From Fourier Analysis to Wavelets (eBook)

Preface 8
Contents 10
1 Introduction 15
1.1 Computational Mathematics 15
1.1.1 Abstraction Levels 16
1.2 Relation Between the Abstraction Levels 18
1.3 Functions and Computational Mathematics 20
1.3.1 Representation and Reconstruction of Functions 20
1.3.2 Specification of Functions 21
1.4 What is the Relation with Graphics? 21
1.4.1 Description of Graphical Objects 22
1.5 Where do Wavelets Fit? 23
1.5.1 Function Representation Using Wavelets 23
1.5.2 Multiresolution Representation 24
1.6 About these Book 24
1.7 Comments and References 25
2 Function Representation and Reconstruction 26
2.1 Representing Functions 26
2.1.1 The Representation Operator 27
2.2 Basis Representation 28
2.2.1 Complete Orthonormal Representation 28
2.3 Representation by Frames 29
2.4 Riesz Basis Representation 31
2.5 Representation by Projection 32
2.6 Galerkin Representation 33
2.7 Reconstruction, Point Sampling and Interpolation 34
2.7.1 Piecewise Constant Reconstruction 34
2.7.2 Piecewise Linear Reconstruction 35
Higher Order Reconstruction 35
2.8 Multiresolution Representation 36
2.9 Representation by Dictionaries 38
2.10 Redundancy in the Representation 39
2.11 Wavelets and Function Representation 39
2.12 Comments and References 40
3 The Fourier Transform 41
3.1 Analyzing Functions 41
3.1.1 Fourier Series 41
3.1.2 Fourier Transform 43
3.1.3 Spatial and Frequency Domain 45
3.2 A Pause to Think 46
3.3 Frequency Analysis 46
3.4 Fourier Transform and Filtering 49
3.4.1 Low-pass Filters 51
3.4.2 High-pass Filter 51
3.4.3 Band-pass Filter 51
3.4.4 Band-stop Filter 52
3.5 Fourier Transform and Function Representation 52
3.5.1 Fourier Transform and Point Sampling 53
3.5.2 The Theorem of Shannon-Whittaker 54
3.6 Point Sampling and Representation by Projection 55
3.7 Point Sampling and Representation Coefficients 56
3.8 Comments and References 57
4 Windowed Fourier Transform 59
4.1 A Walk in The Physical Universe 59
4.2 The Windowed Fourier Transform 60
4.2.1 Invertibility of (t,?) 61
4.2.2 Image of the Windowed Fourier Transform 61
4.2.3 WFT and Function Representation 62
4.3 Time-frequency Domain 62
4.3.1 The Uncertainty Principle 63
4.4 Atomic Decomposition 64
4.5 WFT and Atomic Decomposition 66
4.6 Comments and References 72
5 The Wavelet Transform 73
5.1 The Wavelet Transform 73
5.1.1 Inverse of the Wavelet Transform 75
5.1.2 Image of the Wavelet Transform 76
5.2 Filtering and the Wavelet Transform 76
5.3 The Discrete Wavelet Transform 80
5.3.1 Function Representation 83
5.4 Comments and References 85
6 Multiresolution Representation 86
6.1 The Concept of Scale 86
6.2 Scale Spaces 87
6.2.1 A Remark About Notation 89
6.2.2 Multiresolution Representation 90
6.3 A Pause to Think 91
6.4 Multiresolution Representation and Wavelets 93
6.5 A Pause… to See the Wavescape 96
6.6 Two-Scale Relation 98
6.7 Comments and References 99
7 The Fast Wavelet Transform 100
7.1 Multiresolution Representation and Recursion 100
7.2 Two-Scale Relations and Inner Products 103
7.3 Wavelet Decomposition and Reconstruction 104
7.3.1 Decomposition 104
7.3.2 Reconstruction 105
7.4 The Fast Wavelet Transform Algorithm 106
7.4.1 Forward Transform 107
7.4.2 Inverse Transform 108
7.4.3 Complexity Analysis of the Algorithm 108
7.5 Boundary Conditions 109
7.6 Comments and References 111
8 Filter Banks and Multiresolution 112
8.1 Two-Channel Filter Banks 112
8.1.1 Matrix Representation 114
8.2 Filter Banks and Multiresolution Representation 115
8.3 Discrete Multiresolution Analysis 116
8.3.1 Pause to Review 118
8.4 Reconstruction Bank 121
8.5 Computational Complexity 122
8.6 Comments and References 123
9 Constructing Wavelets 124
9.1 Wavelets in the Frequency Domain 124
9.1.1 The Relations of with m0 124
9.1.2 The Relations of with m1 125
9.1.3 Characterization of m0 126
9.1.4 Characterization of m1 127
9.2 Orthonormalization Method 130
9.3 A Recipe 131
9.4 Piecewise Linear Multiresolution 131
9.5 Shannon Multiresolution Analysis 133
9.6 Where to Go Now? 135
9.7 Comments and References 136
10 Wavelet Design 137
10.1 Synthesizing Wavelets from Filters 137
10.1.1 Conjugate Mirror Filters 138
10.1.2 Conditions for m0 138
10.1.3 Strategy for Computing m0 139
10.1.4 Analysis of P 140
10.1.5 Factorization of P 140
10.1.6 Example (Haar Wavelet) 143
10.2 Properties of Wavelets 145
10.2.1 Orthogonality 145
10.2.2 Support of ? and ? 146
10.2.3 Vanishing Moments and Polynomial Reproduction 146
10.2.4 Regularity 147
10.2.5 Symmetry or Linear Phase 148
10.2.6 Other Properties 149
10.3 Classes of Wavelets 150
10.3.1 Orthogonal Wavelets 150
10.3.2 Biorthogonal Wavelets 150
10.4 Comments and References 151
11 Orthogonal Wavelets 152
11.1 The Polynomial P 152
11.1.1 P as a Product Polynomial 153
11.1.2 P and the Halfband Property 154
11.1.3 The Expression of P 155
11.1.4 The Factorization of P 156
11.1.5 Analysis of P 158
11.2 Examples of Orthogonal Wavelets 159
11.2.1 Daubechies Extremal Phase Wavelets 160
11.2.2 Minimal Phase Orthogonal Wavelets 162
11.2.3 Coiflets 163
11.3 Comments and References 164
12 Biorthogonal Wavelets 165
12.1 Biorthogonal Multiresolution Analysis and Filters 165
12.1.1 Biorthogonal Basis Functions 165
12.1.2 Biorthogonality and Filters 168
12.1.3 Fast Biorthogonal Wavelet Transform 169
12.2 Filter Design Framework for Biorthogonal Wavelets 170
12.2.1 Perfect Reconstruction Filter Banks 170
12.2.2 Conjugate Quadrature Filters 172
12.2.3 The Polynomial P and Wavelet Design 173
12.2.4 Factorization of P for Biorthogonal Filters 174
12.3 Symmetric Biorthogonal Wavelets 175
12.3.1 B-Spline Wavelets 176
12.3.2 Wavelets with Closer Support Width 179
12.3.3 Biorthogonal Bases Closer to Orthogonal Bases 180
12.4 Comments and References 183
13 Directions and Guidelines 186
13.1 History and Motivation 186
13.2 A Look Back 187
13.3 Extending the Basic Wavelet Framework 188
13.3.1 Studying Functions on other Domains 188
13.3.2 Defining other Time-Frequency Decompositions 189
13.3.3 Solving Mathematical Problems 189
13.4 Applications of Wavelets 189
13.5 Comments and References 190
A Systems and Filters 191
A.1 Systems and Filters 191
A.1.1 Spatial Invariant Linear Systems 192
Linearity 192
Spatial Invariance 193
Impulse Response 193
A.1.2 Other Characteristics 194
Finite Impulse Response 194
Causality 194
Stability 195
A.2 Discretization of Systems 195
A.2.1 Discrete Signals 195
A.2.2 Discrete Systems 196
A.3 Upsampling and Downsampling Operators 201
A.4 Filter Banks 203
A.5 Comments and References 204
B The Z Transform 205
B.1 The Z Transform 205
B.1.1 Some Properties 207
B.1.2 Transfer Function 207
B.1.3 The Variable z and Frequency 210
B.2 Subsampling Operations 211
B.2.1 Downsampling in the Frequency Domain 211
B.2.2 Upsampling in the Frequency Domain 212
B.2.3 Upsampling after Downsampling 213
B.3 Comments and References 213
References 214