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

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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.

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Editorial Reviews

From the Publisher
"This book is well organized, and it covers the theory and application of multiscale imaging and image processing. The authors provide Matlab algorithms for wavelet, ridgelet and curvelet transformations, as well as numerical experiments with detailed Matlab and IDL code for each chapter. A detailed list of the references provides further exploration of recent publications in the area.
The book's contents are well prepared for graduate-level students or advanced undergraduates who work in the field of image and signal processing or computer science. The book is also an indispensable resource for professionals looking to adopt innovative concepts for improving the performance of image processing."
Yan Gao, Osterhout Design Group for Optics & Photonics News

"This is an excellent book devoted to an important domain of contemporary science, where the activity includes the following stages: research, theory and publication and the verification of results by the scientific community."
D. Stanomir, Mathematical Reviews

"The book is effective in delivering a concise overview of the field. The book can be highly useful for researchers and graduate students in engineering and science who are looking for research ideas or are interested in applying the techniques to their application domains."
T. Kubota, Computing Reviews

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Product Details

  • ISBN-13: 9780521119139
  • Publisher: Cambridge University Press
  • Publication date: 5/10/2010
  • Pages: 336
  • Product dimensions: 7.20 (w) x 10.10 (h) x 1.00 (d)

Meet the Author

Jean-Luc Starck is a researcher at the Institute of Research into the Fundamental Laws of the Universe (IRFU), CEA-Saclay. He holds a Ph.D. from the University of Nice-Sophia Antipolis and Observatory of Côte d'Azur and a habilitation degree from the University Paris XI. He is a former visiting researcher at the European Southern Observatory (ESO), UCLA, and the Statistics Department at Stanford University. His research interests include image processing, statistical methods in astrophysics, and cosmology. He is also author of two books, entitled Image Processing and Data Analysis: The Multiscale Approach and Astronomical Image and Data Analysis.

Fionn Murtagh directs Ireland's Science Foundation funding programs in Information and Communications Technologies and Energy. He holds a Ph.D. from the Université Paris 6 and a habilitation from Université de Strasbourg. Murtagh held professorial chairs at the University of Ulster, Queen's University Belfast, and now at Royal Holloway, University of London. He is a Fellow of the International Association for Pattern Recognition, a Fellow of the British Computer Society, and an elected Member of the Royal Irish Academy.

Jalal M. Fadili graduated from the Ecole Nationale Supérieure d'Ingénieurs (ENSI) de Caen, France, and received MSc and Ph.D. degrees in signal and image processing from the University of Caen. He was a Research Associate with the University of Cambridge (McDonnell-Pew Fellow) from 1999 to 2000. He has been an Associate Professor of signal and image processing since September 2001 at ENSI. He was a visitor at several universities (QUT-Australia, Stanford University, CalTech, EPFL). His research interests include mathematical signal and image processing, statistics, optimization theory, and sparse representations.

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Table of Contents

Acronyms ix

Notation xiii

Preface xv

1 Introduction to the World of Sparsity 1

1.1 Sparse Representation 1

1.2 From Fourier to Wavelets 5

1.3 From Wavelets to Overcomplete Representations 6

1.4 Novel Applications of the Wavelet and Curvelet Transforms 8

1.5 Summary 15

2 The Wavelet Transform 16

2.1 Introduction 16

2.2 The Continuous Wavelet Transform 16

2.3 Examples of Wavelet Functions 18

2.4 Continuous Wavelet Transform Algorithm 21

2.5 The Discrete Wavelet Transform 22

2.6 Nondyadic Resolution Factor 28

2.7 The Lifting Scheme 31

2.8 Wavelet Packets 34

2.9 Guided Numerical Experiments 38

2.10 Summary 44

3 Redundant Wavelet Transform 45

3.1 Introduction 45

3.2 The Undecimated Wavelet Transform 46

3.3 Partially Decimated Wavelet Transform 49

3.4 The Dual-Tree Complex Wavelet Transform 51

3.5 Isotropic Undecimated Wavelet Transform: Starlet Transform 53

3.6 Nonorthogonal Filter Bank Design 58

3.7 Pyramidal Wavelet Transform 64

3.8 Guided Numerical Experiments 69

3.9 Summary 74

4 Nonlinear Multiscale Transforms 75

4.1 Introduction 75

4.2 Decimated Nonlinear Transform 75

4.3 Multiscale Transform and Mathematical Morphology 77

4.4 Multiresolution Based on the Median Transform 81

4.5 Guided Numerical Experiments 86

4.6 Summary 88

5 The Ridgelet and Curvelet Transforms 89

5.1 Introduction 89

5.2 Background and Example 89

5.3 Ridgelets 91

5.4 Curvelets 100

5.5 Curvelets and Contrast Enhancement 110

5.6 Guided Numerical Experiments 112

5.7 Summary 118

6 Sparsity and Noise Removal 119

6.1 Introduction 119

6.2 Term-By-Term Nonlinear Denoising 120

6.3 Block Nonlinear Denoising 127

6.4 Beyond Additive Gaussian Noise 132

6.5 Poisson Noise and the Haar Transform 134

6.6 Poisson Noise with Low Counts 136

6.7 Guided Numerical Experiments 143

6.8 Summary 145

7 Linear Inverse Problems 149

7.1 Introduction 149

7.2 Sparsity-Regularized Linear Inverse Problems 151

7.3 Monotone Operator Splitting Framework 152

7.4 Selected Problems and Algorithms 160

7.5 Sparsity Penalty with Analysis Prior 170

7.6 Other Sparsity-Regularized Inverse Problems 172

7.7 General Discussion: Sparsity, Inverse Problems, and Iterative Thresholding 174

7.8 Guided Numerical Experiments 176

7.9 Summary 178

8 Morphological Diversity 180

8.1 Introduction 180

8.2 Dictionary and Fast Transformation 183

8.3 Combined Denoising 183

8.4 Combined Deconvolution 188

8.5 Morphological Component Analysis 190

8.6 Texture-Cartoon Separation 198

8.7 Inpainting 204

8.8 Guided Numerical Experiments 210

8.9 Summary 216

9 Sparse Blind Source Separation 218

9.1 Introduction 218

9.2 Independent Component Analysis 220

9.3 Sparsity and Multichannel Data 224

9.4 Morphological Diversity and Blind Source Separation 226

9.5 Illustrative Experiments 237

9.6 Guided Numerical Experiments 242

9.7 Summary 244

10 Multiscale Geometric Analysis on the Sphere 245

10.1 Introduction 245

10.2 Data on the Sphere 246

10.3 Orthogonal Haar Wavelets on the Sphere 248

10.4 Continuous Wavelets on the Sphere 249

10.5 Redundant Wavelet Transform on the Sphere with Exact Reconstruction 253

10.6 Curvelet Transform on the Sphere 261

10.7 Restoration and Decomposition on the Sphere 266

10.8 Applications 269

10.9 Guided Numerical Experiments 272

10.10 Summary 276

11 Compressed Sensing 277

11.1 Introduction 277

11.2 Incoherence and Sparsity 278

11.3 The Sensing Protocol 278

11.4 Stable Compressed Sensing 280

11.5 Designing Good Matrices: Random Sensing 282

11.6 Sensing with Redundant Dictionaries 283

11.7 Compressed Sensing in Space Science 283

11.8 Guided Numerical Experiments 285

11.9 Summary 286

References 289

List of Algorithms 311

Index 313

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