A Primer on WAVELETS and Their Scientific Applications

Transcription

1 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

2 Contents 1 Overview What is wavelet analysis? Notes and references 4 2 Haar wavelet s The Haar transform Haar transform, 1-level Conservation and compaction of energy Conservation of energy Haar transform, multiple levels Justification of conservation of energy Haar wavelets Multiresolution analysis Multiresolution analysis, multiple levels Signal compression A note on quantization Removing noise RMS Error Notes and references Examples and exercises 31 3 Daubechies wavelets The Daub4 wavelets Remarks on small fluctuation values * Conservation and compaction of energy Justification of conservation of energy * How wavelet and scaling numbers are found * Other Daubechies wavelets Coiflets Compression of audio signals Quantization and the significance map Quantization, entropy, and compression Denoising audio signals 69

3 3.6.1 Choosing a threshold value Removing pop noise and background static Biorthogonal wavelets Daub 5/3 System Daub 5/3 inverse MRA for the Daub 5/3 System Daub 5/3 transform, multiple lcvels Daub 5/3 integer-to-integer System The Daub 9/7 System Notes and references Examples and exercises 87 4 Two-dimensional wavelets Two-dimensional wavelet transforms Discrete images D wavelet transforms D wavelets and scaling images Compression of images- fundamentals Error measures in image processing Comparing JPEG with wavelet-based compressors Fingerprint compression The WDR algorithm Bit-plane encoding Difference reduction Arithmetic compression The ASWDR algorithm Arithmetic compression Relation to vision Important image compression features Progressive transniission/reconstruction Lossless compression Region-of-interest JPEG 2000 image compression Compressing color images Denoising images The TAWS algorithm Comparison with Wiener denoising Estimation of noise Standard deviation * Removal of clutter noise Some topics in image processing Edge detcction Edge enhancement Image recognition Notes and references Examples and exercises 147

4 5 Frequency analysis Discrete Fourier analysis Frequency content of wavelets Definition of the DFT and its properties Properties of the DFT z-transforms * Frequency description of wavelet analysis Low-pass and high-pass filtering * Correlation and feature detection DFT method of Computing correlations Proof of DFT effect on correlation * Normalized correlations and fcature detection * Object detection in 2D images Creating scaling Signals and wavelets * Gabor transforms and spectrograms Musical analysis Analysis of Stravinsky's Firebird Suite Analysis of a Chinese folk song Inverting Gabor transforms Gabor transforms and denoising Notes and references Examples and exercises Beyond wavelets Wavelet packet transforms Wavelet packet transforms applications Fingcrprint compression Continuous wavelet transforms Gabor wavelets and speech analysis Musical analysis: formants in song lyrics Percussion scalograms and musical rhythm Analysis of a complex percussive rhythm Multiresolution Principle for rhythm Notes and references Additional references 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

5 B Selected exercise Solutions 259 B.l Introduction 259 B.2 Chapter B.3 Chapter B.4 Chapter B.5 Chapter B.6 Chapter 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