Multiresolution analysis using the wavelet transform has received considerable attention in recent years by researchers in various fields. It is a powerful tool for efficiently representing signals and images at multiple levels of detail with many inherent advantages, including compression, level-of-detail display, progressive transmission, level-of-detail editing, filtering, modeling, fractals and multifractals, etc.
This book aims to provide a simple formalization and new clarity on multiresolution analysis, rendering accessible obscure techniques, and merging, unifying or completing the technique with encoding, feature extraction, compressive sensing, multifractal analysis and texture analysis. It is aimed at industrial engineers, medical researchers, university lab attendants, lecturer-researchers and researchers from various specializations. It is also intended to contribute to the studies of graduate students in engineering, particularly in the fields of medical imaging, intelligent instrumentation, telecommunications, and signal and image processing.
Given the diversity of the problems posed and addressed, this book paves the way for the development of new research themes, such as brain–computer interface (BCI), compressive sensing, functional magnetic resonance imaging (fMRI), tissue characterization (bones, skin, etc.) and the analysis of complex phenomena in general. Throughout the chapters, informative illustrations assist the uninitiated reader in better conceptualizing certain concepts, taking the form of numerous figures and recent applications in biomedical engineering, communication, multimedia, finance, etc.

Introduction  xi

Chapter 1. Introduction to Multiresolution Analysis 1

1.1. Introduction 1

1.2. Wavelet transforms: an introductory review 3

1.2.1. Brief history 3

1.2.2. Continuous wavelet transforms 6

1.2.2.1. Wavelet transform modulus maxima 9

1.2.2.2. Reconstruction 13

1.2.3. Discrete wavelet transforms 14

1.3. Multiresolution 16

1.3.1. Multiresolution analysis and wavelet bases  17

1.3.1.1. Approximation spaces 17

1.3.1.2. Detail spaces 19

1.3.2. Multiresolution analysis: points to remember 21

1.3.3. Decomposition and reconstruction 22

1.3.3.1. Calculation of coefficients 22

1.3.3.2. Implementation of MRA: Mallat algorithm     24

1.3.3.3. Extension to images 26

1.3.4. Wavelet packets 28

1.3.5. Multiresolution analysis summarized 30

1.4. Which wavelets to choose? 33

1.4.1. Number of vanishing moments, regularity,

support (compactness), symmetry, etc 33

1.4.2. Well-known wavelets, scale functions and associated filters 34

1.4.2.1. Haar wavelet 34

vi Signal and Image Multiresolution Analysis

1.4.2.2. Daubechies wavelets 36

1.4.2.3. Symlets 38

1.4.2.4. Coiflets 39

1.4.2.5. Meyer wavelets 41

1.4.2.6. Polynomial spline wavelets 43

1.5. Multiresolution analysis and biorthogonal wavelet bases  48

1.5.1. Why biorthogonal bases? 48

1.5.2. Multiresolution context 48

1.5.3. Example of biorthogonal wavelets, scaling functions

and associated filters 49

1.5.4. The concept of wavelet lifting 51

1.5.4.1. The notion of lifting 51

1.5.4.2. Significance of structure lifting 52

1.6. Wavelet choice at a glance 54

1.6.1. Regularity 54

1.6.2. Vanishing moments 54

1.6.3. Other criteria 55

1.6.4. Conclusion 55

1.7. Worked examples 55

1.7.1. Examples of multiresolution analysis 55

1.7.2. Compression 58

1.7.3. Denoising (reduction of noise) 64

1.8. Some applications 74

1.8.1. Discovery and contributions of wavelets 74

1.8.2. Biomedical engineering 76

1.8.2.1. ECG, EEG and BCI 77

1.8.2.2. Medical imaging 97

1.8.3. Telecommunications 110

1.8.3.1. Adaptive compression for sensor networks 110

1.8.3.2. Masking image encoding and transmission errors 114

1.8.3.3. Suppression of correlated noise 118

1.8.4. “Compressive sensing”, ICA, PCA and MRA    119

1.8.4.1. Principal component analysis 120

1.8.4.2. Independent component analysis 121

1.8.4.3. Compressive sensing 122

1.8.5. Conclusion 128

1.9. Bibliography 129

Chapter 2. Discrete Wavelet Transform-Based Multifractal Analysis 135

2.1. Introduction 135

2.1.1. Fractals and wavelets: a happy marriage? 135

2.1.2. Background 136

2.1.3. Mono/multifractal processes 137

2.1.4. Chapter outline 138

2.2. Fractality, variability and complexity    139

2.2.1. System complexity 139

2.2.2. Complex phenomena properties   141

2.2.2.1. Tendency of autonomous agents to self-organize  141

2.2.2.2. Variability and adaptability 142

2.2.2.3. Bifurcation concept and chaotic model 143

2.2.2.4. Hierarchy and scale invariance 146

2.2.2.5. Self-organized critical phenomena 146

2.2.2.6. Highly optimized tolerance 147

2.2.3. Fractality 148

2.3. Multifractal analysis 150

2.3.1. Point-wise regularity 150

2.3.2. Hölder exponent 150

2.3.3. Signal classification according to the regularity properties 152

2.3.3.1. Monofractal signal 152

2.3.3.2. Multifractal signal 152

2.3.4. Hausdorff dimension   154

2.3.4.1. Theoretic approach 155

2.3.4.2. Qualitative approach and multifractal spectrum    155

2.4. Multifractal formalism 156

2.4.1. Reminder on wavelet decomposition 156

2.4.2. Point-wise regularity characterization  157

2.4.3. Structure function and power law behavior 158

2.4.4. Link between scaling exponents and singularity spectrum  159

2.4.5. Use of wavelet leaders 160

2.4.5.1. Indexing a dyadic square and wavelet leaders 161

2.4.5.2. Polynomial expansion and log-cumulants   162

2.4.6. Wavelet leaders variant: “maximum” coefficients 165

2.5. Algorithm and performances 165

2.5.1. Singularity spectrum estimation algorithm 165

2.5.2. Analysis of a few widely used processes 167

2.5.2.1. fBm: a monofractal process 167

2.5.2.2. CMC: a multifractal process   170

2.5.2.3. BMC: another class of multifractal processes   173

2.5.3. Estimation performances 176

2.5.3.1. fBm and CMC-LN and CMC-LP simulation   176

2.5.3.2. Results 177

2.5.3.3. Interpretation and recommendations 182

2.6. Applications 186

2.6.1. Turbulence 186

viii Signal and Image Multiresolution Analysis

2.6.1.1. From Leonardo da Vinci to Kolmogorov    186

2.6.1.2. Multifractal process in turbulence 191

2.6.2. Multifractal process in finance 194

2.6.2.1. Stock market complexity modeling 194

2.6.2.2. Market turbulence 198

2.6.3. Internet traffic 203

2.6.3.1. The Internet revolution 203

2.6.3.2. Multifractal nature of Internet traffic? 204

2.6.4. Biomedical field 205

2.6.4.1. Analyzing image texture through a multifractal approach 205

2.6.4.2. Multifractality in medical imaging 211

2.7. Conclusion  219

2.8. Bibliography 220

Chapter 3. Multimodal Compression Using JPEG 2000:

Supervised Insertion Approach 225

3.1. Introduction 225

3.2. The JPEG 2000 standard 226

3.3. Multimodal compression by unsupervised insertion 227

3.3.1. Principle of insertion in the wavelet transform domain 228

3.3.2. Principle of insertion in the spatial domain 229

3.4. Multimodal compression by supervised insertion 231

3.4.1. Choice of insertion zone 232

3.4.2. Insertion and separation function 233

3.4.2.1. Insertion function 233

3.4.2.2. Separation function 235

3.5. Criteria for quality evaluation 236

3.5.1. Peak signal-to-noise ratio 236

3.5.2. Percent residual difference 237

3.6. Some preliminary results 238

3.7. Conclusion 242

3.8. Bibliography 243

Chapter 4. Cerebral Microembolism Synchronous Detection

with Wavelet Packets 245

4.1. Issue and stakes 245

4.2. Prior information research 247

4.2.1. Doppler ultrasound blood flow signal 247

4.2.2. Embolic Doppler ultrasound signal 250

4.3. Doppler ultrasound blood emboli signal modeling  251

4.3.1. “Physical” model 251

4.3.2. “Signal” model 255

4.3.3. Statistical tests 258

4.3.3.1. Stationarity test  259

4.3.3.2. Cyclostationarity test  261

4.3.3.3. “Gaussian” test and cardiac cycle regularity 262

4.4. Energy detection 263

4.4.1. State-of-the-art and standard detection   263

4.4.2. Synchronous detection 266

4.5. Wavelet packet energy detection 269

4.5.1. Introduction 269

4.5.2. Multiresolution analysis 271

4.5.3. Wavelet packet subband detection 274

4.5.4. Wavelet packet synchronous detection 277

4.6. Results and discussions 279

4.6.1. In simulation 279

4.6.2. In vivo 282

4.7. Conclusion 285

4.8. Bibliography 285

List of Authors 289

Index 291