A Primer on Wavelets and Their Scientific Applications, Second Edition

by Walker; James S.

• ISBN13: 9781584887454
• ISBN10: 1584887451
• Edition: 2nd
• Format: Nonspecific Binding
• Copyright: 2008-01-29
• Publisher: Chapman & Hall/

Summary

In the first edition of his seminal introduction to wavelets, James S. Walker informed us that the potential applications for wavelets were virtually unlimited. Since that time thousands of published papers have proven him true, while also necessitating the creation of a new edition of his bestselling primer. Updated and fully revised to include the latest developments, this second edition of A Primer on Wavelets and Their Scientific Applicationsguides readers through the main ideas of wavelet analysis in order to develop a thorough appreciation of wavelet applications.Ingeniously relying on elementary algebra and just a smidgen of calculus, Professor Walker demonstrates how the underlying ideas behind wavelet analysis can be applied to solve significant problems in audio and image processing, as well in biology and medicine.Nearly twice as long as the original, this new edition provides·    104 worked examples and 222 exercises, constituting a veritable book of review material· Two sections on biorthogonal wavelets· A mini-course on image compression, including a tutorial on arithmetic compression· Extensive material on image denoising, featuring a rarely covered technique for removing isolated, randomly positioned clutter· Concise yet complete coverage of the fundamentals of time-frequency analysis, showcasing its application to audio denoising, and musical theory and synthesis· An introduction to the multiresolution principle, a new mathematical concept in musical theory· Expanded suggestions for research projects· An enhanced list of references· FAWAV: software designed by the author, which allows readers to duplicate described applications and experiment with other ideas.To keep the book current, Professor Walker has created a supplementary website. This online repository includes ready-to-download software, and sound and image files, as well as access to many of the most important papers in the field.

Table of Contents

Overview	p. 1
What is wavelet analysis?	p. 1
Notes and references	p. 4
Haar wavelets	p. 5
The Haar transform	p. 6
Haar transform, 1-level	p. 7
Conservation and compaction of energy	p. 9
Conservation of energy	p. 10
Haar transform, multiple levels	p. 11
Justification of conservation of energy	p. 12
Haar wavelets	p. 14
Multiresolution analysis	p. 16
Multiresolution analysis, multiple levels	p. 19
Signal compression	p. 21
A note on quantization	p. 26
Removing noise	p. 26
RMS Error	p. 29
Notes and references	p. 30
Examples and exercises	p. 31
Daubechies wavelets	p. 41
The Daub4 wavelets	p. 41
Remarks on small fluctuation values*	p. 49
Conservation and compaction of energy	p. 50
Justification of conservation of energy*	p. 50
How wavelet and scaling numbers are found*	p. 53
Other Daubechies wavelets	p. 54
Coiflets	p. 58
Compression of audio signals	p. 61
Quantization and the significance map	p. 62
Quantization, entropy, and compression	p. 65
Denoising audio signals	p. 69
Choosing a threshold value	p. 70
Removing pop noise and background static	p. 73
Biorthogonal wavelets	p. 75
Daub 5/3 system	p. 76
Daub 5/3 inverse	p. 78
MRA for the Daub 5/3 system	p. 78
Daub 5/3 transform, multiple levels	p. 80
Daub 5/3 integer-to-integer system	p. 82
The Daub 9/7 system	p. 83
Notes and references	p. 85
Examples and exercises	p. 87
Two-dimensional wavelets	p. 97
Two-dimensional wavelet transforms	p. 97
Discrete images	p. 98
2D wavelet transforms	p. 99
2D wavelets and scaling images	p. 102
Compression of images-fundamentals	p. 104
Error measures in image processing	p. 107
Comparing JPEG with wavelet-based compressors	p. 108
Fingerprint compression	p. 110
The WDR algorithm	p. 113
Bit-plane encoding	p. 113
Difference reduction	p. 116
Arithmetic compression	p. 119
The ASWDR algorithm	p. 123
Arithmetic compression	p. 125
Relation to vision	p. 126
Important image compression features	p. 127
Progressive transmission/reconstruction	p. 127
Lossless compression	p. 128
Region-of-interest	p. 130
JPEG 2000 image compression	p. 130
Compressing color images	p. 132
Denosing images	p. 133
The TAWS algorithm	p. 133
Comparison with Wiener denoising	p. 134
Estimation of noise standard deviation*	p. 136
Removal of clutter noise	p. 137
Some topics in image processing	p. 139
Edge detection	p. 139
Edge enhancement	p. 140
Image recognition	p. 141
Notes and references	p. 144
Examples and exercises	p. 147
Frequency analysis	p. 167
Discrete Fourier analysis	p. 168
Frequency content of wavelets	p. 169
Definition of the DFT and its properties	p. 170
Properties of the DFT	p. 171
z-transforms*	p. 173
Frequency description of wavelet analysis	p. 174
Low-pass and high-pass filtering*	p. 178
Correlation and feature detection	p. 180
DFT method of computing correlations	p. 181
Proof of DFT effect on correlation*	p. 183
Normalized correlations and feature detection*	p. 183
Object detection in 2D images	p. 185
Creating scaling signals and wavelets*	p. 188
Gabor transforms and spectrograms	p. 192
Musical analysis	p. 195
Analysis of Stravinsky's Firebird Suite	p. 197
Analysis of a Chinese folk song	p. 199
Inverting Gabor transforms	p. 201
Gabor transforms and denoising	p. 203
Notes and references	p. 206
Examples and exercises	p. 210
Beyond wavelets	p. 223
Wavelet packet transforms	p. 223
Wavelet packet transforms-applications	p. 225
Fingerprint compression	p. 228
Continuous wavelet transforms	p. 228
Gabor wavelets and speech analysis	p. 232
Musical analysis: formants in song lyrics	p. 236
Percussion scalograms and musical rhythm	p. 237
Analysis of a complex percussive rhythm	p. 241
Multiresolution Principle for rhythm	p. 241
Notes and references	p. 241
Additional references	p. 242
Examples and exercises	p. 246
Projects	p. 255
Music	p. 255
Noise removal from audio	p. 256
Wavelet image processing	p. 256
References	p. 257
Selected exercise solutions	p. 259
Introduction	p. 259
Chapter 2	p. 259
Chapter 3	p. 262
Chapter 4	p. 268
Chapter 5	p. 273
Chapter 6	p. 279
Wavelet software	p. 283
Installing the book's software	p. 284
Other software	p. 284
References	p. 285
Bibliography	p. 287
Index	p. 295
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