Introduction to wavelets and wavelet transforms: a primer

Title page......Page 1
Contents......Page 4
Preface......Page 10
1 Introduction to Wavelets......Page 14
What is a Wavelet System?......Page 15
More Specific Characteristics of Wavelet Systems......Page 16
What do Wavelets Look Like?......Page 18
Why is Wavelet Analysis Effective?......Page 19
1.2 The Discrete Wavelet Transform......Page 20
1.3 The Discrete-Time and Continuous Wavelet Transforms......Page 21
1.5 This Chapter......Page 22
2.1 Signal Spaces......Page 23
2.2 The Scaling Function......Page 24
Multiresolution Analysis......Page 25
2.3 The Wavelet Functions......Page 27
2.4 The Discrete Wavelet Transform......Page 30
2.6 Display of the Discrete Wavelet Transform and the Wavelet Expansion......Page 31
2.7 Examples of Wavelet Expansions......Page 33
2.8 An Example of the Haar Wavelet System......Page 36
3.1 Analysis From Fine Scale to Coarse Scale......Page 44
Filtering and Down-Sampling or Decimating......Page 45
Filtering and Up-Sampling or Stretching......Page 49
3.3 Input Coefficients......Page 50
Periodic versus Nonperiodic Discrete Wavelet Transforms......Page 51
The Discrete Wavelet Transform versus the Discrete-Time Wavelet Transform......Page 52
Numerical Complexity of the Discrete Wavelet Transform......Page 53
4.1 Bases, Orthogonal Bases, and Biorthogonal Bases......Page 54
Matrix Examples......Page 56
Sinc Expansion Example......Page 57
4.2 Frames and Tight Frames......Page 58
Matrix Examples......Page 59
Sinc Expansion as a Tight Frame Example......Page 60
4.3 Conditional and Unconditional Bases......Page 61
Signal Classes......Page 63
Fourier Transforms......Page 64
Refinement and Transition Matrices......Page 65
5.2 Necessary Conditions......Page 66
5.3 Frequency Domain Necessary Conditions......Page 67
5.4 Sufficient Conditions......Page 69
Wavelet System Design......Page 70
5.5 The Wavelet......Page 71
5.7 Example Scaling Functions and Wavelets......Page 72
Sinc Wavelets......Page 73
5.8 Further Properties of the Scaling Function and Wavelet......Page 75
General Properties not Requiring Orthogonality......Page 76
Properties that Depend on Orthogonality......Page 77
Length-2 Scaling Coefficient Vector......Page 78
Length-6 Scaling Coefficient Vector......Page 79
Successive Approximations or the Cascade Algorithm......Page 80
Successive approximations in the frequency domain......Page 81
The Dyadic Expansion of the Scaling Function......Page 83
6.1 K-Regular Scaling Filters......Page 86
6.2 Vanishing Wavelet Moments......Page 88
6.3 Daubechies\' Method for Zero Wavelet Moment Design......Page 89
6.5 Relation of Zero Wavelet Moments to Smoothness......Page 96
6.7 Approximation of Signals by Scaling Function Projection......Page 99
6.8 Approximation of Scaling Coefficients by Samples of the Signal......Page 100
6.9 Coiflets and Related Wavelet Systems......Page 101
Generalized Coifman Wavelet Systems......Page 106
6.10 Minimization of Moments Rather than Zero Moments......Page 110
7.1 Tiling the Time-Frequency or Time Scale Plane......Page 111
Nonstationary Signal Analysis......Page 112
Tiling with the Discrete Two-Band Wavelet Transform......Page 113
General Tiling......Page 114
7.2 Multiplicity-M (M-Band) Scaling Functions and Wavelets......Page 115
Properties of M-Band Wavelet Systems......Page 116
M-Band Scaling Function Design......Page 122
Full Wavelet Packet Decomposition......Page 123
Adaptive Wavelet Packet Systems......Page 124
Two-Channel Biorthogonal Filter Banks......Page 127
Biorthogonal Wavelets......Page 129
Comparisons of Orthogonal and Biorthogonal Wavelets......Page 130
Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets with Less Dissimilar Filter Length......Page 131
Lifting Construction of Biorthogonal Systems......Page 132
7.5 Multiwavelets......Page 135
Construction of Two-Band Multiwavelets......Page 136
Support......Page 137
Implementation of Multiwavelet Transform......Page 138
Geronimo-Hardin-Massopust Multiwavelets......Page 139
Other Constructions......Page 140
7.6 Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases......Page 141
A Matrix Example......Page 142
Shift-Invariant Redundant Wavelet Transforms and Nondecimated Filter Banks......Page 145
Adaptive Construction of Frames and Bases......Page 146
7.7 Local Trigonometric Bases......Page 147
Construction of Smooth Windows......Page 149
Folding and Unfolding......Page 150
Local Cosine and Sine Bases......Page 152
7.8 Discrete Multiresolution Analysis, the Discrete-Time Wavelet Transform, and the Continuous Wavelet Transform......Page 154
Discrete Multiresolution Analysis and the Discrete-Time Wavelet Transform......Page 156
Continuous Wavelet Transforms......Page 157
Analogies between Fourier Systems and Wavelet Systems......Page 158
The Filter Bank......Page 161
Direct Characterization of PR......Page 163
Matrix characterization of PR......Page 165
Polyphase (Transform-Domain) Characterization of PR......Page 166
8.2 Unitary Filter Banks......Page 168
8.3 Unitary Filter Banks-Some Illustrative Examples......Page 173
8.4 M-band Wavelet Tight Frames......Page 175
8.5 Modulated Filter Banks......Page 177
Unitary Modulated Filter Bank......Page 180
8.6 Modulated Wavelet Tight Frames......Page 181
8.7 Linear Phase Filter Banks......Page 182
Characterization of Unitary Hp(z) - PS Symmetry......Page 186
Characterization of Unitary Hp(z) - Linear-Phase Symmetry......Page 187
Characterization of Unitary Hp(z) - Linear Phase and PS Symmetry......Page 188
8.8 Linear-Phase Wavelet Tight Frames......Page 189
8.9 Linear-Phase Modulated Filter Banks......Page 190
8.10 Linear Phase Modulated Wavelet Tight Frames......Page 191
8.11 Time-Varying Filter Bank Trees......Page 192
Pruning a Filter Bank Tree......Page 195
Wavelet Bases for L2([0,∞ty))......Page 196
Wavelet Bases for £2((-∞ty,0)......Page 197
Segmented Time-Varying Wavelet Packet Bases......Page 198
8.12 Filter Banks and Wavelets-Summary......Page 199
9.1 Finite Wavelet Expansions and Transforms......Page 201
9.2 Periodic or Cyclic Discrete Wavelet Transform......Page 203
9,3 Filter Bank Structures for Calculation of the DWT and Complexity......Page 204
9.4 The Periodic Case......Page 205
9.5 Structure of the Periodic Discrete Wavelet Transform......Page 207
9.6 More General Structures......Page 208
10.1 Wavelet-Based Signal Processing......Page 209
Introduction......Page 210
Review of the Discrete Fourier Transform and FFT......Page 211
Review of the Discrete Wavelet Transform......Page 213
The Algorithm Development......Page 214
Computational Complexity......Page 216
Summary......Page 217
10.3 Nonlinear Filtering or Denoising with the DWT......Page 218
Denoising by Thresholding......Page 219
Shift-Invariant or Nondecimated Discrete Wavelet Transform......Page 220
Performance Analysis......Page 222
Examples of Denoising......Page 223
l0.4 Statistical Estimation......Page 224
Fundamentals of Data Compression......Page 225
Prototype Transform Coder......Page 226
Improved Wavelet Based Compression Algorithms......Page 228
10.6 Why are Wavelets so Useful?......Page 229
Seismic and Geophysical Signal Processing......Page 230
10.8 Wavelet Software......Page 231
11.1 Properties of the Basic Multiresolution Scaling Function......Page 232
11.2 Types of Wavelet Systems......Page 234
12 References......Page 236
Bibliography......Page 237
Appendix A. Derivations for Chapter 5 on Scaling Functions......Page 259
Appendix B. Derivations for Section on Properties......Page 266
Appendix C. Matlab Programs......Page 271
Index......Page 279