 | Starck J.-L. Sparse image and signal processing: wavelets, curvelets, morphological diversity / J.-L.Starck, F.Murtagh, J.M.Fadili. - Cambridge; New York: Cambridge University Press, 2010. - xvii, 316 p., [16] p. of plates: ill. (some col.). - Ref.: p.289-309. - Ind.: p.313-316. - ISBN 978-0-521-11913-9
|
Acronyms page .................................................. 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
Color Plates follow page ...................................... 148
|
|
