book

Wavelets and their Applications

Name: Wavelets and their Applications
ISBN: 9781905209316

by Michel Misiti, Yves Misiti, Georges Oppenheim, Jean-Michel Poggi

May 2007

Intermediate to advanced

330 pages

7h 23m

English

Wiley

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Coverpage
Titlepage
Copyright
Table of Contents
Notations
Introduction
Chapter 1. A Guided Tour
1.1. Introduction1.2. Wavelets1.2.1. General aspects1.2.2. A wavelet1.2.3. Organization of wavelets1.2.4. The wavelet tree for a signal1.3. An electrical consumption signal analyzed by wavelets1.4. Denoising by wavelets: before and afterwards1.5. A Doppler signal analyzed by wavelets1.6. A Doppler signal denoised by wavelets1.7. An electrical signal denoised by wavelets1.8. An image decomposed by wavelets1.8.1. Decomposition in tree form1.8.2. Decomposition in compact form1.9. An image compressed by wavelets1.10. A signal compressed by wavelets1.11. A fingerprint compressed using wavelet packets
Chapter 2. Mathematical Framework
2.1. Introduction2.2. From the Fourier transform to the Gabor transform2.2.1. Continuous Fourier transform2.2.2. The Gabor transform2.3. The continuous transform in wavelets2.4. Orthonormal wavelet bases2.4.1. From continuous to discrete transform2.4.2. Multi-resolution analysis and orthonormal wavelet bases2.4.3. The scaling function and the wavelet2.5. Wavelet packets2.5.1. Construction of wavelet packets2.5.2. Atoms of wavelet packets2.5.3. Organization of wavelet packets2.6. Biorthogonal wavelet bases2.6.1. Orthogonality and biorthogonality2.6.2. The duality raises several questions2.6.3. Properties of biorthogonal wavelets2.6.4. Semi-orthogonal wavelets
Chapter 3. From Wavelet Bases to the Fast Algorithm
3.1. Introduction3.2. From orthonormal bases to the Mallat algorithm3.3. Four filters3.4. Efficient calculation of the coefficients3.5. Justification: projections and twin scales3.5.1. The decomposition phase3.5.2. The reconstruction phase3.5.3. Decompositions and reconstructions of a higher order3.6. Implementation of the algorithm3.6.1. Initialization of the algorithm3.6.2. Calculation on finite sequences3.6.3. Extra coefficients3.7. Complexity of the algorithm3.8. From 1D to 2D3.9. Translation invariant transform3.9.1. ε -decimated DWT3.9.2. Calculation of the SWT3.9.3. Inverse SWT
Chapter 4. Wavelet Families
4.1. Introduction4.2. What could we want from a wavelet?4.3. Synoptic table of the common families4.4. Some well known families4.4.1. Orthogonal wavelets with compact support4.4.1.1. Daubechies wavelets: dbN4.4.1.2. Symlets: symN4.4.1.3. Coiflets: coifN4.4.2. Biorthogonal wavelets with compact support: bior4.4.3. Orthogonal wavelets with non-compact support4.4.3.1. The Meyer wavelet: meyr4.4.3.2. An approximation of the Meyer wavelet: dmey4.4.3.3. Battle and Lemarié wavelets: btlm4.4.4. Real wavelets without filters4.4.4.1. The Mexican hat: mexh4.4.4.2. The Morlet wavelet: morl4.4.4.3. Gaussian wavelets: gausN4.4.5. Complex wavelets without filters4.4.5.1. Complex Gaussian wavelets: cgau4.4.5.2. Complex Morlet wavelets: cmorl4.4.5.3. Complex frequency B-spline wavelets: fbsp4.5. Cascade algorithm4.5.1. The algorithm and its justification4.5.2. An application4.5.3. Quality of the approximation

Chapter 5. Finding and Designing a Wavelet
5.1. Introduction5.2. Construction of wavelets for continuous analysis5.2.1. Construction of a new wavelet5.2.1.1. The admissibility condition5.2.1.2. Simple examples of admissible wavelets5.2.1.3. Construction of wavelets approaching a pattern5.2.2. Application to pattern detection5.3. Construction of wavelets for discrete analysis5.3.1. Filter banks5.3.1.1. From the Mallat algorithm to filter banks5.3.1.2. The perfect reconstruction condition5.3.1.3. Construction of perfect reconstruction filter banks5.3.1.4. Examples of perfect reconstruction filter banks5.3.2. Lifting5.3.2.1. The lifting method5.3.2.2. Lifting and the polyphase method5.3.3. Lifting and biorthogonal wavelets5.3.4. Construction examples5.3.4.1. Illustrations of lifting5.3.4.2. Construction of wavelets with more vanishing moments5.3.4.3. Approximation of a form by lifting
Chapter 6. A Short 1D Illustrated Handbook
6.1. Introduction6.2. Discrete 1D illustrated handbook6.2.1. The analyzed signals6.2.2. Processing carried out6.2.3. Commented examples6.2.3.1. A sum of sines6.2.3.2. A frequency breakdown6.2.3.3. White noise6.2.3.4. Colored noise6.2.3.5. A breakdown6.2.3.6. Two breakdowns of the derivative6.2.3.7. A breakdown of the second derivative6.2.3.8. A superposition of signals6.2.3.9. A ramp with colored noise6.2.3.10. A first real signal6.2.3.11. A second real signal6.3. The contribution of analysis by wavelet packets6.3.1. Example 1: linear and quadratic chirp6.3.2. Example 2: a sine6.3.3. Example 3: a composite signal6.4. “Continuous” 1D illustrated handbook6.4.1. Time resolution6.4.1.1. Locating a discontinuity in the signal6.4.1.2. Locating a discontinuity in the derivative of the signal6.4.2. Regularity analysis6.4.2.1. Locating a Hölderian singularity6.4.2.2. Analysis of the Hölderian regularity of a singularity6.4.2.3. Study of two Hölderian singularities6.4.3. Analysis of a self-similar signal
Chapter 7. Signal Denoising and Compression
7.1. Introduction7.2. Principle of denoising by wavelets7.2.1. The model7.2.2. Denoising: before and after7.2.3. The algorithm7.2.4. Why does it work?7.3. Wavelets and statistics7.3.1. Kernel estimators and estimators by orthogonal projection7.3.2. Estimators by wavelets7.4. Denoising methods7.4.1. A first estimator7.4.2. From coefficient selection to thresholding coefficients7.4.3. Universal thresholding7.4.4. Estimating the noise standard deviation7.4.5. Minimax risk7.4.6. Further information on thresholding rules7.5. Example of denoising with stationary noise7.6. Example of denoising with non-stationary noise7.6.1. The model with ruptures of variance7.6.2. Thresholding adapted to the noise level change-points7.7. Example of denoising of a real signal7.7.1. Noise unknown but “homogenous” in variance by level7.7.2. Noise unknown and “non-homogenous” in variance by level7.8. Contribution of the translation invariant transform7.9. Density and regression estimation7.9.1. Density estimation7.9.2. Regression estimation7.10. Principle of compression by wavelets7.10.1. The problem7.10.2. The basic algorithm7.10.3. Why does it work?7.11. Compression methods7.11.1. Thresholding of the coefficients7.11.2. Selection of coefficients7.12. Examples of compression7.12.1. Global thresholding7.12.2. A comparison of the two compression strategies7.13. Denoising and compression by wavelet packets7.14. Bibliographical comments
Chapter 8. Image Processing with Wavelets
8.1. Introduction8.2. Wavelets for the image8.2.1. 2D wavelet decomposition8.2.2. Approximation and detail coefficients8.2.2.1. Horizontal, vertical and diagonal details8.2.2.2. Two representations of decomposition8.2.3. Approximations and details8.3. Edge detection and textures8.3.1. A simple geometric example8.3.2. Two real life examples8.4. Fusion of images8.4.1. The problem through a simple example8.4.2. Fusion of fuzzy images8.4.3. Mixing of images8.5. Denoising of images8.5.1. An artificially noisy image8.5.2. A real image8.6. Image compression8.6.1. Principles of compression8.6.2. Compression and wavelets8.6.2.1. Why does it work?8.6.2.2. Why threshold?8.6.2.3. Examples of image compression8.6.3. “True” compression8.6.3.1. The quantization8.6.3.2. EZW coding8.6.3.3. Comments on the JPEG 2000 standard
Chapter 9. An Overview of Applications
9.1. Introduction9.1.1. Why does it work?9.1.2. A classification of the applications9.1.3. Two problems in which the wavelets are competitive9.1.4. Presentation of applications9.2. Wind gusts9.3. Detection of seismic jolts9.4. Bathymetric study of the marine floor9.5. Turbulence analysis9.6. Electrocardiogram (ECG): coding and moment of the maximum9.7. Eating behavior9.8. Fractional wavelets and fMRI9.9. Wavelets and biomedical sciences9.9.1. Analysis of 1D biomedical signals9.9.1.1. Bioacoustic signals9.9.1.2. Electrocardiogram (ECG)9.9.1.3. Electroencephalogram (EEG)9.9.2. 2D biomedical signal analysis9.9.2.1. Nuclear magnetic resonance (NMR)9.9.2.2. fMRI and functional imagery9.10. Statistical process control9.11. Online compression of industrial information9.12. Transitories in underwater signals9.13. Some applications at random9.13.1. Video coding9.13.2. Computer-assisted tomography9.13.3. Producing and analyzing irregular signals or images9.13.4. Forecasting9.13.5. Interpolation by kriging
Appendix. The EZW Algorithm
A.1. CodingA.1.1. Detailed description of the EZW algorithm (coding phase)A.1.2. Example of application of the EZW algorithm (coding phase)A.2. DecodingA.2.1. Detailed description of the EZW algorithm (decoding phase)A.2.2. Example of application of the EZW algorithm (decoding phase)A.3. Visualization on a real image of the algorithm’s decoding phase
Bibliography
Index

Content preview from Wavelets and their Applications

Appendix

The EZW Algorithm

This appendix supplements section 8.6.3.2 dedicated to the EZW algorithm used in wavelet image compression. It details the operation of the algorithm and presents an example of application. We successively tackle coding and decoding (we note by ⌊ x ⌋ the integer part of x).

For other insights on the EZW algorithm, see [CRE 97], [USE 01] and [VAL 99]. In the book by Strang and Nguyen (see [STR 96], p. 362-383), we will find a detailed presentation of the various operations related to compression: quantization, coding, etc.

A.1. Coding

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

(1) Initialization. All the coefficients are placed on the principal list and the threshold is initialized by T₀ = 2^{⌊log2(C_max)⌋}, where C_max is the maximum of the absolute value of the coefficients.

(2) Principal stage. Go through the coefficients on the principal list in an appropriate order (see Figure 8.45) and compare each of them with the current threshold T_n. Attribute one of the four following symbols to each coefficient:

– P, if it is positive and has an absolute value higher than the threshold;

– N, if it is negative and has an absolute value higher than the threshold;

– Z, if its absolute value is lower than the threshold but if one of its children has an absolute value higher than the threshold;

– T if its absolute value is lower than the threshold and if all its children have absolute values lower than the threshold.

Thanks to the zerotree property ...

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