Wavelets and their Applications

Georges Oppenheim, Michel Misiti and Jean-Michel Poggi, members of the Laboratoire de Mathématiques at Paris 11 University, France, are Mathematics Professors at the Ecole Centrale de Lyon, University of Marne-La-Vallée and Paris 5 University, France. Yves Misiti is a research engineer specializing in computer sciences at Paris 11 University, France.

Notations xiii

Introduction xvii

Chapter 1. A Guided Tour 1

1.1. Introduction 1

1.2. Wavelets 2

1.2.1. General aspects 2

1.2.2. A wavelet 6

1.2.3. Organization of wavelets 8

1.2.4. The wavelet tree for a signal 10

1.3. An electrical consumption signal analyzed by wavelets 12

1.4. Denoising by wavelets: before and afterwards 14

1.5. A Doppler signal analyzed by wavelets 16

1.6. A Doppler signal denoised by wavelets 17

1.7. An electrical signal denoised by wavelets 19

1.8. An image decomposed by wavelets 21

1.8.1. Decomposition in tree form 21

1.8.2. Decomposition in compact form 22

1.9. An image compressed by wavelets 24

1.10. A signal compressed by wavelets 25

1.11. A fingerprint compressed using wavelet packets 27

Chapter 2. Mathematical Framework 29

2.1. Introduction 29

2.2. From the Fourier transform to the Gabor transform 30

2.2.1. Continuous Fourier transform 30

2.2.2. The Gabor transform 35

2.3. The continuous transform in wavelets 37

2.4. Orthonormal wavelet bases 41

2.4.1. From continuous to discrete transform 41

2.4.2. Multi-resolution analysis and orthonormal wavelet bases 42

2.4.3. The scaling function and the wavelet 46

2.5. Wavelet packets 50

2.5.1. Construction of wavelet packets 50

2.5.2. Atoms of wavelet packets 52

2.5.3. Organization of wavelet packets 53

2.6. Biorthogonal wavelet bases 55

2.6.1. Orthogonality and biorthogonality 55

2.6.2. The duality raises several questions 56

2.6.3. Properties of biorthogonal wavelets 57

2.6.4. Semi-orthogonal wavelets 60

Chapter 3. From Wavelet Bases to the Fast Algorithm 63

3.1. Introduction. 63

3.2. From orthonormal bases to the Mallat algorithm 64

3.3. Four filters 65

3.4. Efficient calculation of the coefficients 67

3.5. Justification: projections and twin scales 68

3.5.1. The decomposition phase 69

3.5.2. The reconstruction phase 72

3.5.3. Decompositions and reconstructions of a higher order 75

3.6. Implementation of the algorithm 75

3.6.1. Initialization of the algorithm 76

3.6.2. Calculation on finite sequences 77

3.6.3. Extra coefficients 77

3.7. Complexity of the algorithm 78

3.8. From 1D to 2D 79

3.9. Translation invariant transform 81

3.9.1. e-decimated DWT 83

3.9.2. Calculation of the SWT 83

3.9.3. Inverse SWT 87

Chapter 4. Wavelet Families 89

4.1. Introduction 89

4.2. What could we want from a wavelet? 90

4.3. Synoptic table of the common families 91

4.4. Some well known families 92

4.4.1. Orthogonal wavelets with compact support 93

4.4.2. Biorthogonal wavelets with compact support: bior 99

4.4.3. Orthogonal wavelets with non-compact support 101

4.4.4. Real wavelets without filters 104

4.4.5. Complex wavelets without filters 106

4.5. Cascade algorithm 109

4.5.1. The algorithm and its justification 110

4.5.2. An application 112

4.5.3. Quality of the approximation 113

Chapter 5. Finding and Designing a Wavelet 115

5.1. Introduction 115

5.2. Construction of wavelets for continuous analysis 116

5.2.1. Construction of a new wavelet 116

5.2.2. Application to pattern detection 124

5.3. Construction of wavelets for discrete analysis 131

5.3.1. Filter banks 132

5.3.2. Lifting 140

5.3.3. Lifting and biorthogonal wavelets 146

5.3.4. Construction examples 149

Chapter 6. A Short 1D Illustrated Handbook 159

6.1. Introduction 159

6.2. Discrete 1D illustrated handbook 160

6.2.1. The analyzed signals 160

6.2.2. Processing carried out 161

6.2.3. Commented examples 162

6.3. The contribution of analysis by wavelet packets 178

6.3.1. Example 1: linear and quadratic chirp 178

6.3.2. Example 2: a sine181

6.3.3. Example 3: a composite signal 182

6.4. “Continuous” 1D illustrated handbook 183

6.4.1. Time resolution 183

6.4.2. Regularity analysis 187

6.4.3. Analysis of a self-similar signal 193

Chapter 7. Signal Denoising and Compression 197

7.1. Introduction 197

7.2. Principle of denoising by wavelets 198

7.2.1. The model 198

7.2.2. Denoising: before and after 198

7.2.3. The algorithm 199

7.2.4. Why does it work? 200

7.3. Wavelets and statistics 200

7.3.1. Kernel estimators and estimators by orthogonal projection 201

7.3.2. Estimators by wavelets 201

7.4. Denoising methods 202

7.4.1. A first estimator 203

7.4.2. From coefficient selection to thresholding coefficients 204

7.4.3. Universal thresholding 206

7.4.4. Estimating the noise standard deviation 206

7.4.5. Minimax risk 207

7.4.6. Further information on thresholding rules 208

7.5. Example of denoising with stationary noise 209

7.6. Example of denoising with non-stationary noise 212

7.6.1. The model with ruptures of variance 213

7.6.2. Thresholding adapted to the noise level change-points 214

7.7. Example of denoising of a real signal 216

7.7.1. Noise unknown but “homogenous” in variance by level 216

7.7.2. Noise unknown and “non-homogenous” in variance by level 217

7.8. Contribution of the translation invariant transform 218

7.9. Density and regression estimation 221

7.9.1. Density estimation 221

7.9.2. Regression estimation 224

7.10. Principle of compression by wavelets 225

7.10.1. The problem 225

7.10.2. The basic algorithm 225

7.10.3. Why does it work? 226

7.11. Compression methods 226

7.11.1. Thresholding of the coefficients 226

7.11.2. Selection of coefficients 228

7.12. Examples of compression 229

7.12.1. Global thresholding 229

7.12.2. A comparison of the two compression strategies 230

7.13. Denoising and compression by wavelet packets 233

7.14. Bibliographical comments 234

Chapter 8. Image Processing with Wavelets 235

8.1. Introduction 235

8.2. Wavelets for the image 236

8.2.1. 2D wavelet decomposition 237

8.2.2. Approximation and detail coefficients 238

8.2.3. Approximations and details 241

8.3. Edge detection and textures 243

8.3.1. A simple geometric example 243

8.3.2. Two real life examples 245

8.4. Fusion of images 247

8.4.1. The problem through a simple example 247

8.4.2. Fusion of fuzzy images 250

8.4.3. Mixing of images 252

8.5. Denoising of images 256

8.5.1. An artificially noisy image 257

8.5.2. A real image 260

8.6. Image compression 262

8.6.1. Principles of compression 262

8.6.2. Compression and wavelets 263

8.6.3. “True” compression 269

Chapter 9. An Overview of Applications 279

9.1. Introduction 279

9.1.1. Why does it work? 279

9.1.2. A classification of the applications 281

9.1.3. Two problems in which the wavelets are competitive 283

9.1.4. Presentation of applications 283

9.2. Wind gusts 285

9.3. Detection of seismic jolts 287

9.4. Bathymetric study of the marine floor 290

9.5. Turbulence analysis 291

9.6. Electrocardiogram (ECG): coding and moment of the maximum 294

9.7. Eating behavior 295

9.8. Fractional wavelets and fMRI 297

9.9. Wavelets and biomedical sciences 298

9.9.1. Analysis of 1D biomedical signals 300

9.9.2. 2D biomedical signal analysis 301

9.10. Statistical process control 302

9.11. Online compression of industrial information 304

9.12. Transitories in underwater signals 306

9.13. Some applications at random 308

9.13.1. Video coding 308

9.13.2. Computer-assisted tomography 309

9.13.3. Producing and analyzing irregular signals or images 309

9.13.4. Forecasting 310

9.13.5. Interpolation by kriging 310

Appendix. The EZW Algorithm 313

A.1. Coding 313

A.1.1. Detailed description of the EZW algorithm (coding phase) 313

A.1.2. Example of application of the EZW algorithm (coding phase) 314

A.2. Decoding 317

A.2.1. Detailed description of the EZW algorithm (decoding phase) 317

A.2.2. Example of application of the EZW algorithm (decoding phase) 318

A.3. Visualization on a real image of the algorithm’s decoding phase 318

Bibliography 321

Index 329