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<p>Notations xiii</p> <p>Introduction xvii</p> <p><b>Chapter 1. A Guided Tour 1</b></p> <p>1.1. Introduction 1</p> <p>1.2. Wavelets 2</p> <p>1.2.1. General aspects 2</p> <p>1.2.2. A wavelet 6</p> <p>1.2.3. Organization of wavelets 8</p> <p>1.2.4. The wavelet tree for a signal 10</p> <p>1.3. An electrical consumption signal analyzed by wavelets 12</p> <p>1.4. Denoising by wavelets: before and afterwards 14</p> <p>1.5. A Doppler signal analyzed by wavelets 16</p> <p>1.6. A Doppler signal denoised by wavelets 17</p> <p>1.7. An electrical signal denoised by wavelets 19</p> <p>1.8. An image decomposed by wavelets 21</p> <p>1.8.1. Decomposition in tree form 21</p> <p>1.8.2. Decomposition in compact form 22</p> <p>1.9. An image compressed by wavelets 24</p> <p>1.10. A signal compressed by wavelets 25</p> <p>1.11. A fingerprint compressed using wavelet packets 27</p> <p><b>Chapter 2. Mathematical Framework 29</b></p> <p>2.1. Introduction 29</p> <p>2.2. From the Fourier transform to the Gabor transform 30</p> <p>2.2.1. Continuous Fourier transform 30</p> <p>2.2.2. The Gabor transform 35</p> <p>2.3. The continuous transform in wavelets 37</p> <p>2.4. Orthonormal wavelet bases 41</p> <p>2.4.1. From continuous to discrete transform 41</p> <p>2.4.2. Multi-resolution analysis and orthonormal wavelet bases 42</p> <p>2.4.3. The scaling function and the wavelet 46</p> <p>2.5. Wavelet packets 50</p> <p>2.5.1. Construction of wavelet packets 50</p> <p>2.5.2. Atoms of wavelet packets 52</p> <p>2.5.3. Organization of wavelet packets 53</p> <p>2.6. Biorthogonal wavelet bases 55</p> <p>2.6.1. Orthogonality and biorthogonality 55</p> <p>2.6.2. The duality raises several questions 56</p> <p>2.6.3. Properties of biorthogonal wavelets 57</p> <p>2.6.4. Semi-orthogonal wavelets 60</p> <p><b>Chapter 3. From Wavelet Bases to the Fast Algorithm 63</b></p> <p>3.1. Introduction. 63</p> <p>3.2. From orthonormal bases to the Mallat algorithm 64</p> <p>3.3. Four filters 65</p> <p>3.4. Efficient calculation of the coefficients 67</p> <p>3.5. Justification: projections and twin scales 68</p> <p>3.5.1. The decomposition phase 69</p> <p>3.5.2. The reconstruction phase 72</p> <p>3.5.3. Decompositions and reconstructions of a higher order 75</p> <p>3.6. Implementation of the algorithm 75</p> <p>3.6.1. Initialization of the algorithm 76</p> <p>3.6.2. Calculation on finite sequences 77</p> <p>3.6.3. Extra coefficients 77</p> <p>3.7. Complexity of the algorithm 78</p> <p>3.8. From 1D to 2D 79</p> <p>3.9. Translation invariant transform 81</p> <p>3.9.1. e-decimated DWT 83</p> <p>3.9.2. Calculation of the SWT 83</p> <p>3.9.3. Inverse SWT 87</p> <p><b>Chapter 4. Wavelet Families 89</b></p> <p>4.1. Introduction 89</p> <p>4.2. What could we want from a wavelet? 90</p> <p>4.3. Synoptic table of the common families 91</p> <p>4.4. Some well known families 92</p> <p>4.4.1. Orthogonal wavelets with compact support 93</p> <p>4.4.2. Biorthogonal wavelets with compact support: bior 99</p> <p>4.4.3. Orthogonal wavelets with non-compact support 101</p> <p>4.4.4. Real wavelets without filters 104</p> <p>4.4.5. Complex wavelets without filters 106</p> <p>4.5. Cascade algorithm 109</p> <p>4.5.1. The algorithm and its justification 110</p> <p>4.5.2. An application 112</p> <p>4.5.3. Quality of the approximation 113</p> <p><b>Chapter 5. Finding and Designing a Wavelet 115</b></p> <p>5.1. Introduction 115</p> <p>5.2. Construction of wavelets for continuous analysis 116</p> <p>5.2.1. Construction of a new wavelet 116</p> <p>5.2.2. Application to pattern detection 124</p> <p>5.3. Construction of wavelets for discrete analysis 131</p> <p>5.3.1. Filter banks 132</p> <p>5.3.2. Lifting 140</p> <p>5.3.3. Lifting and biorthogonal wavelets 146</p> <p>5.3.4. Construction examples 149</p> <p><b>Chapter 6. A Short 1D Illustrated Handbook 159</b></p> <p>6.1. Introduction 159</p> <p>6.2. Discrete 1D illustrated handbook 160</p> <p>6.2.1. The analyzed signals 160</p> <p>6.2.2. Processing carried out 161</p> <p>6.2.3. Commented examples 162</p> <p>6.3. The contribution of analysis by wavelet packets 178</p> <p>6.3.1. Example 1: linear and quadratic chirp 178</p> <p>6.3.2. Example 2: a sine181</p> <p>6.3.3. Example 3: a composite signal 182</p> <p>6.4. “Continuous” 1D illustrated handbook 183</p> <p>6.4.1. Time resolution 183</p> <p>6.4.2. Regularity analysis 187</p> <p>6.4.3. Analysis of a self-similar signal 193</p> <p><b>Chapter 7. Signal Denoising and Compression 197</b></p> <p>7.1. Introduction 197</p> <p>7.2. Principle of denoising by wavelets 198</p> <p>7.2.1. The model 198</p> <p>7.2.2. Denoising: before and after 198</p> <p>7.2.3. The algorithm 199</p> <p>7.2.4. Why does it work? 200</p> <p>7.3. Wavelets and statistics 200</p> <p>7.3.1. Kernel estimators and estimators by orthogonal projection 201</p> <p>7.3.2. Estimators by wavelets 201</p> <p>7.4. Denoising methods 202</p> <p>7.4.1. A first estimator 203</p> <p>7.4.2. From coefficient selection to thresholding coefficients 204</p> <p>7.4.3. Universal thresholding 206</p> <p>7.4.4. Estimating the noise standard deviation 206</p> <p>7.4.5. Minimax risk 207</p> <p>7.4.6. Further information on thresholding rules 208</p> <p>7.5. Example of denoising with stationary noise 209</p> <p>7.6. Example of denoising with non-stationary noise 212</p> <p>7.6.1. The model with ruptures of variance 213</p> <p>7.6.2. Thresholding adapted to the noise level change-points 214</p> <p>7.7. Example of denoising of a real signal 216</p> <p>7.7.1. Noise unknown but “homogenous” in variance by level 216</p> <p>7.7.2. Noise unknown and “non-homogenous” in variance by level 217</p> <p>7.8. Contribution of the translation invariant transform 218</p> <p>7.9. Density and regression estimation 221</p> <p>7.9.1. Density estimation 221</p> <p>7.9.2. Regression estimation 224</p> <p>7.10. Principle of compression by wavelets 225</p> <p>7.10.1. The problem 225</p> <p>7.10.2. The basic algorithm 225</p> <p>7.10.3. Why does it work? 226</p> <p>7.11. Compression methods 226</p> <p>7.11.1. Thresholding of the coefficients 226</p> <p>7.11.2. Selection of coefficients 228</p> <p>7.12. Examples of compression 229</p> <p>7.12.1. Global thresholding 229</p> <p>7.12.2. A comparison of the two compression strategies 230</p> <p>7.13. Denoising and compression by wavelet packets 233</p> <p>7.14. Bibliographical comments 234</p> <p><b>Chapter 8. Image Processing with Wavelets 235</b></p> <p>8.1. Introduction 235</p> <p>8.2. Wavelets for the image 236</p> <p>8.2.1. 2D wavelet decomposition 237</p> <p>8.2.2. Approximation and detail coefficients 238</p> <p>8.2.3. Approximations and details 241</p> <p>8.3. Edge detection and textures 243</p> <p>8.3.1. A simple geometric example 243</p> <p>8.3.2. Two real life examples 245</p> <p>8.4. Fusion of images 247</p> <p>8.4.1. The problem through a simple example 247</p> <p>8.4.2. Fusion of fuzzy images 250</p> <p>8.4.3. Mixing of images 252</p> <p>8.5. Denoising of images 256</p> <p>8.5.1. An artificially noisy image 257</p> <p>8.5.2. A real image 260</p> <p>8.6. Image compression 262</p> <p>8.6.1. Principles of compression 262</p> <p>8.6.2. Compression and wavelets 263</p> <p>8.6.3. “True” compression 269</p> <p><b>Chapter 9. An Overview of Applications 279</b></p> <p>9.1. Introduction 279</p> <p>9.1.1. Why does it work? 279</p> <p>9.1.2. A classification of the applications 281</p> <p>9.1.3. Two problems in which the wavelets are competitive 283</p> <p>9.1.4. Presentation of applications 283</p> <p>9.2. Wind gusts 285</p> <p>9.3. Detection of seismic jolts 287</p> <p>9.4. Bathymetric study of the marine floor 290</p> <p>9.5. Turbulence analysis 291</p> <p>9.6. Electrocardiogram (ECG): coding and moment of the maximum 294</p> <p>9.7. Eating behavior 295</p> <p>9.8. Fractional wavelets and fMRI 297</p> <p>9.9. Wavelets and biomedical sciences 298</p> <p>9.9.1. Analysis of 1D biomedical signals 300</p> <p>9.9.2. 2D biomedical signal analysis 301</p> <p>9.10. Statistical process control 302</p> <p>9.11. Online compression of industrial information 304</p> <p>9.12. Transitories in underwater signals 306</p> <p>9.13. Some applications at random 308</p> <p>9.13.1. Video coding 308</p> <p>9.13.2. Computer-assisted tomography 309</p> <p>9.13.3. Producing and analyzing irregular signals or images 309</p> <p>9.13.4. Forecasting 310</p> <p>9.13.5. Interpolation by kriging 310</p> <p><b>Appendix. The EZW Algorithm 313</b></p> <p>A.1. Coding 313</p> <p>A.1.1. Detailed description of the EZW algorithm (coding phase) 313</p> <p>A.1.2. Example of application of the EZW algorithm (coding phase) 314</p> <p>A.2. Decoding 317</p> <p>A.2.1. Detailed description of the EZW algorithm (decoding phase) 317</p> <p>A.2.2. Example of application of the EZW algorithm (decoding phase) 318</p> <p>A.3. Visualization on a real image of the algorithm’s decoding phase 318</p> <p>Bibliography 321</p> <p>Index 329</p>

<b>Georges Oppenheim, Michel Misiti and Jean-Michel Poggi</b>, 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. <p><b>Yves Misiti</b> is a research engineer specializing in computer sciences at Paris 11 University, France.</p>

The last 15 years have seen an explosion of interest in wavelets with applications in fields such as image compression, turbulence, human vision, radar and earthquake prediction.<br /> Wavelets represent an area that combines signal in image processing, mathematics, physics and electrical engineering.<br /> As such, this title is intended for the wide audience that is interested in mastering the basic techniques in this subject area, such as decomposition and compression.

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