Sparse image and signal processing : wavelets, curvelets, morphological diversity

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 are available for download at the associated web site.

Author Notes

Jean-Luc Starck 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 Cte 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 Processing 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 cole Nationale Suprieured'Ingnieurs (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.

Acronyms	p. ix
Notation	p. xiii
Preface	p. xv
1 Introduction to the World of Sparsity	p. 1
1.1 Sparse Representation	p. 1
1.2 From Fourier to Wavelets	p. 5
1.3 From Wavelets to Overcomplete Representations	p. 6
1.4 Novel Applications of the Wavelet and Curvelet Transforms	p. 8
1.5 Summary	p. 15
2 The Wavelet Transform	p. 16
2.1 Introduction	p. 16
2.2 The Continuous Wavelet Transform	p. 16
2.3 Examples of Wavelet Functions	p. 18
2.4 Continuous Wavelet Transform Algorithm	p. 21
2.5 The Discrete Wavelet Transform	p. 22
2.6 Nondyadic Resolution Factor	p. 28
2.7 The Lifting Scheme	p. 31
2.8 Wavelet Packets	p. 34
2.9 Guided Numerical Experiments	p. 38
2.10 Summary	p. 44
3 Redundant Wavelet Transform	p. 45
3.1 Introduction	p. 45
3.2 The Undecimated Wavelet Transform	p. 46
3.3 Partially Decimated Wavelet Transform	p. 49
3.4 The Dual-Tree Complex Wavelet Transform	p. 51
3.5 Isotropic Undecimated Wavelet Transform: Starlet Transform	p. 53
3.6 Nonorthogonal Filter Bank Design	p. 58
3.7 Pyramidal Wavelet Transform	p. 64
3.8 Guided Numerical Experiments	p. 69
3.9 Summary	p. 74
4 Nonlinear Multiscale Transforms	p. 75
4.1 Introduction	p. 75
4.2 Decimated Nonlinear Transform	p. 75
4.3 Multiscale Transform and Mathematical Morphology	p. 77
4.4 Multiresolution Based on the Median Transform	p. 81
4.5 Guided Numerical Experiments	p. 86
4.6 Summary	p. 88
5 The Ridgelet and Curvelet Transforms	p. 89
5.1 Introduction	p. 89
5.2 Background and Example	p. 89
5.3 Ridgelets	p. 91
5.4 Curvelets	p. 100
5.5 Curvelets and Contrast Enhancement	p. 110
5.6 Guided Numerical Experiments	p. 112
5.7 Summary	p. 118
6 Sparsity and Noise Removal	p. 119
6.1 Introduction	p. 119
6.2 Term-By-Term Nonlinear Denoising	p. 120
6.3 Block Nonlinear Denoising	p. 127
6.4 Beyond Additive Gaussian Noise	p. 132
6.5 Poisson Noise and the Haar Transform	p. 134
6.6 Poisson Noise with Low Counts	p. 136
6.7 Guided Numerical Experiments	p. 143
6.8 Summary	p. 145
7 Linear Inverse Problems	p. 149
7.1 Introduction	p. 149
7.2 Sparsity-Regularized Linear Inverse Problems	p. 151
7.3 Monotone Operator Splitting Framework	p. 152
7.4 Selected Problems and Algorithms	p. 160
7.5 Sparsity Penalty with Analysis Prior	p. 170
7.6 Other Sparsity-Regularized Inverse Problems	p. 172
7.7 General Discussion: Sparsity, Inverse Problems, and Iterative Thresholding	p. 174
7.8 Guided Numerical Experiments	p. 176
7.9 Summary	p. 178
8 Morphological Diversity	p. 180
8.1 Introduction	p. 180
8.2 Dictionary and Fast Transformation	p. 183
8.3 Combined Denoising	p. 183
8.4 Combined Deconvolution	p. 188
8.5 Morphological Component Analysis	p. 190
8.6 Texture-Cartoon Separation	p. 198
8.7 Inpainting	p. 204
8.8 Guided Numerical Experiments	p. 210
8.9 Summary	p. 216
9 Sparse Blind Source Separation	p. 218
9.1 Introduction	p. 218
9.2 Independent Component Analysis	p. 220
9.3 Sparsity and Multichannel Data	p. 224
9.4 Morphological Diversity and Blind Source Separation	p. 226
9.5 Illustrative Experiments	p. 237
9.6 Guided Numerical Experiments	p. 242
9.7 Summary	p. 244
10 Multiscale Geometric Analysis on the Sphere	p. 245
10.1 Introduction	p. 245
10.2 Data on the Sphere	p. 246
10.3 Orthogonal Haar Wavelets on the Sphere	p. 248
10.4 Continuous Wavelets on the Sphere	p. 249
10.5 Redundant Wavelet Transform on the Sphere with Exact Reconstruction	p. 253
10.6 Curvelet Transform on the Sphere	p. 261
10.7 Restoration and Decomposition on the Sphere	p. 266
10.8 Applications	p. 269
10.9 Guided Numerical Experiments	p. 272
10.10 Summary	p. 276
11 Compressed Sensing	p. 277
11.1 Introduction	p. 277
11.2 Incoherence and Sparsity	p. 278
11.3 The Sensing Protocol	p. 278
11.4 Stable Compressed Sensing	p. 280
11.5 Designing Good Matrices: Random Sensing	p. 282
11.6 Sensing with Redundant Dictionaries	p. 283
11.7 Compressed Sensing in Space Science	p. 283
11.8 Guided Numerical Experiments	p. 285
11.9 Summary	p. 286
References	p. 289
List of Algorithms	p. 311
Index	p. 313

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