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Wavelets: A Concise Guide Book

Wavelets: A Concise Guide
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Wavelets: A Concise Guide, Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction p, Wavelets: A Concise Guide
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  • Wavelets: A Concise Guide
  • Written by author Amir-Homayoon Najmi
  • Published by Johns Hopkins University Press, 2/24/2012
  • Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction p
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Authors

List of Figures xiii

List of Acronyms xix

Preface xxi

Acknowledgments xxix

1 Analysis in Vector and Function Spaces 1

1.1 Introduction 1

1.2 The Lebesgue Integral 3

1.3 Discrete Time Signals 5

1.4 Vector Spaces 5

1.5 Linear Independence 6

1.6 Bases and Basis Vectors 7

1.7 Normed Vector Spaces 8

1.8 Inner Product 9

1.9 Banach and Hilbert Spaces 12

1.10 Linear Operators, Operator Norm, the Adjoint Operator 13

1.11 Reproducing Kernel Hilbert Space 15

1.12 The Dirac Delta Distribution 18

1.13 Orthonormal Vectors 20

1.14 Orthogonal Projections 21

1.15 Multi-Resolution Analysis Subspaces 22

1.16 Complete and Orthonormal Bases in L2 (R) 25

1.17 The Dirac Notation 28

1.18 The Fourier Transform 31

1.19 The Fourier Series Expansion 34

1.20 The Discrete Time Fourier Transform 36

1.21 The Discrete Fourier Transform 37

1.22 Band-Limited Functions and the Sampling Theorem 38

1.23 The Basis Operator in L2(R) 41

1.24 Biorthogonal Bases and Representations in L2 (R) 43

1.25 Frames in a Finite Dimensional Vector Space 45

1.26 Frames in L2 (R) 50

1.27 Dual Frame Construction Algorithm 54

1.28 Exercises 56

2 Linear Time-Invariant Systems 59

2.1 Introduction 59

2.2 Convolution in Continuous Time 59

2.3 Convolution in Discrete Time 60

2.4 Convolution of Finite Length Sequences 61

2.5 Linear Time-Invariant Systems and the Z Transform 63

2.6 Spectral Factorization for Finite Length Sequences 66

2.7 Perfect Reconstruction Quadrature Mirror Filters 68

2.8 Exercises 73

3 Time, Frequency, and Scale Localizing Transforms 75

3.1 Introduction 75

3.2 The Windowed Fourier Transform 79

3.3 The Windowed Fourier Transform Inverse 81

3.4 The Range Space of the Windowed Fourier Transform 81

3.5 The Discretized Windowed Fourier Transform 83

3.6 Time-Frequency Resolution of the Windowed Fourier Transform 88

3.7 The Continuous Wavelet Transform 90

3.8 The Continuous Wavelet Transform Inverse 93

3.9 The Range Space of the Continuous Wavelet Transform 95

3.10 The Morlet, the Mexican Hat, and the Haar Wavelets 96

3.11 Discretizing the Continuous Wavelet Transform 101

3.12 Algorithm A' Trous 104

3.13 The Morlet Scalogram 107

3.14 Exercises 110

4 The Haar and Shannon Wavelets 111

4.1 Introduction 111

4.2 Haar Multi-Resolution Analysis Subspaces 112

4.3 Summary and Generalization of Results 119

4.4 The Spectra of the Haar Filter Coefficients 122

4.5 Half-Band Finite Impulse Response Filters 124

4.6 The Shannon Scaling Function 125

4.7 The Spectrum of the Shannon Filter Coefficients 130

4.8 Meyer's Wavelet 131

4.9 Exercises 133

5 General Properties of Scaling and Wavelet Functions 135

5.1 Introduction 135

5.2 Multi-Resolution Analysis Spaces 135

5.3 The Inverse Relations 140

5.4 The Shift-Invariant Discrete Wavelet Transform 143

5.5 Time Domain Properties 145

5.6 Examples of Finite Length Filter Coefficients 149

5.7 Frequency Domain Relations 150

5.8 Orthogonalization of a Basis Set: b1 Spline Wavelet 157

5.9 The Cascade Algorithm 159

5.10 Biorthogonal Wavelets 163

5.11 Multi-Resolution Analysis Using Biorthogonal Wavelets 167

5.12 Exercises 170

6 Discrete Wavelet Transform of Discrete Time Signals 173

6.1 Introduction 173

6.2 Discrete Time Data and Scaling Function Expansions 174

6.3 Implementing the DWT for Even Length h0 Filters 179

6.4 Denoising and Thresholding 185

6.5 Biorthogonal Wavelets of Compact Support 187

6.6 The Lazy Filters 191

6.7 Exercises 191

7 Wavelet Regularity and Daubechies Solutions 193

7.1 Introduction 193

7.2 Zero Moments of the Mother Wavelet 194

7.3 The Form of H0(z) and the Decay Rate of ¦(É) 199

7.4 Daubechies Orthogonal Wavelets of Compact Support 200

7.5 Wavelet and Scaling Function Vanishing Moments 204

7.6 Biorthogonal Wavelets of Compact Support 207

7.7 Biorthogonal Spline Wavelets 211

7.8 The Lifting Scheme 215

7.9 Exercises 217

8 Orthogonal Wavelet Packets 221

8.1 Introduction 221

8.2 Review of the Orthogonal Wavelet Transform 221

8.3 Packet Functions for Orthonormal Wavelets 224

8.4 Discrete Orthogonal Packet Transform of Finite Length Se-quences 231

8.5 The Best Basis Algorithm 236

8.6 Exercises 239

9 Wavelet Transform in Two Dimensions 241

9.1 Introduction 241

9.2 The Forward Transform 242

9.3 The Inverse Transform 247

9.4 Implementing the Two-Dimensional Wavelet Transform 248

9.5 Application to Image Compression 249

9.6 Image Fusion 257

9.7 Wavelet Descendants 258

9.8 Exercises 259

Bibliography 261

Index 267


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Wavelets: A Concise Guide, Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction p, Wavelets: A Concise Guide

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Wavelets: A Concise Guide, Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction p, Wavelets: A Concise Guide

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Wavelets: A Concise Guide, Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction p, Wavelets: A Concise Guide

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