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 xiiiIntroduction xviiChapter 1. A Guided Tour 11.1. Introduction 11.2. Wavelets 21.2.1. General aspects 21.2.2. A wavelet 61.2.3. Organization of wavelets 81.2.4. The wavelet tree for a signal 101.3. An electrical consumption signal analyzed by wavelets 121.4. Denoising by wavelets: before and afterwards 141.5. A Doppler signal analyzed by wavelets 161.6. A Doppler signal denoised by wavelets 171.7. An electrical signal denoised by wavelets 191.8. An image decomposed by wavelets 211.8.1. Decomposition in tree form 211.8.2. Decomposition in compact form 221.9. An image compressed by wavelets 241.10. A signal compressed by wavelets 251.11. A fingerprint compressed using wavelet packets 27Chapter 2. Mathematical Framework 292.1. Introduction 292.2. From the Fourier transform to the Gabor transform 302.2.1. Continuous Fourier transform 302.2.2. The Gabor transform 352.3. The continuous transform in wavelets 372.4. Orthonormal wavelet bases 412.4.1. From continuous to discrete transform 412.4.2. Multi-resolution analysis and orthonormal wavelet bases 422.4.3. The scaling function and the wavelet 462.5. Wavelet packets 502.5.1. Construction of wavelet packets 502.5.2. Atoms of wavelet packets 522.5.3. Organization of wavelet packets 532.6. Biorthogonal wavelet bases 552.6.1. Orthogonality and biorthogonality 552.6.2. The duality raises several questions 562.6.3. Properties of biorthogonal wavelets 572.6.4. Semi-orthogonal wavelets 60Chapter 3. From Wavelet Bases to the Fast Algorithm 633.1. Introduction. 633.2. From orthonormal bases to the Mallat algorithm 643.3. Four filters 653.4. Efficient calculation of the coefficients 673.5. Justification: projections and twin scales 683.5.1. The decomposition phase 693.5.2. The reconstruction phase 723.5.3. Decompositions and reconstructions of a higher order 753.6. Implementation of the algorithm 753.6.1. Initialization of the algorithm 763.6.2. Calculation on finite sequences 773.6.3. Extra coefficients 773.7. Complexity of the algorithm 783.8. From 1D to 2D 793.9. Translation invariant transform 813.9.1. e-decimated DWT 833.9.2. Calculation of the SWT 833.9.3. Inverse SWT 87Chapter 4. Wavelet Families 894.1. Introduction 894.2. What could we want from a wavelet? 904.3. Synoptic table of the common families 914.4. Some well known families 924.4.1. Orthogonal wavelets with compact support 934.4.2. Biorthogonal wavelets with compact support: bior 994.4.3. Orthogonal wavelets with non-compact support 1014.4.4. Real wavelets without filters 1044.4.5. Complex wavelets without filters 1064.5. Cascade algorithm 1094.5.1. The algorithm and its justification 1104.5.2. An application 1124.5.3. Quality of the approximation 113Chapter 5. Finding and Designing a Wavelet 1155.1. Introduction 1155.2. Construction of wavelets for continuous analysis 1165.2.1. Construction of a new wavelet 1165.2.2. Application to pattern detection 1245.3. Construction of wavelets for discrete analysis 1315.3.1. Filter banks 1325.3.2. Lifting 1405.3.3. Lifting and biorthogonal wavelets 1465.3.4. Construction examples 149Chapter 6. A Short 1D Illustrated Handbook 1596.1. Introduction 1596.2. Discrete 1D illustrated handbook 1606.2.1. The analyzed signals 1606.2.2. Processing carried out 1616.2.3. Commented examples 1626.3. The contribution of analysis by wavelet packets 1786.3.1. Example 1: linear and quadratic chirp 1786.3.2. Example 2: a sine1816.3.3. Example 3: a composite signal 1826.4. “Continuous” 1D illustrated handbook 1836.4.1. Time resolution 1836.4.2. Regularity analysis 1876.4.3. Analysis of a self-similar signal 193Chapter 7. Signal Denoising and Compression 1977.1. Introduction 1977.2. Principle of denoising by wavelets 1987.2.1. The model 1987.2.2. Denoising: before and after 1987.2.3. The algorithm 1997.2.4. Why does it work? 2007.3. Wavelets and statistics 2007.3.1. Kernel estimators and estimators by orthogonal projection 2017.3.2. Estimators by wavelets 2017.4. Denoising methods 2027.4.1. A first estimator 2037.4.2. From coefficient selection to thresholding coefficients 2047.4.3. Universal thresholding 2067.4.4. Estimating the noise standard deviation 2067.4.5. Minimax risk 2077.4.6. Further information on thresholding rules 2087.5. Example of denoising with stationary noise 2097.6. Example of denoising with non-stationary noise 2127.6.1. The model with ruptures of variance 2137.6.2. Thresholding adapted to the noise level change-points 2147.7. Example of denoising of a real signal 2167.7.1. Noise unknown but “homogenous” in variance by level 2167.7.2. Noise unknown and “non-homogenous” in variance by level 2177.8. Contribution of the translation invariant transform 2187.9. Density and regression estimation 2217.9.1. Density estimation 2217.9.2. Regression estimation 2247.10. Principle of compression by wavelets 2257.10.1. The problem 2257.10.2. The basic algorithm 2257.10.3. Why does it work? 2267.11. Compression methods 2267.11.1. Thresholding of the coefficients 2267.11.2. Selection of coefficients 2287.12. Examples of compression 2297.12.1. Global thresholding 2297.12.2. A comparison of the two compression strategies 2307.13. Denoising and compression by wavelet packets 2337.14. Bibliographical comments 234Chapter 8. Image Processing with Wavelets 2358.1. Introduction 2358.2. Wavelets for the image 2368.2.1. 2D wavelet decomposition 2378.2.2. Approximation and detail coefficients 2388.2.3. Approximations and details 2418.3. Edge detection and textures 2438.3.1. A simple geometric example 2438.3.2. Two real life examples 2458.4. Fusion of images 2478.4.1. The problem through a simple example 2478.4.2. Fusion of fuzzy images 2508.4.3. Mixing of images 2528.5. Denoising of images 2568.5.1. An artificially noisy image 2578.5.2. A real image 2608.6. Image compression 2628.6.1. Principles of compression 2628.6.2. Compression and wavelets 2638.6.3. “True” compression 269Chapter 9. An Overview of Applications 2799.1. Introduction 2799.1.1. Why does it work? 2799.1.2. A classification of the applications 2819.1.3. Two problems in which the wavelets are competitive 2839.1.4. Presentation of applications 2839.2. Wind gusts 2859.3. Detection of seismic jolts 2879.4. Bathymetric study of the marine floor 2909.5. Turbulence analysis 2919.6. Electrocardiogram (ECG): coding and moment of the maximum 2949.7. Eating behavior 2959.8. Fractional wavelets and fMRI 2979.9. Wavelets and biomedical sciences 2989.9.1. Analysis of 1D biomedical signals 3009.9.2. 2D biomedical signal analysis 3019.10. Statistical process control 3029.11. Online compression of industrial information 3049.12. Transitories in underwater signals 3069.13. Some applications at random 3089.13.1. Video coding 3089.13.2. Computer-assisted tomography 3099.13.3. Producing and analyzing irregular signals or images 3099.13.4. Forecasting 3109.13.5. Interpolation by kriging 310Appendix. The EZW Algorithm 313A.1. Coding 313A.1.1. Detailed description of the EZW algorithm (coding phase) 313A.1.2. Example of application of the EZW algorithm (coding phase) 314A.2. Decoding 317A.2.1. Detailed description of the EZW algorithm (decoding phase) 317A.2.2. Example of application of the EZW algorithm (decoding phase) 318A.3. Visualization on a real image of the algorithm’s decoding phase 318Bibliography 321Index 329