A Primer on Wavelets and Their Scientific Applications

A Primer on Wavelets and their Scientific Applications......Page 1
Dedication......Page 2
Table of Contents......Page 3
Preface......Page 5
Acknowledgments......Page 8
1.1 The Haar transform......Page 9
Table of Contents......Page 0
Haar transform, 1-level......Page 11
Conservation of energy......Page 14
Haar transform, multiple levels......Page 16
Justification of Energy Conservation......Page 17
1.3 Haar wavelets......Page 18
1.4 Multiresolution analysis......Page 21
1.5 Compression of audio signals......Page 26
1.6 Removing noise from audio signals......Page 31
1.7 Notes and references......Page 36
2.1 The Daub4 wavelets......Page 37
Further remarks on Property I......Page 45
2.2 Conservation and compaction of energy......Page 46
Justification of conservation of energy......Page 47
How wavelet and scaling numbers are found......Page 49
2.3 Other Daubechies wavelets......Page 50
Coiflets......Page 54
2.4 Compression of audio signals......Page 57
Quantizing signal values, compressing the significance map......Page 58
2.5 Quantization, entropy, and compression......Page 61
2.6 Denoising audio signals......Page 66
Choosing a threshold value......Page 67
Removing pop noise and background static......Page 70
Discrete images......Page 73
2D wavelet transforms......Page 74
2D wavelets and scaling images......Page 77
2.8 Compression of images......Page 80
2.9 Fingerprint compression......Page 83
Remarks on the significance map......Page 85
2.10 Denoising images......Page 87
Quantitative measures of error......Page 90
Further aspects of noise removal......Page 92
Edge detection......Page 95
Edge enhancement......Page 96
Image recognition......Page 97
2.12 Notes and references......Page 100
3.1 Discrete Fourier analysis......Page 103
3.2 Definition of the DFT and its properties......Page 106
z-transforms......Page 109
3.3 Frequency description of wavelet analysis......Page 111
Low-pass and high-pass filtering......Page 114
3.4 Correlation and feature detection......Page 116
DFT method of computing correlations......Page 119
Proof of (3.40)......Page 120
Normalized correlations and feature detection......Page 121
3.5 Object detection in 2D images......Page 122
3.6 Creating scaling signals and wavelets......Page 126
3.7 Notes and references......Page 130
4.1 Wavelet packet transforms......Page 131
4.2 Applications of wavelet packet transforms......Page 134
4.3 Continuous wavelet transforms......Page 137
4.4 Gabor wavelets and speech analysis......Page 142
4.5 Notes and references......Page 146
Appendix A: Software for wavelet analysis......Page 147
Wavelet series, wavelet packet series......Page 148
Audio and image editors......Page 149
A.3 Other software......Page 150
References......Page 152