A Primer on WAVELETS and Their Scientific Applications SECOND EDITION James S. Walker University of Wisconsin Eau Ciaire, Wisconsin, U.S.A. ^LlChapman & Hall/CRC ^^ Taylor & Francis Group Boca Raton London New York Chapman Sc Hall/CRC is an imprint of the Taylor St Francis Croup, an informa business
Contents 1 Overview 1 1.1 What is wavelet analysis? 1 1.2 Notes and references 4 2 Haar wavelet s 5 2.1 The Haar transform 6 2.1.1 Haar transform, 1-level 7 2.2 Conservation and compaction of energy 9 2.2.1 Conservation of energy 10 2.2.2 Haar transform, multiple levels 11 2.2.3 Justification of conservation of energy 12 2.3 Haar wavelets 14 2.4 Multiresolution analysis 16 2.4.1 Multiresolution analysis, multiple levels 19 2.5 Signal compression 21 2.5.1 A note on quantization 26 2.6 Removing noise 26 2.6.1 RMS Error 29 2.7 Notes and references 30 2.8 Examples and exercises 31 3 Daubechies wavelets 41 3.1 The Daub4 wavelets 41 3.1.1 Remarks on small fluctuation values * 49 3.2 Conservation and compaction of energy 50 3.2.1 Justification of conservation of energy * 50 3.2.2 How wavelet and scaling numbers are found * 53 3.3 Other Daubechies wavelets 54 3.3.1 Coiflets 58 3.4 Compression of audio signals 61 3.4.1 Quantization and the significance map 62 3.5 Quantization, entropy, and compression 65 3.6 Denoising audio signals 69
3.6.1 Choosing a threshold value 70 3.6.2 Removing pop noise and background static 73 3.7 Biorthogonal wavelets 75 3.7.1 Daub 5/3 System 76 3.7.2 Daub 5/3 inverse 78 3.7.3 MRA for the Daub 5/3 System 78 3.7.4 Daub 5/3 transform, multiple lcvels 80 3.7.5 Daub 5/3 integer-to-integer System 82 3.8 The Daub 9/7 System 83 3.9 Notes and references 85 3.10 Examples and exercises 87 4 Two-dimensional wavelets 97 4.1 Two-dimensional wavelet transforms 97 4.1.1 Discrete images 98 4.1.2 2D wavelet transforms 99 4.1.3 2D wavelets and scaling images 102 4.2 Compression of images- fundamentals 104 4.2.1 Error measures in image processing 107 4.2.2 Comparing JPEG with wavelet-based compressors... 108 4.3 Fingerprint compression 110 4.4 The WDR algorithm 113 4.4.1 Bit-plane encoding 113 4.4.2 Difference reduction 116 4.4.3 Arithmetic compression 119 4.5 The ASWDR algorithm 123 4.5.1 Arithmetic compression 125 4.5.2 Relation to vision 126 4.6 Important image compression features 127 4.6.1 Progressive transniission/reconstruction 127 4.6.2 Lossless compression 128 4.6.3 Region-of-interest 130 4.7 JPEG 2000 image compression 130 4.7.1 Compressing color images 132 4.8 Denoising images 133 4.8.1 The TAWS algorithm 133 4.8.2 Comparison with Wiener denoising 134 4.8.3 Estimation of noise Standard deviation * 136 4.8.4 Removal of clutter noise 137 4.9 Some topics in image processing 139 4.9.1 Edge detcction 139 4.9.2 Edge enhancement 140 4.9.3 Image recognition 141 4.10 Notes and references 144 4.11 Examples and exercises 147
5 Frequency analysis 167 5.1 Discrete Fourier analysis 168 5.1.1 Frequency content of wavelets 169 5.2 Definition of the DFT and its properties 170 5.2.1 Properties of the DFT 171 5.2.2 z-transforms * 173 5.3 Frequency description of wavelet analysis 174 5.3.1 Low-pass and high-pass filtering * 178 5.4 Correlation and feature detection 180 5.4.1 DFT method of Computing correlations 181 5.4.2 Proof of DFT effect on correlation * 183 5.4.3 Normalized correlations and fcature detection * 183 5.5 Object detection in 2D images 185 5.6 Creating scaling Signals and wavelets * 188 5.7 Gabor transforms and spectrograms 192 5.8 Musical analysis 195 5.8.1 Analysis of Stravinsky's Firebird Suite 197 5.8.2 Analysis of a Chinese folk song 199 5.9 Inverting Gabor transforms 201 5.10 Gabor transforms and denoising 203 5.11 Notes and references 206 5.12 Examples and exercises 210 6 Beyond wavelets 223 6.1 Wavelet packet transforms 223 6.2 Wavelet packet transforms applications 225 6.2.1 Fingcrprint compression 228 6.3 Continuous wavelet transforms 228 6.4 Gabor wavelets and speech analysis 232 6.4.1 Musical analysis: formants in song lyrics 236 6.5 Percussion scalograms and musical rhythm 237 6.5.1 Analysis of a complex percussive rhythm 241 6.5.2 Multiresolution Principle for rhythm 241 6.6 Notes and references 241 6.6.1 Additional references 242 6.7 Examples and exercises 246 A Projects 255 A.l Music 255 A.2 Noise removal from audio 256 A.3 Wavelet image processing 256 A.4 References 257
B Selected exercise Solutions 259 B.l Introduction 259 B.2 Chapter 2 259 B.3 Chapter 3 262 B.4 Chapter 4 268 B.5 Chapter 5 273 B.6 Chapter 6 279 C Wavelet Software 283 C.l Installing the book's Software 284 C.2 Other Software 284 C.3 References 285 Bibliography 287 Index 295