Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity

by Jean-Luc Starck , Fionn Murtagh , Jalal M. Fadili

• ISBN13: 9780521119139
• ISBN10: 0521119138
• Format: Hardcover
• Copyright: 2010-05-10
• Publisher: Cambridge University Press

Summary

This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelet transforms, and non-linear multiscale transforms based on the median and mathematical morphology operators. Recent concepts of sparsity and morphological diversity are described and exploited for various problems such as denoising, inverse problem regularization, sparse signal decomposition, blind source separation, and compressed sensing. This book weds theory and practice in examining applications in areas such as astronomy, biology, physics, digital media, and forensics. A final chapter explores a paradigm shift in signal processing, showing that previous limits to information sampling and extraction can be overcome in very significant ways. Matlab and IDL code accompany these methods and applications to reproduce the experiments and illustrate the reasoning and methodology of the research available for download at the associated Web site.

Author Biography

Jean-LucStarck is Senior Scientist at the Fundamental Laws of the Universe Research Institute, CEA-Saclay. He holds a PhD from the University of Nice Sophia Antipolis and the Observatory of the Cote d'Azur, and a Habilitation degree from the University Paris 11. He has held visiting appointments at the European Southern Observatory, the University of California Los Angeles, and the Statistics Department, Stanford University. He is author of the following books: Image Pro-cessing and Data Analysis: The Multiscale Approach and Astronomical Image and Data Analysis. In 2009, he won a European Research Council Advanced Investigator award.
Fionn Murtagh directs Science Foundation Ireland's national funding programs in Information and Communications Technologies, and in Energy. He holds a PhD in Mathematical Statistics from the University of Paris 6, and a Habilitation from the University of Strasbourg. He has held professorial chairs in computer science at the University of Ulster, Queen's University Belfast, and now in the University of London at Royal Holloway. He is a Member of the Royal Irish Academy, a Fellow of the International Association for Pattern Recognition, and a Fellow of the British Computer Society.
Jalal M. Fadili graduated from the Ecole Nationale Superieured'lngenieurs (ENSI), Caen, France and received MSc and PhD degrees in signal processing, and a Habilitation, from the University of Caen. He was McDonnell-Pew Fellow at the University of Cambridge in 1999-2000. Since 2001 he is Associate Professor of Signal and Image Processing at ENSI. He has held visiting appointments at Queensland University of Technology, Stanford University, Caltech, and EPFL.

Table of Contents

Acronyms	p. ix
Notation	p. xiii
Preface	p. xv
Introduction to the World of Sparsity	p. 1
Sparse Representation	p. 1
From Fourier to Wavelets	p. 5
From Wavelets to Overcomplete Representations	p. 6
Novel Applications of the Wavelet and Curvelet Transforms	p. 8
Summary	p. 15
The Wavelet Transform	p. 16
Introduction	p. 16
The Continuous Wavelet Transform	p. 16
Examples of Wavelet Functions	p. 18
Continuous Wavelet Transform Algorithm	p. 21
The Discrete Wavelet Transform	p. 22
Nondyadic Resolution Factor	p. 28
The Lifting Scheme	p. 31
Wavelet Packets	p. 34
Guided Numerical Experiments	p. 38
Summary	p. 44
Redundant Wavelet Transform	p. 45
Introduction	p. 45
The Undecimated Wavelet Transform	p. 46
Partially Decimated Wavelet Transform	p. 49
The Dual-Tree Complex Wavelet Transform	p. 51
Isotropic Undecimated Wavelet Transform: Starlet Transform	p. 53
Nonorthogonal Filter Bank Design	p. 58
Pyramidal Wavelet Transform	p. 64
Guided Numerical Experiments	p. 69
Summary	p. 74
Nonlinear Multiscale Transforms	p. 75
Introduction	p. 75
Decimated Nonlinear Transform	p. 75
Multiscale Transform and Mathematical Morphology	p. 77
Multiresolution Based on the Median Transform	p. 81
Guided Numerical Experiments	p. 86
Summary	p. 88
The Ridgelet and Curvelet Transforms	p. 89
Introduction	p. 89
Background and Example	p. 89
Ridgelets	p. 91
Curvelets	p. 100
Curvelets and Contrast Enhancement	p. 110
Guided Numerical Experiments	p. 112
Summary	p. 118
Sparsity and Noise Removal	p. 119
Introduction	p. 119
Term-By-Term Nonlinear Denoising	p. 120
Block Nonlinear Denoising	p. 127
Beyond Additive Gaussian Noise	p. 132
Poisson Noise and the Haar Transform	p. 134
Poisson Noise with Low Counts	p. 136
Guided Numerical Experiments	p. 143
Summary	p. 145
Linear Inverse Problems	p. 149
Introduction	p. 149
Sparsity-Regularized Linear Inverse Problems	p. 151
Monotone Operator Splitting Framework	p. 152
Selected Problems and Algorithms	p. 160
Sparsity Penalty with Analysis Prior	p. 170
Other Sparsity-Regularized Inverse Problems	p. 172
General Discussion: Sparsity, Inverse Problems, and Iterative Thresholding	p. 174
Guided Numerical Experiments	p. 176
Summary	p. 178
Morphological Diversity	p. 180
Introduction	p. 180
Dictionary and Fast Transformation	p. 183
Combined Denoising	p. 183
Combined Deconvolution	p. 188
Morphological Component Analysis	p. 190
Texture-Cartoon Separation	p. 198
Inpainting	p. 204
Guided Numerical Experiments	p. 210
Summary	p. 216
Sparse Blind Source Separation	p. 218
Introduction	p. 218
Independent Component Analysis	p. 220
Sparsity and Multichannel Data	p. 224
Morphological Diversity and Blind Source Separation	p. 226
Illustrative Experiments	p. 237
Guided Numerical Experiments	p. 242
Summary	p. 244
Multiscale Geometric Analysis on the Sphere	p. 245
Introduction	p. 245
Data on the Sphere	p. 246
Orthogonal Haar Wavelets on the Sphere	p. 248
Continuous Wavelets on the Sphere	p. 249
Redundant Wavelet Transform on the Sphere with Exact Reconstruction	p. 253
Curvelet Transform on the Sphere	p. 261
Restoration and Decomposition on the Sphere	p. 266
Applications	p. 269
Guided Numerical Experiments	p. 272
Summary	p. 276
Compressed Sensing	p. 277
Introduction	p. 277
Incoherence and Sparsity	p. 278
The Sensing Protocol	p. 278
Stable Compressed Sensing	p. 280
Designing Good Matrices: Random Sensing	p. 282
Sensing with Redundant Dictionaries	p. 283
Compressed Sensing in Space Science	p. 283
Guided Numerical Experiments	p. 285
Summary	p. 286
References	p. 289
List of Algorithms	p. 311
Index	p. 313
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