Introduction to wavelets and wavelet transforms: a primer
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Upper Saddle River, NJ
Prentice-Hall
1998
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 268 S. Ill., graph. Darst. |
| ISBN: | 0134896009 |
Staff View
MARC
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| 100 | 1 | |a Burrus, C. S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to wavelets and wavelet transforms |b a primer |c C. Sidney Burrus ; Ramesh A. Gopinath and Haitao Guo |
| 264 | 1 | |a Upper Saddle River, NJ |b Prentice-Hall |c 1998 | |
| 300 | |a XIV, 268 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 700 | 1 | |a Gopinath, Ramesh A. |e Verfasser |4 aut | |
| 700 | 1 | |a Guo, Haitao |e Verfasser |4 aut | |
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Record in the Search Index
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| adam_text | Contents
Preface xi
1 Introduction to Wavelets 1
1.1 Wavelets and Wavelet Expansion Systems 2
What is a Wavelet Expansion or a Wavelet Transform? 2
What is a Wavelet System? 2
More Specific Characteristics of Wavelet Systems 3
Haar Scaling Functions and Wavelets 5
What do Wavelets Look Like? 5
Why is Wavelet Analysis Effective? 6
1.2 The Discrete Wavelet Transform 7
1.3 The Discrete Time and Continuous Wavelet Transforms 8
1.4 Exercises and Experiments 9
1.5 This Chapter 9
2 A Multiresolution Formulation of Wavelet Systems 10
2.1 Signal Spaces 10
2.2 The Scaling Function 11
Multiresolution Analysis 12
2.3 The Wavelet Functions 14
2.4 The Discrete Wavelet Transform 17
2.5 A Parseval s Theorem 18
2.6 Display of the Discrete Wavelet Transform and the Wavelet Expansion 18
2.7 Examples of Wavelet Expansions 20
2.8 An Example of the Haar Wavelet System 23
3 Filter Banks and the Discrete Wavelet Transform 31
3.1 Analysis From Fine Scale to Coarse Scale 31
Filtering and Down Sampling or Decimating 32
3.2 Synthesis From Coarse Scale to Fine Scale 36
Filtering and Up Sampling or Stretching 36
3.3 Input Coefficients 37
3.4 Lattices and Lifting 38
v
vj Contents
3.5 Different Points of View 38
Multiresolution versus Time Frequency Analysis 38
Periodic versus Nonperiodic Discrete Wavelet Transforms 38
The Discrete Wavelet Transform versus the Discrete Time Wavelet Transform 39
Numerical Complexity of the Discrete Wavelet Transform 40
4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Un¬
conditional Bases 41
4.1 Bases, Orthogonal Bases, and Biorthogonal Bases 41
Matrix Examples 43
Fourier Series Example 44
Sine Expansion Example 44
4.2 Frames and Tight Frames 45
Matrix Examples 46
Sine Expansion as a Tight Frame Example 47
4.3 Conditional and Unconditional Bases 48
5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coeffi¬
cients 50
5.1 Tools and Definitions 50
Signal Classes 50
Fourier Transforms 51
Refinement and Transition Matrices 52
5.2 Necessary Conditions 53
5.3 Frequency Domain Necessary Conditions 54
5.4 Sufficient Conditions 56
Wavelet System Design 57
5.5 The Wavelet 58
5.6 Alternate Normalizations 59
5.7 Example Scaling Functions and Wavelets 59
Haar Wavelets 60
Sine Wavelets 60
Spline and Battle Lemarie Wavelet Systems
5.8 Further Properties of the Scaling Function and Wavelet ^
General Properties not Requiring Orthogonality 63
Properties that Depend on Orthogonality 64
5.9 Parameterization of the Scaling Coefficients 65
Length 2 Scaling Coefficient Vector 65
Length 4 Scaling Coefficient Vector 66
Length 6 Scaling Coefficient Vector 66
5.10 Calculating the Basic Scaling Function and Wavelet 67
Successive Approximations or the Cascade Algorithm
Iterating the Filter Bank 68
Successive approximations in the frequency domain
The Dyadic Expansion of the Scaling Function
CONTENTS vii
6 Regularity, Moments, and Wavelet System Design 73
6.1 K Regular Scaling Filters 73
6.2 Vanishing Wavelet Moments 75
6.3 Daubechies Method for Zero Wavelet Moment Design 76
6.4 Non Maximal Regularity Wavelet Design 83
6.5 Relation of Zero Wavelet Moments to Smoothness 83
6.6 Vanishing Scaling Function Moments 86
6.7 Approximation of Signals by Scaling Function Projection 86
6.8 Approximation of Scaling Coefficients by Samples of the Signal 87
6.9 Coiflets and Related Wavelet Systems 88
Generalized Coifman Wavelet Systems 93
6.10 Minimization of Moments Rather than Zero Moments 97
7 Generalizations of the Basic Multiresolution Wavelet System 98
7.1 Tiling the Time Frequency or Time Scale Plane 98
Nonstationary Signal Analysis 99
Tiling with the Discrete Time Short Time Fourier Transform 100
Tiling with the Discrete Two Band Wavelet Transform 100
General Tiling 101
7.2 Multiplicity M (M Band) Scaling Functions and Wavelets 102
Properties of M Band Wavelet Systems 103
M Band Scaling Function Design 109
M Band Wavelet Design and Cosine Modulated Methods 110
7.3 Wavelet Packets 110
Full Wavelet Packet Decomposition 110
Adaptive Wavelet Packet Systems 111
7.4 Biorthogonal Wavelet Systems 114
Two Channel Biorthogonal Filter Banks 114
Biorthogonal Wavelets 116
Comparisons of Orthogonal and Biorthogonal Wavelets 117
Example Families of Biorthogonal Systems 118
Cohen Daubechies Feauveau Family of Biorthogonal Spline Wavelets 118
Cohen Daubechies Feauveau Family of Biorthogonal Wavelets with Less Dissimilar
Filter Length 118
Tian Wells Family of Biorthogonal Coiflets 119
Lifting Construction of Biorthogonal Systems 119
7.5 Multiwavelets 122
Construction of Two Band Multiwavelets 123
Properties of Multiwavelets 124
Approximation, Regularity and Smoothness 124
Support 124
Orthogonality 125
Implementation of Multiwavelet Transform 125
Examples 126
Geronimo Hardin Massopust Multiwavelets 126
Spline Multiwavelets 127
viii Contents
Other Constructions 127
Applications 128
7.6 Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases 128
Overcomplete Representations 129
A Matrix Example 129
Shift Invariant Redundant Wavelet Transforms and Nondecimated Filter Banks 132
Adaptive Construction of Frames and Bases 133
7.7 Local Trigonometric Bases 134
Nonsmooth Local Trigonometric Bases 136
Construction of Smooth Windows 136
Folding and Unfolding 137
Local Cosine and Sine Bases 139
Signal Adaptive Local Trigonometric Bases 141
7.8 Discrete Multiresolution Analysis, the Discrete Time Wavelet
Transform, and the Continuous Wavelet Transform 141
Discrete Multiresolution Analysis and the Discrete Time Wavelet Transform 143
Continuous Wavelet Transforms 144
Analogies between Fourier Systems and Wavelet Systems 145
8 Filter Banks and Transmultiplexers 148
8.1 Introduction 148
The Filter Bank 148
Transmultiplexer 150
Perfect Reconstruction—A Closer Look 150
Direct Characterization of PR 150
Matrix characterization of PR 152
Polyphase (Transform Domain) Characterization of PR 153
8.2 Unitary Filter Banks 155
8.3 Unitary Filter Banks—Some Illustrative Examples 160
8.4 M band Wavelet Tight Frames 162
8.5 Modulated Filter Banks 164
Unitary Modulated Filter Bank 167
8.6 Modulated Wavelet Tight Frames 168
8.7 Linear Phase Filter Banks 169
Characterization of Unitary Hp(z) — PS Symmetry 173
Characterization of Unitary Hp(z) — PCS Symmetry 174
Characterization of Unitary Hp(z) — Linear Phase Symmetry 174
Characterization of Unitary Hp(z) — Linear Phase and PCS Symmetry 175
Characterization of Unitary Hp{z) — Linear Phase and PS Symmetry 175
8.8 Linear Phase Wavelet Tight Frames 176
8.9 Linear Phase Modulated Filter Banks 177
DCT/DST I/II based 2M Channel Filter Bank 178
8.10 Linear Phase Modulated Wavelet Tight Frames 178
8.11 Time Varying Filter Bank Trees 179
Growing a Filter Bank Tree 182
Pruning a Filter Bank Tree 182
CONTENTS ix
Wavelet Bases for the Interval 183
Wavelet Bases for L2([0, oo)) 183
Wavelet Bases for L2({ oo,0]) 184
Segmented Time Varying Wavelet Packet Bases 185
8.12 Filter Banks and Wavelets—Summary 186
9 Calculation of the Discrete Wavelet Transform 188
9.1 Finite Wavelet Expansions and Transforms 188
9.2 Periodic or Cyclic Discrete Wavelet Transform 190
9.3 Filter Bank Structures for Calculation of the DWT and Complexity 191
9.4 The Periodic Case 192
9.5 Structure of the Periodic Discrete Wavelet Transform 194
9.6 More General Structures 195
10 Wavelet Based Signal Processing and Applications 196
10.1 Wavelet Based Signal Processing 196
10.2 Approximate FFT using the Discrete Wavelet Transform 197
Introduction 197
Review of the Discrete Fourier Transform and FFT 198
Review of the Discrete Wavelet Transform 200
The Algorithm Development 201
Computational Complexity 203
Fast Approximate Fourier Transform 203
Computational Complexity 203
Noise Reduction Capacity 204
Summary 204
10.3 Nonlinear Filtering or Denoising with the DWT 205
Denoising by Thresholding 206
Shift Invariant or Nondecimated Discrete Wavelet Transform 207
Combining the Shensa Beylkin Mallat a trous Algorithms and Wavelet Denoising 209
Performance Analysis 209
Examples of Denoising 210
10.4 Statistical Estimation 211
10.5 Signal and Image Compression 212
Fundamentals of Data Compression 212
Prototype Transform Coder 213
Improved Wavelet Based Compression Algorithms 215
10.6 Why are Wavelets so Useful? 216
10.7 Applications 217
Numerical Solutions to Partial Differential Equations 217
Seismic and Geophysical Signal Processing 217
Medical and Biomedical Signal and Image Processing 218
Application in Communications 218
Fractals 218
10.8 Wavelet Software 218
X Contents
11 Summary Overview 219
11.1 Properties of the Basic Multiresolution Scaling Function 219
11.2 Types of Wavelet Systems 221
12 References 223
Bibliography 224
Appendix A. Derivations for Chapter 5 on Scaling Functions 246
Appendix B. Derivations for Section on Properties 253
Appendix C. Matlab Programs 258
Index 266
|
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| language | English |
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| spelling | Burrus, C. S. Verfasser aut Introduction to wavelets and wavelet transforms a primer C. Sidney Burrus ; Ramesh A. Gopinath and Haitao Guo Upper Saddle River, NJ Prentice-Hall 1998 XIV, 268 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wavelet (DE-588)4215427-3 gnd rswk-swf Wavelet (DE-588)4215427-3 s DE-604 Gopinath, Ramesh A. Verfasser aut Guo, Haitao Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007842079&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Burrus, C. S. Gopinath, Ramesh A. Guo, Haitao Introduction to wavelets and wavelet transforms a primer Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4215427-3 |
| title | Introduction to wavelets and wavelet transforms a primer |
| title_auth | Introduction to wavelets and wavelet transforms a primer |
| title_exact_search | Introduction to wavelets and wavelet transforms a primer |
| title_full | Introduction to wavelets and wavelet transforms a primer C. Sidney Burrus ; Ramesh A. Gopinath and Haitao Guo |
| title_fullStr | Introduction to wavelets and wavelet transforms a primer C. Sidney Burrus ; Ramesh A. Gopinath and Haitao Guo |
| title_full_unstemmed | Introduction to wavelets and wavelet transforms a primer C. Sidney Burrus ; Ramesh A. Gopinath and Haitao Guo |
| title_short | Introduction to wavelets and wavelet transforms |
| title_sort | introduction to wavelets and wavelet transforms a primer |
| title_sub | a primer |
| topic | Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Wavelet |
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