Theoretical Foundations and Numerical Methods for Sparse Recovery (eBook)

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Massimo Fornasier, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria.

Preface 6
Table of Contents 8
Compressive Sensing and Structured Random Matrices 12
1 Introduction 12
2 Recovery via l1-Minimization 14
3 Structured Random Matrices 27
4 Random Sampling in Bounded Orthonormal Systems 29
5 Partial Random Circulant Matrices 38
6 Tools from Probability Theory 40
7 Proof of Nonuniform Recovery Result for Bounded Orthonormal Systems 70
8 Proof of Uniform Recovery Result for Bounded Orthonormal Systems 76
9 Proof of Recovery Theorem for Partial Circulant Matrices 88
10 Appendix 94
Bibliography 96
Numerical Methods for Sparse Recovery 104
1 Introduction 104
2 An Introduction to Sparse Recovery 106
3 Numerical Methods for Compressed Sensing 118
4 Numerical Methods for Sparse Recovery 148
5 Large Scale Computing 166
Bibliography 206
Sparse Recovery in Inverse Problems 212
1 Introduction 212
2 Classical Inverse Problems 215
3 Nonlinear Approximation for Linear Ill-Posed Problems 223
4 Tikhonov Regularization with Sparsity Constraints 232
5 Iterated Shrinkage for Nonlinear Ill-Posed Problems 241
6 Projected Accelerated Steepest Descent for Nonlinear Ill- Posed Problems 251
Bibliography 270
An Introduction to Total Variation for Image Analysis 274
1 The Total Variation 274
2 Some Functionals Where the Total Variation Appears 294
3 Algorithmic Issues 308
4 Applications 328
A A Proof of Convergence 342
Bibliography 346