,111 m =00 I
2
:::;
BE
(3.4)
= 00
where T m ,11 are the discrete wavelet coefficients, A and B are the frame bounds, and E is the energy of the original signal given by equation (2.19) in chapter 2: E = f~oo Ix(t)1 2 dt = Ilx(t)11 2 , where our signal, x(t), is defined to have finite energy. The values of the frame bounds A and B depend upon both the parameters ao and bo chosen for the analysis and the wavelet function used. (For details of how to determine A and B see Daubechies (1992).) If A = B the frame is known as 'tight'. Such tightframes have a simple reconstruction formula given by the infinite series 1 00 00 (3.5) x(t) = Tm,I1'I/Jm,I1(t) m=oo 11=00
ALL
A tight frame with A (= B) > 1 is redundant, with A being a measure of the redundancy. However, when A = B = 1 the wavelet family defined by the frame forms an orthonormal basis. If A is not equal to B a reconstruction formula can still be written: I 2 ~ x(t)= A+B L
~ Tm,I1'I/Jm,11 (t) L m=oo 11=00
(3.6)
Frames and orthogonal wavelet bases
67
O.5a
scale m a
= (V2)m
b
width
xV2
O.5a
b
Figure 3.1. The nearly tight Mexican hat wavelet frame with ao = 2'/2 and bo = 0.5. Three consecutive locations of the Mexican hat wavelet for scale indices m (top) and m + 1 (lower) and location indices n, n + I, n + 2. That is, a = 2111 and a = 2111 + 1 respectively, and three consecutive b locations separated by aj2.
where x' (t) is the reconstruction which differs from the original signal x( t) by an error which depends on the values of the frame bounds. The error becomes acceptably small for practical purposes when the ratio B/A is near unity. It has been shown, for the case of the Mexican hat wavelet, that if we use ao = 2 1/ v , where v ~ 2 and bo ::; 0.5, the frame is nearly tight or 'snug' and for practical purposes it may be considered tight. (This fractional discretization, v, of the poweroftwo scale is known as a voice.) For example, setting ao = 2 1/ 2 and bo = 0.5 for the Mexican hat leads to A = 13.639 and B = 13.673 and the ratio B/A equals 1.002. The closer this ratio is to unity, the tighter the frame. Thus discretizing a Mexican hat wavelet transform using these scale and location parameters results in a highly redundant representation of the signal but with very little difference between x(t) and x'(t). The nearly tight Mexican hat wavelet frame with these parameters (ao = 2 1/ 2 and b o = 0.5) is shown in figure 3.1 for two consecutive scales m and m + 1 and at three consecutive locations, n = 0, 1 and 2. 3.2.2
Dyadic grid scaling and orthonormal wavelet transforms
Common choices for discrete wavelet parameters ao and b o are 2 and 1 respectively. This poweroftwo logarithmic scaling of both the dilation and translation steps is known as the dyadic grid arrangement. The dyadic grid is perhaps the simplest and most efficient discretization for practical purposes and lends itself to the construction
68
The discrete wavelet transform
of an orthonormal wavelet basis. Substituting ao = 2 and b o = 1 into equation (3.2), we see that the dyadic grid wavelet can be written as (3.7a) or, more compactly, as (3.7b) Note that this has the same notation as the general discrete wavelet given by equation (3.2). From here on in this chapter we will use 1/Jm,n(t) to denote only dyadic grid scaling with ao = 2 and b o = 1. Discrete dyadic grid wavelets are commonly chosen to be orthonormal. These wavelets are both orthogonal to each other and normalized to have unit energy. This is expressed as
oo f
1/Jm,n(t)1/Jm',n,(t) dt
• 00
=
{I
0
if m = m' and n . otherwIse
=
n'
(3.8)
That is to say, the product of each wavelet with all others in the same dyadic system (i.e. those which are translated and/or dilated versions of each other) are zero. This means that the information stored in a wavelet coefficient T m ,n is not repeated elsewhere and allows for the complete regeneration of the original signal without redundancy. In addition to being orthogonal, orthonormal wavelets are normalized to have unit energy. This can be seen from equation (3.8) as, when m = m' and n = n', the integral gives the energy of the wavelet function equal to unity. Orthonormal wavelets have frame bounds A = B = 1 and the corresponding wavelet family is an orthonormal basis. (A basis is a set of vectors, a combination of which can completely define the signal, x(t). An orthonormal basis has component vectors which, in addition to being able to completely define the signal, are perpendicular to each other.) The discrete dyadic grid wavelet lends itself to a fast computer algorithm, as we shall see later. Using the dyadic grid wavelet of equation (3.7a), the discrete wavelet transform (DWT) can be written as:
T,n,n =
J:oo x(t)1/Jm,n(t) dt
(3.9)
By choosing an orthonormal wavelet basis, 1/Jm,n(t), we can reconstruct the original signal in terms of the wavelet coefficients, T,n ,11' using the inverse discrete wavelet transform as follows: 00
x(t)
=
00
L L
Tm,n1/Jm,n(t)
(3.10a)
m=oo 11=00
requiring the summation over all integers m and n. This is actually equation (3.5), with A = 1 due to the orthonormality of the chosen wavelet. Equation (3. lOa) is often seen
Frames and orthogonal wavelet bases
69
written in terms of the inner product 00
00
x(t)
L L
=
(x,7/Jm,n)7/Jm,n(t)
(3.lOb)
m=oo n=oo
where the combined decomposition and reconstruction processes are clearly seen: going from x(t) to Tm,n via the inner product (x, 7/Jm,n) then back to x(t) via the infinite summations. In addition, as A = B and A = 1, we can see from equation (3.4) that the energy of the signal may be expressed as (3.11 )
Before continuing it is important to make clear the distinct difference between the DWT and the discretized approximations of the CWT covered in chapter 2. The discretization of the continuous wavelet transform, required for its practical implementation, involves a discrete approximation of the transform integral (i.e. a summation) computed on a discrete grid of a scales and b locations. The inverse continuous wavelet transform is also computed as a discrete approximation. How close an approximation to the original signal is recovered depends mainly on the resolution of the discretization used and, with care, usually a very good approximation can be recovered. On the other hand, for the DWT, as defined in equation (3.9), the transform integral remains continuous but is determined only on a discretized grid of a scales and b locations. We can then sum the DWT coefficients (equation (3. lOa)) to get the original signal back exactly. We will see later in this chapter how, given an initial discrete input signal, which we treat as an initial approximation to the underlying continuous signal, we can compute the wavelet transform and inverse transform discretely, quickly and without loss of signal information. 3.2.3
The scaling function and the multiresolution representation
Orthonormal dyadic discrete wavelets are associated with scaling functions and their dilation equations. The scaling function is associated with the smoothing of the signal and has the same form as the wavelet, given by I, ~m~
(t)
=
2 m / 2 ~ I'(2 m t  n )
(3.12)
They have the property
J:oo ¢o,o(t) dt
=
1
(3.13)
where ¢o,o(t) = ¢(t) is sometimes referred to as the father scaling function or father wavelet (cf. mother wavelet). (Remember from chapter 2 that the integral of a wavelet function is zero.) The scaling function is orthogonal to translations of itself, but not to dilations of itself The scaling function can be convolved with the signal to produce approximation coefficients as follows:
Sm,n
=
J:oo x(t)¢m"Jt) dt
(3.14)
70
The discrete wavelet transform
From the last three equations, we can see that the approximation coefficients are simply weighted averages of the continuous signal factored by 2111 / 2. The approximation coefficients at a specific scale m are collectively known as the discrete approximation of the signal at that scale. A continuous approximation of the signal at scale m can be generated by summing a sequence of scaling functions at this scale factored by the approximation coefficients as follows: 00
x l11 (t)
L
=
SI11,I1¢I11,I1(t)
(3.15)
11=00
where x l11 (t) is a smooth, scalingfunctiondependent, version of the signal x(t) at scale index m. This continuous approximation approaches x(t) at small scales, i.e. as m + 00. Figure 3.2(a) shows a simple scaling function, a block pulse, at scale index 0 and location index 0: ¢o,o(t) = ¢(t)the father functiontogether with two of its corresponding dilations at that location. It is easy to see that the convolution of the block pulse with a signal (equation (3.14» results in a local weighted averaging of the signal over the nonzero portion of the pulse. Figure 3.2(b) shows one period of a sine wave, x(t) contained within a window. Figure 3.2(c) shows various approximations of the sine wave generated using equations (3.14) and (3.15) with the scaling function set to a range of widths, 2° to 27 . These widths are indicated by the vertical lines and arrows in each plot. Equation (3.14) computes the approximation coefficients Sml1, which are, as mentioned above for this simple block scaling function, the weighted average of the signal over the pulse width. The approximation coefficients are then used in equation (3.15) to produce an approximation of the signal which is simply a sequence of scaling functions placed side by side, each factored by their corresponding approximation coefficient. This is obvious from the blocky nature of the signal approximations. The approximation at the scale of the window (=2 7 ) is simply the average over the whole sine wave which is zero. As the scale decreases, the approximation is seen to approach the original waveform. This simple block pulse scaling function used in this example here is associated with the Haar wavelet, which we will come to shortly. We can represent a signal x(t) using a combined series expansion using both the approximation coefficients and the wavelet (detail) coefficients as follows: 00 mo 00 (3.16) x(t) = SI110,n¢1110)t) + ~n.I11/JI11,I1(t)
L
L L
11=00
111=00 11=00
We can see from this equation that the original continuous signal is expressed as a combination of an approximation of itself, at arbitrary scale index mo, added to a succession of signal details from scales mo down to negative infinity. The signal detail at scale m is defined as 00
dm(t)
L
=
~11.I11/Jm,I1(t)
(3.17)
11=00
hence we can write equation (3.16) as
1110
x(t)
=
xmo(t)
+
L
111=00
dm(t)
(3.18)
71
Frames and orthogonal wavelet bases 2(il2). J.
J
0.12':''....4.L...1..1.~6.L...1..1.~8 ........~1O~~12~.u14 log2 (MYJ)
Figure 4.24. Number fractions of wavelet transforms with U},k > (u 2 )j and U},k > 2(u 2 )j. Filled circles are longitudinal components. Open circles are for transverse components. Dotted lines indicate values expected from Gaussian distribution. Note that the authors use the nomenclature il],k to mean the squared wavelet coefficient at location k and level j. In addition, (il 2)) is the variance (average energy) of the coefficients at level j. After Mouri et al 1999 Journal of Fluid Mechanics 389 229254. With kind permission from Cambridge University Press.
the regular structures in images associated with vortex pairs. Yilmaz and Kodal (2000) have investigated turbulent coaxial jet flows using Morlet wavelet transforms, as have Walker et al (1997), who used the Morlet wavelet to investigate multiple acoustic modes and shear layer instabilities which characterize a supersonic jet. Jordan and Miksad (1998) have examined intermittent events in a wake and Jordan et al (2000) have used a method based on wavelet ridges to demodulate transitional wake instability modes where there is no well defined carrier frequency. Gordeyev and Thomas (1999) have found interesting phase shifting behaviour of the subharmonic instability within a forced laminar jet shear layer using a Morlet wavelet decomposition of velocity signals. They provided a Hamiltonian formulation of the problem, and found good agreement with the experimentally observed phenomena and this model. Hangan et al (1999) have developed a wavelet pattern recognition method and used it to investigate the relationship between small (incoherent) and large (coherent) turbulent scales. Specifically, they employed the method in the study of velocity data fields taken in the near region of two wake generatorsa solid circular cylinder and a porous mesh strip. Poggie and Smits (1997) have analysed the wall pressure fluctuations in a Mach 3 flow over a blunt fin using the continuous wavelet transform. They partitioned the signal into two parts: one associated with the characteristic timescale of the shock crossing events and the other associated with the relatively smaller scale turbulent fluctuations. 4.3.2
Fluidstructure interaction
The interaction of fluids with structures is a common phenomenon which manifests itself, within an engineering context, as a number of problems including wind loading on buildings, flowinduced vibration of bridge decks and both the loading and scouring at bridge piers. Gurley et al (1997) have discussed the use of the wavelet transform in a more general paper concerning analysis and simulation tools in wind engineering. They illustrated the potential uses of waveletbased methods for Copyright © 2002 lOP Publishing Ltd.
transient and evolutionary phenomena, using the specific example of a wind velocity signal measured just after the hurricane eye has passed the measuring instrument. In
another related paper, Gurley and Kareem (1999) have discussed the applications of both the discrete and continuous wavelet transform to earthquake, wind and ocean engineering, including fluids engineering problems such as the transient response of buildings to wind storms, the analysis of bridge responses to vortex shedding and the correlation between pressure measured at a building rooftop and upstream. Both papers give details of two wavelet methods for the simulation of nonstationary wind velocity signals. An example of a measured and corresponding synthetic signal is shown in figure 4.25. Hajj and Tieleman (1996) have suggested using the wavelet transform to characterize the intermittent nature of wind events to model pressure variations on low rise
structures. The authors detailed the advantages of using a waveletbased approach over a conventional Fourier approach to the problem. They illustrated their ideas briefly using the Daubechies D4 wavelet to decompose a sample wind velocity time series. In later work using the Morlet wavelet transform, Hajj el al (1998) found correlations between energetic events in the atmospheric wind and low pressure
25 20 t5 to
50
to
20
30
40
50
60
to
20
30
40
50
60
(aJ
25 20 t5 to
50 (b)
Figure 4.25. Measured and synthesized wind velocity signals. (a) Measured and (b) synthesized using W< :"
u 0 0
800
400
600 400
200
"c
~
(b)
0.05 0.1 0.15 0.2 time I seconds
800 600 400 200
200
0
0.25
1000
1600
800
0.2
0.25
0.05 (c)
0.1
0.15
0.2
0.25
time I seconds
Figure 5.20. Harmonic wavelet analysis of the response at one end of a freely supported elastic beam subjected to an impulse input. (a) Sample acceleration time history. (b) Original harmonic wavelet timefrequency map for the sample timehistory. (c) Corresponding ridge diagram. After Newland (1999a). Reproduced with kind permission of the ASME.
suggests the use of the method to determine the velocity and attenuation of ultrasonic pulse echo signals used in nondestructive testing. In a subsequent related paper, again using Morlet wavelets, Inoue er al (1996) reported on a timefrequency analysis of the flexural waves set up by central impacts on a simply supported beam. A method was developed by the authors to determine both the group velocity of the structural waves and the impact sites on the beams by utilizing the arrival times extracted from the wavelet analysis. In a similar study, this time for steel plate elements, Gaul and Hurlebaus (1997) have determined impact site locations using the wavelet decomposition of strain sensor data. A waveletbased study of transient waves propagating in composite laminate plates has been described by Jeong and Jang (2000). An analysis of the nonstationary response of a rigid block resting on a moving plane has been performed by Basu and Gupta (1999) using a waveletbased stochastic linearization technique. This simple model was used to show the potential application of waveletbased analytical tools to the response of slipping structures to earthquake excitation. See also the earlier papers by Basu and Gupta (1997,1998). Gurley and Kareem (1999) mention briefly the application of wavelet transforms to ground motion analysis during earthquakes and building responses to wind events (including vortex shedding) in their paper concerning analysis and simulation tools for wind engineering. Robertson er al (1998a,b) have detailed a discrete wavelet transformbased method for extracting the temporal impulse response functions of structures, and Chen and Wu (1995) have developed a spline wavelet expansionbased finite element method for frame structure vibration analysis. Copyright © 2002 lOP Publishing Ltd.
A study by Dalpiaz and Rivola (1997) compared the effectiveness and reliability of different vibration analysis techniques for fault detection and diagnostics in cam mechanisms used in highperformance packing machines. They compared traditional analysis methodsamplitude probability density (APD), power spectral density (PSD) and time synchronous averaging (TSA)with a wavelet transform method based on the Morlet wavelet. They found the timefrequency analysis of the waveletbased method well suited to detecting and precisely locating transient dynamic phenomena from the signal. Mastroddi and Bcttoli (1999) have performcd a wavelet analysis on the output signal of a nonlinear system in the neighbourhood of a Hopf bifurcation. They used the Morlet waveletbased analysis to point out the linear and nonlinear signatures of the system and suggested its use for aeroelastic applications. Karshenas el al (1999) compared the wavelet power spectrum smoothing method with the Welch method in the random vibration control algorithm of an electrodynamic shaker. They found that the wavelet method achieved twice the power spectrum resolution of the Welch method. Sjoberg el al (1995) included wavelet transformbased methods in a comprehensive paper concerning 'blackbox' models of nonlinear dynamical systems. A new waveletbased method for denoising transient dynamical signals by first projecting them into a multidimensional state space has been described by Effern el al (2000) and applied to both model data and eventrelated potentials (medical EEG signals).
5.5
Chaos
Nonlinear oscillator systems are capable of the most fascinating behaviour known as chaotic motion, or simply chaos, whereby even simple nonlinear systems can, under
certain operating conditions, behave in a seemingly unpredictable manner (Addison, 1997). The realization that real systems can exhibit this type of nonperiodic response has prompted much research work in the area over the past two decades. The ability of waveletbased methods to characterize chaotic oscillations has received attention from a variety of workers in the field. Both Daubechies and Morlet wavelets have been employed by Staszewski and Worden (1999) to analyse timeseries data sets containing a variety of features including coherent structures (fluid turbulcnce), fractal structures (devi!'s staircase and MandelbrotWeierstrass function), chaos (Duffing, Henon, Lorenz and Rossler systems) and noise (Gaussian white). Their paper provides a wide ranging overview at an introductory level of the application of wavelets to signals from these types of systems. Figure 5.2l(a) shows the time series of a Duffing oscillator in chaotic mode. This oscillator is a sinusoidally driven, damped nonlinear oscillator with the nonlinearity contained in the cubic spring tenn. The Duffing oscillator investigated by Staszewski and Worden has the form
x + 0.05': + x 3 =
7.5 cos 1
(5.2)
where ,i: and x are, respectively, the first and second derivatives of x (the displacement if we think of it as a physical massspringdamping system). For the set of parameter Copyright © 2002 lOP Publishing Ltd.

!.~
6.0
~ 0.0 .~
Q.
~ 6·°0
512
1024 data samples
(a)
1536
2048
Poincare Map 2.0 1.0
q 0
0
0;
>
o.ol ,
1.02.0 ....
• 3()25
1.5
0.5 displacement {l.5
1.5
(b)
Figure 5.21. Chaotic time series of the Duffing osciUator and corresponding Poincare map. After Staszewski and Worden (1999). Reproduced with kind permission of World Scientific Publishing Co Pte Ltd and the authors.
values given in equation (5.2) Ihis forced nonlinear oscillator produces a chaotic response. The Poincare map of figure 5.21(b) was generated by plolting the velocity against displacement once every period of forcing of the oscillator. This map is useful in highlighting the fractal structure of the strange altractor associated with the chaotic system. Figure 5.22 contains the phase portrail of a Gaussian noise signal logether wilh that for the Duffing oscillator. The similarity between the Iwo
2.2
• '" .."
2.2
• '" .."
2.1
0
2.1
0
T
2
•0
1.9
T 0
•
1.9 1.8
o (a)
o 20
20 40 60 80 100120 140 time [s]
(b)
40 60 80 100120 140 time [5]
Figure 5.22. Wa\'elet phase for (a) Gaussian noise and (b) Duffing oscillator. After Staszewski and Worden (1999). Reproduced with kind permission of World Scientific Publishing Co Pte Ltd and the authors.
Copyright © 2002 lOP Publishing Ltd.
at small scales is evident, highlighting the selfsimilarity contained within the two systems. Wong and Chen (2001) provide a clear, well illustrated introduction to the Morlet wavelet transform of the nonlinear and chaotic behaviour of multidegreeoffreedom systems. They introduce the transform explaining its use, using a variety of simple test signals before considering single, then multiple, oscillator systems based on coupled Duffing oscillators. These systems are considered when subjected to both single impulses and continuous forcing. They conclude by examining the chaotic response of the singledegreeoffreedom Duffing oscillator, contrasting its modulus and phase plots with nonchaotic cases. Their paper is well worth consulting for the many clearly presented diagrams used to illustrate the discussion. In their comprehensive paper on adaptive strategies for recognition, noise filtering, control, synchronization and targeting of chaos, Arrecchi and Boccaletti (1997) have shown how to employ the Daubechies D20 wavelet in a noise reduction strategy to separate noisy contributions from deterministic parts of chaotic data sets. As an example (Boccaletti et aI, 1997), they used the MackeyGlass delay differential equation configured to produce 7.5dimensional dynamics with both white and coloured noise added separately. They detailed the effect of wavelet threshold level on the ability of their method to determine the underlying dynamics of the system and recommended it for easy implementation in experimental situations as it does not require information on the correlation properties of the additive noise. Grzesiak (2000) has also employed a waveletbased denoising technique to filter chaotic data. Grzesiak determined the efficiency of the technique by comparing the correlation dimension of noisy and clean data generated for a variety of chaotic dynamical systems and found that the wavelet method was comparable with other methods commonly used to filter chaotic data. Permann and Hamilton (1992) have performed a wavelet analysis of the time series from a Duffing oscillator in both periodic and chaotic mode. They were able to detect smallamplitude harmonic forcing terms, even when the data were highly nonstationary and of short duration. See also the paper by Permann and Hamilton (1994) who investigated the chaotic behaviour of a weakly damped and weakly forced Morse oscillator using Daubechies D8 wavelets. Lamarque and Malasoma (1996) have constructed waveletbased exponents, similar to Lyapunov exponents, for the identification of chaotic behaviour. Cao et al (1995) have used wavelet networks to make both short and longterm predictions of the time series from chaotic systems. Systems they investigated include the MackeyGlass equation, Lorenz system, and the U shiki and Ikeda maps. In addition, they modified the Ikeda map by using one of its parameters as a variable. In this way they were able to investigate parametervarying systems. Allingham et al (1998) have used a hybrid system to model time series from chaotic dynamical systems. Their system combines continuous optimization with a wavelet matching pursuit method. Gamero et al (1997) have analysed the Lorenz equations displaying chaotic motion using their multiresolutionbased information measures for dynamical signals. They also considered the Henon map and an EEG signal displaying an epileptic seizure. Heidari et al (1996) have studied the wavelet transform of deterministic selfsimilar signals, suggesting its use as a method to interrogate the noisy strange attractor of the Henon map. Masoller et al (1998) have interrogated experimentally measured lowfrequency intensity fluctuations from a semiconductor Copyright © 2002 lOP Publishing Ltd.
laser operated near threshold. Using the discrete wavelet transform, they compared their results with the wavelet analysis of a theoretical model, showing that the differences between the results were confined to the 'fast and short' components of the signal. Another experimental study by Russo el al (2000) has concentrated on the dynamical regimes optically induced in a nematic liquidcrystal as the intensity of an incident laser beam increases. Their paper contains a series of Morlet wavelet transform plots generated at increasing values of the control parameter, which
illustrates the jump to stochastic behaviour above a known threshold. Their waveletbased analysis suggests the presence of a transition towards a chaotic state.
5.6
Nondestructive testing
Nondestructive testing (NDT) is concerned with the interrogation of underlying structural integrity using procedures which do not impair in any way the intended performance of the structure during and after examination. Sonic echo testing is a common method employed in the NDT of structural elements. It involves striking the test specimen (e.g. structural element or material specimen) with an instrumented hammer which records both the input pulse (strike) and subsequent response of the specimen. This response is interpreted as an indirect measurement of the integrity of the specimen. A typical velocity trace from such a test on a foundation pile is shown in figure 5.23 (Watson el ai, 1999). A schematic of the sonic pulse transmission through the pile is shown in the figure. For such a heavily damped system there are rarely multiple longitudinal reflections and the frequency dependences of the group velocities are negligible. Therefore, the temporal isolation of the signal features is more important than their frequency decoupling and hence a Mexican hat was used which is more temporally compact than the standard Morlet (5 < Wo S; 6) or harmonic wavelets used in the study of free beam and plate vibrations. The scalogram corresponding to this velocity trace is shown below. The pile is II m long and the velocity of the stress wave through the pile is 3800 m s'. Thus we would expect to see the reflection of the end of the pile occur 2 x 11/3800 = 0.0058 s after the initial impulse. The pile toe reflection shows up particularly well in the top righthand quadrant of the scalogram, as it has a distinctively different shape and appears lower down the scalogram from the initial oscillations which occur at a dilation around a = 10 4 These oscillations, seen to occur in the top lefthand quadrant of the scalogram just after the input initial pulse, are known as ringdown and are in fact the surface oscillations of the pile head due to the hammer impact. Figure 5.24 contains the reconstructed traces of both wavelet and Fourier filtered traces. Simple scale dependent wavelet filtering was employed where the transform components at a scales less than 0.0001 were set to zero and an inverse wavelet transform performed. The wavelet filtered trace is shown in the top left quadrant of figure 5.24. The Fourier low pass filter cutoff frequency was set to 2.25 kHz, as is the case in practice. The Fourier filtered trace is shown in the top right quadrant of figure 5.24. The two lower plots in the figure show a zoomedin section of the upper traces in region of the pile toe reflection. Comparing the filtered traces of figure 5.24, it can be seen that the wavelet filtering separates the ringdown oscillations Copyright © 2002 lOP Publishing Ltd.
........
,
0.01
j ~.
~
...
input pulse echo from pile toe
0.005 f
~
'EE ">
f1
0
0.005
~
'v\fVvvv~
0
I
I
I
0.002
0.004
0.006
0.008
0.006
0.0070.008
time (s)
0.0001
0.001
o
0.001
0.002
0.003
0.004
0.005
location 'b' (seconds)
Figure 5.23. Wavelet transform decomposition of a sonic echo signal. Schematic of sonic echo testing of a foundation pile (top left). A Fourier filtered velocity trace (top right) and corresponding wavelet transform plot (bottom) of an 11 In pile in stiff/very stiff clay.
0.004 O'OO6~tim:';s~! :::~~: °llo.OO2
0.006
0.008
time (5)
~ 2o W]
_
>. 0 100 0
_2 103 0
0.006
0.004
0.007
0.D08
I
0.004
5'1O~
time . (5) E Jolo'_3'101 _5'104
/
Figure 5.24. Wa\'elet and Fourier filtering of the sonic echo pile signal. W
'
,,,
IA
0 {l.01
0
2 3 time (s)
1
(a)
4
1
0
x Ill'
~
2 3 time (s)
(b)
4
xla'
Figure 5.27. The reconstruction from only the scalogram modulus maxima lines. (a) Modulus maxima plot derived lines from the scalogram in previous figure. (b) Reconstructed trace using only the modulus maxima lines in (a). Reprinted from Watson and Addison (2002), Mechanics Research COlIIlJlunications (in press at time of publication), with permission from Elsevier Science.
0.04
0.04
o
o
om 0
2
(a)
0.01 ':....lL:__c:__;:__~:__l
4 xlO 3
time (s)
iI

3

lO'
3
4
time (s)
10'
\
~ ~ >
u
~ >u
xlO 3
\
,
u
""go "
"" "
~
'""
lO2
0 (b)
2
I
~
'"
o
(d)
2 3 time (5)
4
10 2
0 xlO 3
(e)
2 3 time (5)
4 xlO 3
Figure 5.28. The partitioning of the modulus maxima lines for signal filtering. (a) Original data. (b) Discarded modulus maxima lines. (Threshold shown as horizontal dotted line.) (c) ReLained modulus maxima lines. (d) Filtered data. Reprinted from Watson and Addison (2002), Mechanics Research Communications (in press at time of publication), wiLh permission from Elsevier Science. CopyrighL © 2002 lOP Publishing Ltd.
threshold and be removed. In addition, noise, which also manifests itself as modulus maxima restricted to high frequency regions, is also removed from the signal. The resultant reconstructed trace shown in figure 5.28(d) illustrates how all ringdown has been eliminated whilst retaining the pertinent signal features. In addition, the retained features still contain their high frequency components and are not excessively smoothed, as would be the case if bandpass filtered using Fourier techniques. Figure 5.29 illustrates the use of the Morlet wavelet with low central frequencies in the analysis of a highly oscillatory sonic echo signal where the pile toe is not obvious in the time domain (Addison ef ai, 2002a). When using central frequencies, wo, less than 5 (io < 0.8) the complete Morlet wavelet given by equation (2.36) in chapter 2 must be used. Morlet wavelets with low central frequencies result in analyses that are more 'temporal' than 'spectral' in that they are better at locating short duration temporal features than those with higher values of Wo (refer back to section 2.12, chapter 2). This can be seen in figure 5.29, where the pile toe can be located in the wavelet transform scalogram plots only at lower values of woo The location of the pile toe is indicated both in the time signal and the lowest scalogram plot which corresponds to the complete Morlet wavelet with Wo = 1.5 (i.e. fo = 0.238). It is interesting to note that, although they do seem to be particularly useful for certain tasks, the literature contains surprisingly very little on the use of complex wavelets with few oscillations such as the complete Morlet wavelet of low central frequency or the complex Mexican hat. A variety of continuous wavelets were used by Abbate ef al(1997) in a study of the signal detection and noise reduction properties of the wavelet transform when used to elucidate ultrasonic pulseecho traces in steel specimens. They used a combination of 'pruning' and soft thresholding to reduce the noise from the signals. Pruning simply sets wavelet coefficients outside a certain ascale range to zero, i.e.
T(a,b)
=
{;(a'b)
for
a
for
(II
for
{/2
(5.3)
where the coefficients outside the range set by al and a, are thought most likely to come from noise. Pruning is effectively scale dependent thresholding within a band of limits al and a,. The soft thresholding employed by Abbate and coworkers had the usual form:
T(a, b) =
o { sgn[T(a, b)] x (IT(a,b)l A)
for for
IT(a,b)1 < A IT(a,b)I;o: A
(5.4)
where A is the threshold. Figure 5.30(a) shows an input signal used in the ultrasonic testing of steel specimens. Figure 5.30(b) shows the same signal with added noise and its wavelet transform plot is given in figure 5.30(c). The reconstructed signal after pruning and thresholding is shown in figure 5.31 (a) with the filtered wavelet transform plot used for the reconstruction shown below in figure 5.31(b). Figure 5.32 contains the signal from a cast iron sample together with its waveletfiltered version. A Morlet wavelet was employed and the filtering used both pruning and thresholding. Both echoes are clearly detected without any other contribution from the acoustic noise. Copyright © 2002 lOP Publishing Ltd.
pile toe location
o
2
4
6
8
10
12
14
16
time (ms)
Figure 5.29. A complete Morlet wavelet analysis of a sonic echo signal. The figure contains a sonic echo signal taken from a pile, together with a sequence of scalograms generated using a complete Morlet wavelet decomposition of the signal with the central frequency set to (from top to bottom) Wo = 5.5, 4.5, 3.5, 2.5 and 1.5. After Addison el 01 (2002a). Reproduced with kind permission of Academic Press Ltd.
Copyright © 2002 lOP Publishing Ltd.
flaw signal y (f) 1.0,,,
0.5
o r.1IIIIV.....   0.5 (a)
 1.0 ::c:::::c=:::::c=:c=:c:' o 500 1000 1500 2000 2500 3000 r~.~~~__,
1.5 1.0 0.5 _ ';i;'
(b)
0 III\/ln,
0.5 1.0 _1.5L
o c
.0
~
:a
(e)
E E
••u
15 8u
2 1
o
~
500
1000
@.o
1500

~'
2000
2500
3000
2000
2500
3000
'CQ?
I
2 3
4
0
500
1000
1500 time
Figure 5.30. \Val'elef analysis of a pulse echo reflection signal. (a) Original signal. (b) Signal in (a) with added acoustic noise generated by spherical voids in steel. (c) The wavelet transform contour plot of the signal in (b). After Abbate et al 1997 IEE£ Transactions 011 Ulrrasonics alld Frequency Conrrol 44(1) 1426. © IEEE 1997.
Chen ef al (1996) have proposed a waveletbased technique to improve the signal to noise ratio of ultrasonic inspection signals obtained from coarsegrained stainless steel specimens. Chen ef al (1999c, 2000) have detailed both wavelet and wavelet packet methods for denoising ultrasonic signals for the nondestructive evaluation
'" ~
~
(a)
0.5 0 0.5 1.0 0
c
E E 0
.~ '5
:a (b)
15 8u
500
1000
1500
2000
2500
3000
2000
2500
3000
2 I 0 I 2
o
3
0
500
1000
1500 time
Figure 5.31. Filtered pulse echo reflection signal. (a) The reconstructed signal obtained from the filtered wavelet transform plot shown in (b). After Abbate et af 1997 IEEE 7i'(l1lsactions Oil Ultrasonics and Frequency C0ll1rol44(I) 1426. © IEEE 1997.
Copyright © 2002 lOP Publishing Ltd.
40
r~~~~~~~,
20
o ~Mlil 20 (a)
40
~ e
L_=,~,,"c_,.,._,.J
o
1000
2000
3000
4000
SOOO
20
0 f;Jlllr"I\",V/\rI 20
(b)
time
Figure 5.32. Wavelet of a pulse echo reflection signal taken from a cast iron sample. (a) Ultrasonic signal. (b) Output after wavelet filtering clearly showing the two echoes. After Abbate et al 1997 IEEE Transactiol/s 01/ Ultrasonics and Frequency Control 44(1) 1426. © IEEE 1997.
of steel samples with known defects. Cho et a/ (1996) employed Morlet wavelets to detect subsurface defects in steel test specimens from noncontact laser ultrasonic signals. Staszewski et a/ (1997) have described a waveletbased signal processing
method to enhance defect detection in a carbon fibre composite plate interrogated using ultrasonic Lamb waves and incorporating an optical fibre receiver. Wu and
Chen (1999) have also used the Morlet wavelet to analyse noncontact laser ultrasonic signals from epoxybonded copperaluminium layered specimens. Their study focused on the detection of unbonded regions. A Mexican hat wavelet was employed by Guilbaud and Audoin (1999) in the interrogation of laserinduced ultrasonic signals used to measure stiffness coefficients in a viscoelastic composite material.
They found that the reliability of their method may justify its use in the field of material behaviour characterization.
A number of other authors have developed waveletbased tools to aid the interpretation of NOT signals. Shyu and Pai (1997) have performed impactecho tests on free standing concrete cylinders using Daubechies wavelets. Tang and Shi (1997) used wavelet techniques to detect and classify a variety of welding defects from NOT signals. Doyle (1997) has employed a wavelet technique to identify the impact force on structures using a knowledge of the structure and its response to the force. Hamelin et a/ (1996) used wavelets to analyse eddy current signals. They analysed the complex and real parts of the signal separately in order to develop a classification scheme to inspect the transformed signals for the location of distinct maxima and associated phase. Pierri et a/ (1998) investigated the use of twodimensional Haar wavelet transforms in eddy current NOT, and Lingvall and Stepinski (2000) have described an automatic method for the detection and classification of cracks located in aircraft riveted lapjoints during eddy current inspection which employs Coiflet wavelets. Qi (2000) has described a waveletbased method to analyse acoustic emission (AE) signals in a study of material fracture behaviour. Qi employed Daubechies wavelets to analyse experimental AE signals and found that Copyright © 2002 lOP Publishing Ltd.
waveletbased techniques better approximate the relationship between the stress and stress intensity factor than do classical techniques. The AE behaviour of reinforced concrete beams tested under flexural loading was investigated by Yoon et al (2000) using both Fourier and Morletbased wavelet methods. They found that the Fourier spectra and wavelet transforms of the AE signals gave useful information about the relationship between the damage mechanisms of the concrete (e.g. microcracking, localized cracking, flexural cracking and shear bond cracking) and the AE response. An examination of the integrity of thin coated foil used in the food packaging industry was conducted by Futatsugi et al (1996) using both AE monitoring and microscopic observation. AE signals were omitted by the cracking of the surface coating (SiO x film) and also from its delamination from the foil. The threshold tensile strain necessary to cause the first fracture, estimated from the AE signals, agreed well with the strain determined using a laser microscope. An experimental study of crack detection in metallic structural elements using fourthorder Daubechies wavelets has been carried out by Biemans et al (1999). They instrumented a precracked rectangular metal plate with piezoceramic sensors and subjected it to both static and dynamic tensile loading. A statistical measurement based on the logarithmic wavelet variance of the strain data was used as a damage index. This parameter provided an insight into the scale dependent changes in energy of the strain data from the piezoceramic sensors. Wang and Deng (1999) have used wavelets to probe the spatial profiles of damaged cantilevered beams under static and dynamic loading. Their results clearly show that the location of a structural crack in the beam can be pinpointed through the Haar wavelet transform coefficients. They found similar results using Morletbased transforms. See also the related papers by Deng and Wang (1998), Liew and Wang (1998) and Quan et al (1999). Morlet wavelet transforms have been employed by Li and Berthelot (2000) to analyse pulseecho signals from thick annular waveguides. They developed a local spectraltemporal wavelet energy measure by integrating the energy density scalogram over a box of limited extent in scale and location. They found this energy measure to be particularly good at localizing cracks in faulty annular components and tested it successfully on data from both an annular waveguide with a machined crack and on a partially annular component of the pitch shaft of an H46 helicopter. Marwala (2000) has developed a 'committee of neural networks' technique which employs frequency response functions, modal properties and wavelet transform data simultaneously to identify damage in structures. The method was tested on synthetic data from three coupled oscillators and then used to identify the damage in seamwelded cylindrical shells. Gros et al (2000) have presented results concerning the fusion of images from multiple NDT sources gathered during the inspection of a composite material damaged by impact. The images were fused using a variety of techniques, including one based on Daubechies D8 wavelet transforms, in order to improve the defect detection and provide a more accurate measurement of defect dimensionality. Twodimensional Morlet wavelets have been used by Li (2000) to detect partial fringe patterns generated in NDT interferometry induced by defects. The author found that the wavelet method was suitable for both holographic interferometry and electron speckle pattern interferometry. Chan et al (2000b) have Copyright © 2002 lOP Publishing Ltd.
used wavelet packet denoising within a digital speckle correlation method to detect defects in multilayer ceramic capacitors in surfacemounted printed circuit boards.
5.7
Surface characterization
The characterization of engineering surfaces is pertinent to a number of engineering fields providing quantitative information on the formation process of the surface, for example the manufacturing process used to form a machine component or the fracture process causing a rugged crack surface. Surface topography is one of the most important factors affecting the performance of manufactured components. It can be related to a number of pertinent engineering aspects such as wear, lubrication,
friction, corrosion, fatigue, coating, paintability, etc. Twodimensional biorthogonal wavelet transforms have been used by Jiang e1 al (1999) to probe the surface topography of orthopaedic joint prostheses. They used three waveletbased parameters to characterize the surface: roughness, waviness and form. The roughness of the surface was defined as the detailed surface found through the inverse transform of the thresholded wavclet coefficients within a band of the smallest scale indices. Similarly, the waviness of the surface was defined as the detailed surface found through the inverse transform of the thresholded wavelet coefficients within a band of the next smallest scale indices. The thresholding method employed in both cases simply limited the maximum absolute value of the coefficients to three times their standard deviation at each scale. This was carried out to disassociate the
overall surface topographic features from localized peaks, pits and scratches, as it was assumed that each detail coefficient belonging to the roughness or waviness follows a Gaussian distribution. (Hence, a coefficient is very unlikely to appear outside three standard deviations of the distribution.) The form of the surface was defined to be the original surface minus the detail signals over the roughness and waviness scales.
Figure 5.33 contains the original centre profile of a ceramic femoral head together with the multi resolution approximations of the profile after wavelet decomposition with the biorthogonal wavelet pair. The detail signals at scales I to 4 are shown on the righthand column. These scales cover the roughness wavelengths and the addition of these four detailed signals gives the roughness profile shown at the top righthand of the figure. Figure 5.34 shows three threedimensional plots of the same surface separated into its roughness, waviness and form components.
Multiscalar topographical features such as peaks, pits and scratches on the surface were then defined by Jiang and his colleagues by hard thresholding the coefficients using two standard deviations of the coefficient amplitude at each scale as the threshold. An example of the multiscalar topographical features defined in this way is given in figure 5.35. Tn a later paper, Jiang ef al (2000) have provided details of a waveletbased analysis of both rolled steel sheets and ceramic femoral heads. See also Chen ef al (I999d), who have developed a similar waveletbased method to decompose surface data sets into a surface roughness component and a wavelet reference surface.
Chen e1 al (1995) have analysed surfaces produced by typical manufacturing processes using Daubechies wavelets, citing the advantage of the spacescale Copyright © 2002 lOP Publishing Ltd.
~m
~m
1~1z::sJ
0.70
o
0.0605
0.1210 0.1815
0.2420
the original centre profile
nun
0006~
0.003 0.000
0.003 0.006
o
0.0605 0.1210 0.1815 0.2420 the roughness centre profile nun (D 1+D2+D3+D4 )
j=1
j=2
D, j=3
j=4
A4=reference profile
0.0605
0.1210
0.1815
0.2420
the approximations
o
0.0605
0.1210 the details
0.1815
0.2420 mm
Figure 5.33. Multiscalar decomposition of a ceramic femoral head using a twodimensional biorthogonal wavelet pair: centre profile. This material has been reproduced from the Proceedings of rhe Illstill/riol/ of Mechanical Engineers, Parr fI, Journal of Engineering ill Medicine 19992134968, figure 6A, by Jiang et ai, by permission of the council of the Institution of Mechanical Engineers.
localization properties of the wavelet transform over other techniques. A number of continuous wavelets are considered by Lee et al (1998) in a study of the morphological characterization of engineering surfaces. They employed the Morlet and Mexican hat (second derivative of Gaussian) wavelets, as well as the less common Barrat wavelet and an eighth derivative of a Gaussian function, to investigate the potential applications of wavelet decomposition in assessing the muItiscale features of the engineered 0.10 0.05
o
o.o~j
I!m
Figure 5.34. Reconstrucled surface lopographies of a ceramic femoral head in different transmission bands: rough (left), wavy (centre) and form (right). This material has been reproduced from the Proceedings of the II/srittlrion of Mechanical Engineers, Part H, Jotlmal of Engineering in Medicine 1999 213 4968, figure 7A, by Jiang et aI, by permission of the council of the Institution of Mechanical Engineers.
Copyright © 2002 lOP Publishing Ltd.
slope intensity image
Figure 5.35. Multiscalar topographical features (peaks, pits and scratches) in the equatorial region of the ceramic femoral head. This material has been reproduced from the Proceedings a/the /nstitlltion 0/ Mecl/{micaf Engilleers, Part /1, Joumaf o/Engineering ill Medicine 1999 2134968, figure 8A, by Jiang et af, by permission of the council of the Institution of Mechanical Engineers.
surface, considering both its manufacturing and functional aspects. Song el al (2000) have developed a technique for the inspection of surface mount devices in the electronic industry using modified Haar wavelets. Oogariu el al (1994) have described the light scattered by a slightly rough surface in terms of twodimensional wavelet transforms, and Zhuang el al (1998) have developed a laserbased noncontact system for the inspection of pipe inner walls where waveletbased decomposition of the reflected signal from the pipe wall is used to characterize the surface texture as form, error, wavincss, roughness and sporadic scratches. Jasper el al (1996) have cmployed adaptive wavelet bases to capture texture information and locate defects in woven fabrics from their images, and Lin and Xu (2000) have introduced a waveletbased method to quantify the fuzziness of woven and knitted fabrics from their images. The fractal structure of fractured granite surface profiles is examined using wavelets by Simonsen el al (1998), and Addison and Watson (1997) have detailed the use of wavelet transforms to analyse rough surfaces modelled using fractional Brownian motions. In an analysis of the fractal properties of cracked concrete surfaces, Dougan el al (2000) have compared both wavelet and Fourierbased power spectral methods with traditional methods for determining fractal parameters. The multiscale characterization of pitting corrosion damage has been carried out by Frantziskonis el al (2000) using Oaubechies wavelets with four vanishing moments (i.e. 08s) to determine the Hurst exponent associated with the geometry of the corrosion pits. The submicron surface roughness of anisotropically etched silicon has been analysed using Meyer wavelets by Moktadir and Sato (2000) who found a scaling exponent for the surface close to 0.5. Srinivasan and Wood (1997) used Oaubechies 012 wavelet transforms as a tool to compute the relevant fractal parameters in a fractalbased approach to geometrical tolerancing (see also Tumer el ai, 1995). The authors focused on mechanical tolerances applied to physicalproduct features, such as machined parts and assemblies. They used the roundness of ball bearing elements as a design example. Chapter 7 contains more information concerning the use of wavelet Copyright © 2002 lOP Publishing Ltd.
transformbased methods for the analysis of both geophysical topographic features and surfaces exhibiting fractal structure.
5.8 5.8.1
Other applications in engineering and further resources Impacting
The impacting of detector tubes in boiling water nuclear reactors has been detected by Ritcz and Pazsit (1998) using Haar wavelets. They thresholded the Haar wavelet coefficients to remove noise and reconstructed the signals to produce a time series of the impacts. The threshold was set to four times the standard deviation of the coefficients whereby all coefficients below this level were set to zero. Reconstruction using only the remaining large coefficients gave an indication of the impact events in the signal. RilCZ and Pazsit investigated both modelled data and in situ measurements. The top trace in figure 5.36 shows a detector signal taken from the Swedish Barsebackl reactor where no noticeable vibrations were observed to occur. The fuel box vibration signal, after filtering out the low frequency detector string vibrations, is shown in the middle of the figure. This signal is then transformed and filtered using the Haar transform as described above. The reconstructed signal is shown in the bottom plot. A few intermittent spikes can be seen to occur over the signal length. The paucity of spikes in this waveletfiltered signal can be contrasted to the 5 ,_~_ _~_d",e",tec",>;to""r""si,,n",a,,"1!=L,",PRM=,,18~3,_~ _ _~_,
o 5
__ 1000 1500
L._~.,_~
o
500
__ 2000 2500
~_~
......J.._ _ 3000 3500
~.,_
J~ __
4000
time fuel box vibration (after stop filtering)
0.1 r~':'::;:'=.:.c.:;==:;====~,
o 0.1
_ _~ _ ~_ _~ _ ~ _ ~ ~ _ . J
"_~_~
o
500
1000
1500
2000
2500
3000
3500
4000
time fuel box vibration; severity =0.04069
0.1 r~"':':;:=::"':';==':;:::':=~=::':""~,
o H4+++ 0.1
+tfIHtflt+HI 1+.. ..
"_~:_:~;___;=~=:=~
o
500
1000
1500
2000
2500
L..
3000
. _ J  .._ _ ..__
3500
4000
time
Figure 5.36. Signal from detector tube experiencing no \'ibrations. The filtered detector signal (middle trace) is shown below the original signal (LOp). The reconstructed signal from the Ihresholded Haae coefficients is shown in the bottom trace. Reprinted from Rilcz and Pilzsit 1998 AIII/als of Nuclear Energy 25(6) 387400. Copyright (1998), with permission from Elsevier Science. Copyright © 2002 lOP Publishing Ltd.
detector signal LPRM033
0.5~£U:~
_O:c::·=_=.J.....~_~_~ o 500
1000
1500
2000 2500 3000 time fuel box vibration (after stop filtering)
3500
4000
,~~.::.~,''e.'cc.,:.'=~~,
0.5
o 0.5 0~:5~00:1000:':c15~00:'2~000::2~5OO~:3:000~3::5~00:4000""· Time fuel box vibration; severity = 0.765 _ _~_, 0.5 ,~ _ _.;cc:'"c",==:;:..:==.,==~
o 0.5
'~_,_'~~~~~'
o
500
1000
1500
2000 time
2500
3000
3500
4000
Figure 5.37. Signal from detector tube which is known to ha\'e experienced impacting. Reprinted from Racz and Pazsit 1998 Allllals of Nuclear Energy 25(6) 387400. Copyright (1998), with permission from Elsevier Science.
high occurrence of spikes in the bottom plot of figure 5.37. This signal was taken from a dctcctor tubc which was found from subscqucnt inspcction to have cxperienced impacting. The authors determined a parameter, 5, for the severity of impacting based on a normalized value of the estimated impact rate. High values of the severity 5 indicate the occurrence of strong vibrations and the possibility of impacting. 5.8.2
Data compression
An investigation of the use of waveletbased methods for the compression of vibration signals has becn carried out by Staszewski (I998b). Hc compared Fourierbased compression with waveletbased compression (using Oaubechies 04 and 020 wavelets) for a variety of signals. Figurc 5.38 shows a periodic signal with its associated Fourier frequency spectrum and wavelet coefficients. We can see by looking at the Fourier and wavelet domain representations of the signal that the Fourier case is more compact. [n fact, we would expect Fourier representation to favour the concise representation of a periodic signaL Figure 5.39 shows the normalized mean square error (MSE) of the reconstructed signal as a function of the number of coefficients (both Fourier and wavelet) used in its reconstruction. This plot confirms what we can see by eye. That is, for a periodic signal, a much lower error results from using the same number of Fourier coefficients as wavelet coefficients. The nOlmalized MSE is defined as N
MSE(x) =
100", "  , L. (x;  x,)Na;.
Copyright © 2002 lOP Publishing Ltd.
;=]
(5.5)
1.0 "0 .g" 0.5
0
§ 0.0
" c
.~O.5 ~
1.0 128
0
256
384
512
192
258
data samples
Cal
"
"0
B
;g, 0.40 E
~
~ u
0.20 ~
"
Co
JJ
A
~
0
A 128
64
(b)
frequency orders
2.0
" 1.5 g
"0
\ \ \
0
\
§ 1.0
\
;;
"" > ~
•
\
,
0.5 0
128
(el
256
384
512
wave coefficients
Figure 5.38. Periodic data used for compression. (a) Time domain. (b) Frequency domain. (c) Wavelet domain; dashed line indicates wavelet coefficients decreasing according 10 the amplitude level. (Note that Daubechies 020 wavelets are used.) After Staszewski (1998b). Reproduced with kind permission or Academic Press Ltd.
where X; are the components of the original signal of length N, ,,; is the variance of the signal, x~ is the reconstructed signal using selected coefficients, and the factor of 100 gives MSE(x) as a percentage. The situation depicted in figures 5.38 and 5.39 changes when a transient signal is compressed. An example of such a signal is given in figure 5.40(a). We can see from figures 5.40(b) and (c) that the Fourier spectrum for this signal is relatively broad band and the dominant wavelet coefficients are relatively localized. For this transient signal, the mean square error plot indicates that far fewer wavelet coefficients are required to form a good approximation to the original signal (see figure 5.41). Staszewski went on to use wavelet compression as a method for feature dctection in two data sets: ultrasonic flaw detection signals and gear vibration data. He used a variety of methods to select the most appropriate coefficients for this task including: simple thresholding; a priori knowledge of the Copyright © 2002 lOP Publishing Ltd.
90.0 r                  ,
. _
60.0
30.0
o
,, ,, ,, ,
,,
,,
_.
  
100
200
300
number of coefficients
Figure 5.39. Compression performance for periodic data. MSE plotted as a function of number of wavelet coefficients used for compression. Solid line = D4; dashed line= 020; dash