Wavelet Methods in Mathematical Analysis and Engineering (Series in Contemporary Applied Mathematics)

✍ Scribed by Alain Damlamian

• Publisher: World Scientific Publishing Company
• Year: 2010
• Tongue: English
• Leaves: 190
• Category: Library

✦ Synopsis

This book gives a comprehensive overview of both the fundamentals of wavelet analysis and related tools, and of the most active recent developments towards applications. It offers a state-of-the-art in several active areas of research where wavelet ideas, or more generally multiresolution ideas have proved particularly effective. The main applications covered are in the numerical analysis of PDEs, and signal and image processing. Recently introduced techniques such as Empirical Mode Decomposition (EMD) and new trends in the recovery of missing data, such as compressed sensing, are also presented. Applications range for the reconstruction of noisy or blurred images, pattern and face recognition, to nonlinear approximation in strongly anisotropic contexts, and to the classification tools based on multifractal analysis.

✦ Table of Contents

Contents......Page 10
Preface......Page 6
Abstract......Page 12
1 High-resolution image reconstruction model......Page 13
2 Preliminaries on tight framelets......Page 17
3.1 Filter design......Page 19
3.2 Matrix representation of filters with periodic boundary conditions......Page 21
3.3 Matrix representation of filters with symmetric boundary conditions......Page 22
3.4 Multi-level frame let decomposition and reconstruction......Page 27
4 Algorithms......Page 28
4.1 Basic algorithm......Page 29
4.2 Algorithms with tight frame denoising scheme......Page 31
4.2.2 Algorithm II......Page 32
4.2.3 Algorithm III......Page 33
5.1 Proximal forward-backward splitting......Page 34
5.2 Convergence of Algorithm I......Page 35
5.3 Minimization property of Algorithm I......Page 38
6 Numerical experiments......Page 40
References......Page 44
1 Introduction......Page 48
2 Best N -term approximation......Page 49
2.1 The case of an orthonormal basis......Page 50
2.3 Approximation results......Page 51
2.4 Open questions and related topics......Page 53
3 Adaptive triangulations......Page 54
3.1 From uniform to adaptive isotropic triangulations......Page 55
3.2 Approximation theory of anisotropic triangulations......Page 57
3.3 A greedy algorithm for anisotropic triangulations......Page 58
3.4 Convergence properties of the algorithm......Page 59
References......Page 60
Abstract......Page 62
1 Kolmogorov's scaling law and function spaces......Page 63
2.1 Holder exponents......Page 66
2.2 Other notions of pointwise regularity......Page 67
2.3 Brownian motion and related noncentered stochastic processes......Page 69
3 Lacunary Fourier series......Page 72
3.1 A pointwise irregularity criterium......Page 73
3.2 Application to nonharrnonic Fourier series......Page 74
3.3 Everywhere irregularity of solutions of Schrodinger's equation......Page 75
1.1 Signal......Page 0
4.1 Orthonormal and biorthogonal wavelet bases......Page 77
4.2 Wavelets and function spaces......Page 79
4.3 Wavelet characterizations of pointwise regularity......Page 81
4.4 Application to decentered Fractional Brownian Motions......Page 87
5 The multifractal formalism......Page 89
5.1 Fractal dimensions and spectrums of singularities......Page 92
5.2 Derivation of the multifractal formalism......Page 94
5.3 Properties of the scaling function......Page 95
5.4 Upper bound of the spectrums......Page 99
5.5 Validity of the multifractal formalism......Page 101
5.6 Some open questions......Page 105
References......Page 106
1 Introduction......Page 110
2 Face recognition task......Page 111
2.1 Why face recognition is studied......Page 113
2.2 Challenge of face recognition......Page 114
2.3 History of face recognition......Page 116
3 The structure of a Pattern Recognition System (PRS)......Page 117
4.1 The one dimensional wavelet transform......Page 120
4.2 The two-dimensional wavelet transform......Page 121
4.3 The wavelet-packet transform......Page 122
4.4 Properties of wavelets for image processing......Page 123
5 Preprocessing: wavelets for noise removal......Page 124
6 Wavelet for feature extraction......Page 126
6.1 Feature extraction based on the separable wavelet transform......Page 127
6.2.1 The dual-tree CWT......Page 135
6.2.2 Complex-WT-face......Page 138
7 Conclusion and discussion......Page 139
References......Page 141
Abstract......Page 149
1.2 Signal transmission and modulation......Page 151
1.3 The existence of instantaneous frequency for aperiodic signals......Page 153
1.4 Complex extension and demodulation by Hilbert transform......Page 154
1.5 Paradoxes regarding the instantaneous frequency defined by the analytic signal......Page 158
2.1 Intrinsic mode functions......Page 160
2.2 Empirical mode decomposition......Page 161
2.3 Hilbert spectrum......Page 164
3 Some relevant questions and our recent researches......Page 165
3.1.1 Empirical AMjFM demodulation......Page 166
3.1.2 Riding waves and an improvement to the algorithm......Page 167
3.2 Hilbert transform and the Bedrosian identity......Page 171
3.3 Hilbert transform on distribution spaces......Page 174
4.1 HHT -based detection of spindles in sleep EEGs......Page 175
4.2 Pitch period detection algorithm based on Hilbert- Huang transform......Page 179
4.3 Chinese font recognition based on empirical mode decompositon......Page 182
References......Page 186