Wavelets and their Applications

Table of Contents

Notations

Introduction

Chapter 1. A Guided Tour

1.1. Introduction

1.2. Wavelets

1.2.1. General aspects

1.2.2. A wavelet

1.2.3. Organization of wavelets

1.2.4. The wavelet tree for a signal

1.3. An electrical consumption signal analyzed by wavelets

1.4. Denoising by wavelets: before and afterwards

1.5. A Doppler signal analyzed by wavelets

1.6. A Doppler signal denoised by wavelets

1.7. An electrical signal denoised by wavelets

1.8. An image decomposed by wavelets

1.8.1. Decomposition in tree form

1.8.2. Decomposition in compact form

1.9. An image compressed by wavelets

1.10. A signal compressed by wavelets

1.11. A fingerprint compressed using wavelet packets

Chapter 2. Mathematical Framework

2.1. Introduction

2.2. From the Fourier transform to the Gabor transform

2.2.1. Continuous Fourier transform

2.2.2. The Gabor transform

2.3. The continuous transform in wavelets

2.4. Orthonormal wavelet bases

2.4.1. From continuous to discrete transform

2.4.2. Multi-resolution analysis and orthonormal wavelet bases

2.4.3. The scaling function and the wavelet

2.5. Wavelet packets

2.5.1. Construction of wavelet packets

2.5.2. Atoms of wavelet packets

2.5.3. Organization of wavelet packets

2.6. Biorthogonal wavelet bases

2.6.1. Orthogonality and biorthogonality

2.6.2. The duality raises several questions

2.6.3. Properties of biorthogonal wavelets

2.6.4. Semi-orthogonal wavelets

Chapter 3. From Wavelet Bases to the Fast Algorithm

3.1. Introduction

3.2. From orthonormal bases ...