Page 1

The Illustrated Wavelet

Transform Handbook

SECOND EDITION

Introductory Theory and Applications in

Science, Engineering, Medicine and Finance

The Illustrated Wavelet

Transform Handbook

SECOND EDITION

Paul S. Addison

Edinburgh, Scotland

Introductory Theory and Applications in

Science, Engineering, Medicine and Finance

Boca Raton London New York

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

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Library of Congress Cataloging‑in‑Publication Data

Names: Addison, Paul S., author.

Title: e illustrated wavelet transform handbook : introductory theory and

applications in science, engineering, medicine and finance / Paul S.

Addison.

Description: Second edition. | Boca Raton, FL : CRC Press, Taylor & Francis

Group, [2016] | Includes bibliographical references and index.

Identifiers: LCCN 2016033578| ISBN 9781482251326 (hardback ; alk. paper) |

ISBN 1482251329 (hardback ; alk. paper) | ISBN 9781482251333 (e-book) |

ISBN 1482251337 (e-book)

Subjects: LCSH: Wavelets (Mathematics) | Fourier analysis. | Biomedical

engineering.

Classification: LCC QA403.3 .A35 2016 | DDC 515/.2433--dc23

LC record available at https://lccn.loc.gov/2016033578

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

To Hannah, Stephen, Anthony, Michael and, this time,

David; to my dear parents, Josephine and Stanley, gone

but still with us; and to my loving wife, Stephanie.

Thank you.

vii

Contents

Preface to Second Edition, xiii

Preface to First Edition, xv

CHAPTER 1 ▪ Getting Started

1

1.1 INTRODUCTION

1

1.2 WAVELET TRANSFORM

2

1.3 READING THE BOOK

4

CHAPTER 2 ▪ The Continuous Wavelet Transform

7

2.1 INTRODUCTION

7

2.2 THE WAVELET

7

2.3 REQUIREMENTS FOR THE WAVELET

10

2.4 THE ENERGY SPECTRUM OF THE WAVELET

10

2.5 WAVELET TRANSFORM

12

2.6 IDENTIFICATION OF COHERENT STRUCTURES

14

2.7 EDGE DETECTION

22

2.8 INVERSE WAVELET TRANSFORM

24

2.9 SIGNAL ENERGY: WAVELET-BASED ENERGY AND POWER SPECTRA 28

2.10 WAVELET TRANSFORM IN TERMS OF THE FOURIER TRANSFORM

33

2.11 COMPLEX WAVELETS: THE MORLET WAVELET

34

2.12 WAVELET TRANSFORM, SHORT-TIME FOURIER TRANSFORM AND

HEISENBERG BOXES

44

2.13 ADAPTIVE TRANSFORMS: MATCHING PURSUITS

51

2.14 WAVELETS IN TWO OR MORE DIMENSIONS

55

2.15 THE CWT: COMPUTATION, BOUNDARY EFFECTS AND VIEWING

56

2.16 RIDGE FOLLOWING AND SECONDARY WAVELET FEATURE

DECOUPLING

61

2.17 RIDGE HEIGHTS

66

2.18 RUNNING WAVELET ARCHETYPING

67

viii ◾ Contents

2.19 WAVELET TRANSFORM REPHASING

69

2.20 REASSIGNMENT AND SYNCHROSQUEEZING

73

2.21 COMPARING TWO SIGNALS USING WAVELET TRANSFORMS

75

2.21.1 Transform Differences and Ratios

76

2.21.2 Cross-Wavelet Transform

78

2.21.3 Wavelet Cross-Correlation

81

2.21.4 Phase Comparison Measures

83

2.21.5 Wavelet Coherence

85

2.22 BICOHERENCE AND CROSS-BICOHERENCE

86

2.23 ENDNOTES

88

2.23.1 Chapter Key Words and Phrases

88

2.23.2 Additional Notes and Resources

88

2.23.3 ings to Try

89

2.23.4 Final Note: e CWT as a ‘Soft Tool’ for Algorithm Development

91

CHAPTER 3 ▪ The Discrete Wavelet Transform

93

3.1 INTRODUCTION

93

3.2 FRAMES AND ORTHOGONAL WAVELET BASES

93

3.2.1 Frames

93

3.2.2 Dyadic Grid Scaling and Orthonormal Wavelet Transforms

95

3.2.3 Scaling Function and Multiresolution Representation

98

3.2.4 Scaling Equation, Scaling Coefficients and Associated Wavelet

Equation

101

3.2.5 Haar Wavelet

102

3.2.6 Coefficients from Coefficients: Fast Wavelet Transform

105

3.3 DISCRETE INPUT SIGNALS OF FINITE LENGTH

107

3.3.1 Approximations and Details

107

3.3.2 Multiresolution Algorithm: An Example

111

3.3.3 Wavelet Energy

114

3.3.4 Alternative Indexing of Dyadic Grid Coefficients

115

3.3.5 A Simple Worked Example: e Haar Wavelet Transform

117

3.4 EVERYTHING DISCRETE

122

3.4.1 Discrete Experimental Input Signals

122

3.4.2 Smoothing, resholding and Denoising

127

3.5 DAUBECHIES WAVELETS

135

3.5.1 Filtering

143

Contents    ◾    ix

3.5.2 Symmlets and Coiflets

148

3.6 TRANSLATION INVARIANCE

149

3.7 BIORTHOGONAL WAVELETS

150

3.8 TWO-DIMENSIONAL WAVELET TRANSFORMS

151

3.9 ADAPTIVE TRANSFORMS: WAVELET PACKETS

163

3.10 ‘X-LETS’: CONTOURLETS, RIDGELETS, CURVELETS, SHEARLETS

AND SO ON

168

3.11 ENDNOTES

173

3.11.1 Chapter Key Words and Phrases

173

3.11.2 Further Resources

173

CHAPTER 4 ▪ Fluids

177

4.1 INTRODUCTION

177

4.2 STATISTICAL MEASURES FOR FLUID TURBULENCE

178

4.2.1 Moments, Energy and Power Spectra

178

4.2.2 Intermittency and Correlation

185

4.2.3 Wavelet resholding

186

4.2.4 Wavelet Selection Using Entropy Measures

191

4.3 ENGINEERING FLOWS

194

4.3.1 Experimental Flows: Jets, Wakes, Turbulence and Coherent

Structures

194

4.3.2 Computational Fluid Dynamics: Simulation and Analysis

200

4.3.3 Fluid–Structure Interaction

206

4.4 GEOPHYSICAL FLOWS

210

4.4.1 Atmospheric Processes: Wind, Boundary Layers and Turbulence

211

4.4.2 Ocean Processes: Waves, Large-Scale Oscillations,

Ocean–Atmosphere Interactions and Biological Processes

215

4.4.3 Rainfall and River Flows

221

4.5 TWO-PHASE FLOWS

224

4.6 OTHER APPLICATIONS IN FLUIDS

227

CHAPTER 5 ▪ Engineering Testing, Monitoring and Characterization

231

5.1 INTRODUCTION

231

5.2 DYNAMICS

232

5.2.1 Fundamental Behaviour

232

5.2.2 Chaos

236

x ◾ Contents

5.3 NON-DESTRUCTIVE TESTING OF STRUCTURAL ELEMENTS

238

5.4 CONDITION MONITORING OF ROTATING MACHINERY

249

5.4.1 Gears

249

5.4.2 Shafts, Bearings and Blades

255

5.5 MACHINING PROCESSES

259

5.6 CHARACTERIZATION OF SURFACES AND FIBROUS MATERIALS

261

5.7 OTHER APPLICATIONS IN ENGINEERING

263

5.7.1 Compression

263

5.7.2 Control

264

5.7.3 Electrical Systems and Circuits

265

5.7.4 Miscellaneous

265

CHAPTER 6 ▪ Medicine

267

6.1 INTRODUCTION

267

6.2 ELECTROCARDIOGRAM

267

6.2.1 ECG Beat Detection and Timings

268

6.2.2 Detection of Abnormalities

271

6.2.3 Heart Rate Variability

275

6.2.4 Cardiac Arrhythmias

278

6.2.5 ECG Data Compression

289

6.2.6 Hardware Implementation

289

6.3 NEUROELECTRIC WAVEFORMS

290

6.3.1 Evoked Potentials and Event-Related Potentials

291

6.3.2 Epileptic Seizures and Epileptogenic Foci

297

6.3.3 Sleep Studies

299

6.3.4 Other Areas

302

6.4 PHOTOPLETHYSMOGRAM

302

6.4.1 Respiratory Modulations, Respiratory Rate and Respiratory Effort 303

6.4.2 Oxygen Saturation

307

6.4.3 e Video Photoplethysmogram (Video-PPG)

309

6.4.4 Other Areas

312

6.5 PATHOLOGICAL SOUNDS, ULTRASOUNDS AND VIBRATIONS

315

6.5.1 Cardiovascular System

315

6.5.2 Lung Sounds, Swallowing, Snoring and Speech

318

6.5.3 Acoustic Response

321

Contents    ◾    xi

6.6 BLOOD FLOW AND BLOOD PRESSURE

323

6.7 MEDICAL IMAGING

328

6.7.1 Optical Imaging

328

6.7.2 Ultrasonic Images

330

6.7.3 Computed Tomography, Magnetic Resonance Imaging and Other

Radiographic Images

330

6.8 OTHER APPLICATIONS IN MEDICINE

332

6.8.1 Electromyographic Signals

332

6.8.2 Posture, Gait and Activity

333

6.8.3 Analysis of Multiple Biosignals

334

6.8.4 Miscellaneous

334

CHAPTER 7 ▪ Fractals, Finance, Geophysics, Astronomy and Other Areas 337

7.1 INTRODUCTION

337

7.2 FRACTALS

337

7.2.1 Exactly Self-Similar Fractals

338

7.2.2 Stochastic Fractals

341

7.2.3 Multifractals

350

7.3 FINANCE

355

7.4 GEOPHYSICS

361

7.4.1 Properties of Subsurface Media: Well Logging, Cores and Seismic

Methods

361

7.4.2 Remote Sensing

366

7.5 ASTRONOMY: SIGNALS AND IMAGES

371

7.6 OTHER AREAS

378

REFERENCES, 381

APPENDIX: USEFUL BOOKS, PAPERS AND WEBSITES, 441

INDEX, 443

xiii

Preface to Second Edition

S

the first edition in 2002, there has been an explosion of

interest in wavelet transform methods. In the intervening period, I have moved on

from academia to a start-up company, and then to a global medical device company; and

a significant part of this time has been spent working with wavelet methods. Working in

a small start-up requires rolling up your sleeves and get involved in all aspects of the busi-

ness. I was lucky enough to get the opportunity work closely with many interesting indi-

viduals from a wide range of diverse backgrounds across disciplines and across the world:

amazing engineers and scientists from many sub-specialities; dedicated clinicians working

at the forefront of medical science in a variety of areas of care; skilful lawyers practicing

corporate law and the management of intellectual property; and some quite remarkably

astute businessmen and women. is has given me a deep appreciation of what consti-

tutes real professionalism in other disciplines; many of which were previously a complete

mystery to me. I have made many friends and met many good folk along the way: smart,

dedicated professionals with integrity. None more so, nor more enjoyable company, than

Jamie Watson, with whom I began the rollercoaster ride that was our start-up company

CardioDigital Ltd, and Peter Galen, who joined us en route with still quite a few peaks and

troughs to ride out. anks guys.

I have been fortunate enough to be inventor on around 100 US patents to date, with

around 200 US patent applications in the system stemming from approximately 400 sub-

mitted ideas (dyadic scaling!) – and many of these concerning wavelet methods. I find that

grasping the concepts before tackling the mathematics is my best way forward when it

comes to understanding and innovation, and I know no better way of achieving the former

than through the use of an illustration: a quickly scribbled diagram, flow chart or cartoon

drawing. I often write instructions for colleagues in terms of diagrams rather than text and

I like drawing mind maps as I explain things. Hence, The Illustrated Wavelet Transform

Handbook. Wavelet transform methods produce distinct morphologies in ‘wavelet space’-

a concept that is so much easier to grasp, and more importantly feel comfortable ‘within’,

when viewing a well-constructed plot: whether it be a modulus or phase plot or something

that combines both. Once familiar with these mathematical ‘landscapes’, we may begin to

explore unchartered territory. at’s when the real fun begins!

I must thank a large number of people for their considerable time and effort in provid-

ing comment on various sections of the draft text, without all of whom I really could not

have got to the end of this task. ese are, in roughly the order of the text: Dr. Ge, CFD

xiv ◾ Preface to Second Edition

group lead, American Bureau of Shipping, Houston, Texas, USA; Dr Christopher Torrence,

Harris Geospatial Solutions, Boulder, Colorado; and Professor Stephen Payne of the

University of Oxford for reading over various parts of the new theory sections in Chapter 2;

Dr Andr� Atunes, a colleague and multi-talented research scientist, for providing detailed

comment on the mathematics and explanatory text for the whole of Chapter 2; Professor

James Walker, Department of Mathematics at the University of Wisconsin-Eau Claire for

his comments on various sections in the theory chapters, 2 and 3; Associate Professor Paul

Brandner of the Cavitation Research Laboratory within the Australian Maritime College

at the University of Tasmania for his assiduous reading and most useful comments with

respect to all aspects of fluid phenomena in Chapter 4; Professor Wieslaw Staszewski of

the AGH University of Science & Technology in Krakow for his comments on various

engineering sections in Chapter 5; James Ochs, an erstwhile colleague and excellent bio-

medical engineer, for reading over the whole of Chapter 6; Dr Paul Mannheimer, another

former colleague and a recognised world expert in pulse oximetry, for his comments on

the section concerning the analysis of the photoplethysmogram in Chapter 6; Dr Dean

Montgomery, a colleague and outstanding scientist and engineer, for reading through the

medical images section of Chapter 6; Professor Leontios Hadjileontiadis of the Aristotle

University of essaloniki, Greece and the Khalifa University, Abu Dhabi who provided

comment on the wavelet coherence theory in Chapter 2 and the section on physiologi-

cal sounds in Chapter 6. Finally, a number of people were extremely helpful in checking

Chapter 7, with its very wide ranging selection of topics. ese included Dr Maria Haase of

the Institut f�r H�chstleistungsrechnen, Universit�t Stuttgart for reading over the fractal

and multifractal sections of Chapter 7, as she did for the first edition with such enthusiasm

and rigor; Dr Ginanjar Dewandaru of INCEIF, e Global University of Islamic Finance,

Lorong University A, Kuala Lumpur, Malaysia and Claudiu Albulescu, Associate Professor

in the Management Department at the Politehnica University of Timisoara, Romania for

comments regarding the finance section of Chapter 7; Mathieu J. Duchesne, Research

Scientist, Geological Survey of Canada, Quebec City for his thoughts on the geophysics

section, and Matthew S. Tiscareno, Senior Research Scientist at the SETI Institute and Dr.

Florent Mertens, Kapteyn Astronomical Institute, Groningen, e Netherlands for proof

reading the astronomy section in Chapter 7 in meticulous detail.

I’d also like to take the opportunity to express my gratitude to the Wellcome Trust for

their unwavering support for our university spin-off company, CardioDigital Ltd, over the

years. In the same vein, thank you to the Scottish Government and Scottish Enterprise for

their invaluable grant funding.

Finally, thanks to my daughter Hannah for her unbelievable attention to detail while

proof reading over the whole manuscript from a non-technical vantage point; my sons,

Michael, Anthony, Stephen and David for their interest in what I do (and help picking the

new cover); and my wife Stephanie, for everything.

Paul S. Addison

xv

Preface to First Edition

O

or so, wavelet transform analysis has emerged as a major new

time–frequency decomposition tool for data analysis. is book is intended to pro-

vide the reader with an overview of the theory and practical applications of wavelet trans-

form methods. It is designed specifically for the ‘applied’ reader, whether he or she is a

scientist, engineer, medic, financier or other.

e 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. e 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. e final chapter, Chapter 7, covers a number of subject areas. In

this chapter, three main topics are considered initially – fractals, finance and geophysics –

and then these are followed by a general discussion which includes many other areas that

are not covered in the rest of the book.

e chapters which address theory (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. e key words and phrases are listed at the end of each chapter and the reader

who is 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. e chapters which address the various

applications of the theory (Chapters 4 through 7) are at the same level, although a consid-

erable amount of useful information can be gained without an in-depth knowledge of the

theory in Chapters 2 and 3, especially in providing an overview of these applications.

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 to 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 informa-

tion. e book is very much ‘figure driven’ as I believe that figures are extremely useful for

illustrating the mathematics and conveying the concepts. e application chapters of the

book aim to make the reader aware of the similarities that exist in the uses of wavelet trans-

form analysis across disciplines. In addition, and perhaps more importantly, it is intended

to make the reader aware of wavelet-based methods in use in unfamiliar disciplines which

may be transferred to his or her own area – thus promoting an interchange of ideas across

discipline boundaries.

xvi ◾ Preface to First Edition

e application chapters are essentially a whistle-stop tour of work by a large num-

ber 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. e examples and

illustrations used have been chosen to provide an appropriate range to best illustrate the

wavelet-based work being carried out in each subject area. It is not intended to delve deeply

into each subject but rather to provide a brief overview. It is then left to the reader to fol-

low up the relevant references cited in the text for himself or herself in order to delve more

deeply into each particular topic as he or she requires.

I refer to over 700 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 apologize in advance. Of course, all comments are wel-

come – my e-mail address can be found below.

e book stems from my own interest in wavelet transform analysis over the past few

years. is interest has led to a number of research projects concerning the wavelet-based

analysis of both engineering and medical signals; including non-destructive testing sig-

nals, 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 pat-

ent applications and a university spin-off company, Cardiodigital Ltd.

Quite a mixed bag, at first appearance, but with a common thread of wavelet analy-

sis 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 work – although

the temptation was high for a number of reasons, including knowledge of the work, ease of

reproduction, etc. I hope that 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 weighs more heavily towards 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, non-medical biology, power systems

analysis) are detailed briefly in the final chapter.

ere are some idiosyncrasies in the text which are worth pointing out. I am an f person

not an ɷ 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; however, ɷ

I have to convert. Hence the frequencies in the text are in the form of 1/time either in

hertz or non-dimensionalized. e small downside is that the mathematics, in general,

contains a few more terms – mostly 2s and πs. I have devoted a whole chapter to the con-

tinuous 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 wavelet transform. I believe that

the continuous wavelet transform has a wide variety of data analysis tasks to offer, and

I attempt, through this text, to redress the balance somewhat. (Actually, the proportion

Preface to First Edition    ◾    xvii

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 textbooks.) e book is focused on the wavelet transform and makes only passing

reference in the application chapters to some of the other time–frequency methods now

available. However, I have added sections on the short-time Fourier transform and match-

ing 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 vari-

ous 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 Heriot-Watt 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 of the 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 the 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

1

CHAPTER 1

Getting Started

1.1 INTRODUCTION

The wavelet transform (WT) has been found to be particularly useful for analyzing

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 an ever-increasing number of sophisticated wavelet-based 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 astronomi-

cal signals and images, from surface characterization to the characterization of turbu-

lent 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 mid-

1980s, when it was 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. There are now thousands of refereed journal papers

concerning the wavelet transform, and these cover all numerate disciplines. The wavelet

transform is a mathematical tool which is now an indispensible part of many data ana-

lysts’ toolboxes. This book aims to provide the reader with both an introduction to the

theory of wavelet transforms and a wide-ranging overview of their use in practice. The

two remaining sections of this short introductory chapter contain, respectively, a brief

non-mathematical description of the wavelet transform and a guide to subsequent chap-

ters of the book.

2 ◾ The Illustrated Wavelet Transform Handbook

1.2 WAVELET TRANSFORM

Wavelet transform analysis uses little wavelike functions known as wavelets. Actually,

localized wavelike function is a more accurate description of a wavelet. Figure�1.1a shows a

few examples of wavelets commonly encountered in practice. Wavelets are used to trans-

form the signal under investigation into another representation which presents the sig-

nal information in a more useful form. This transformation of the signal is known as the

wavelet transform. Mathematically speaking, the wavelet transform may be interpreted as

a convolution of the signal with a wavelet function, 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�1.1b) and it can be stretched or squeezed (Figure�1.1c). Figure�1.2

shows a schematic of the wavelet transform which basically quantifies the local matching

(a)

(b)

(c)

FIGURE�1.1 The little wave: (a) Some wavelets, (b) Location and (c) Scale.

Getting Started    ◾    3

of the wavelet with the signal. If 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�1.2, then a large

transform value is obtained. If, however, the wavelet and the signal do not correlate

well, a low transform value is obtained. The transform value is then located in the two-

dimensional transform plane shown at the bottom of Figure�1.2 (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 continuous wavelet transform (CWT) or in discrete steps for the discrete

wavelet transform (DWT). Plotting the wavelet transform allows a picture to be built up

of the correlation between the wavelet – at various scales and locations�– and the sig-

nal. In subsequent chapters, we will cover the wavelet transform in more mathematical

detail.

Signal

Local matching of

wavelet and signal leads

to a large transform value

Location

Scale

Wavelet

transform

plot

Current

wavelet

location

Current

wavelet

scale

Wavelet

transform

FIGURE�1.2 The wavelet, the signal and the transform.

4 ◾ The Illustrated Wavelet Transform Handbook

1.3 READING THE BOOK

The purpose of the book is both to introduce the wavelet transform and to convey its mul-

tidisciplinary nature. This is achieved in subsequent chapters by first providing an ele-

mentary introduction to wavelet transform theory and then presenting a wide range of

examples of its application. It will quickly become apparent that very often the same wave-

let 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 – comprising Chapters�2 and 3 –

deals, respectively, with continuous wavelet transform theory and discrete transform

theory, while the second – comprising Chapters�4 through 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 own area of interest. The reader is, however, also advised to look outwith his

or her own area to see how wavelets are being employed elsewhere. (The author cannot

emphasize this enough!) Details of further resources concerning the theory and appli-

cations 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 sig-

nal. The continuous wavelet transform is compared with both the short-time Fourier

transform and the matching pursuit method. Transform features such as ridges and

local phase cycling are then considered before examining various useful manipula-

tions of the transform such as rephasing, the production of a transform archetype,

synchrosqueezing and reassignment. Finally, the relationship between two trans-

formed signals is considered via transform differences and ratios, the cross-wavelet

transform, wavelet cross-correlation, phase comparison measures, wavelet coherence

and bicoherence.

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 dis-

crete 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 also briefly covers wavelet packets – a generalization of the discrete wavelet

transform which allows for adaptive partitioning of the time–frequency plane – prior

to ending with a section on time–frequency functions that stem from the wavelet

method and allow efficient representation of directionally dependent two-dimen-

sional data.

Getting Started    ◾    5

Chapter�4: This chapter deals with fluid mechanics, a subject that is always open to new

mathematical techniques. The time-frequency localization properties of the wavelet

transform have been employed extensively in the study of a wide variety of fluid phe-

nomena, including the intermittent nature of fluid turbulence, the characteristics of

turbulent jets, the nature of fluid–structure interactions and the behaviour of large-

scale geophysical flows. Chapter�4 also contains the mathematics for discrete wave-

let statistics and power spectra following on from some of the basic theory given in

Chapter�3. Thus, it is worthwhile reading Section�4.2 of this chapter even if fluids is

not your area of interest.

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

analysis of fundamental dynamical behaviour and chaotic motions, non-destructive

testing of structural elements and the condition monitoring of machinery, machin-

ing processes, and the characterization of surfaces and fibrous materials.

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 electrocardiogram

(ECG), electroencephalogram (EEG), electromyogram (EMG) photoplethysmogram

(PPG), pathological sounds (heart murmurs, lung sounds, swallowing sounds, snor-

ing, speech, otoacoustic emissions), blood flows, blood pressures, medical images

(optical, x-ray, NMR, ultrasound, etc.), and posture, gait and activity signals.

Chapter�7: This final chapter covers a variety of areas of application. Most of the chap-

ter is devoted to four main subjects – fractal geometry, finance, geophysics and

astronomy – with a separate section given over to each one of them. The final part of

the chapter provides a brief account of the role that wavelet transform analysis has

played in a number of other areas, including quantum mechanics, chemistry, ecologi-

cal processes and patterns, and more.

Appendix 1: The appendix contains a brief list of useful books and websites concerning

wavelet transform theory and its application. These have been chosen by the author

for their extensive content and/or clarity of presentation.

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