 | Variational methods in imaging / O.Scherzer et al.; ed. by S.S.Antman, J.E.Marsden, L.Sirovich. - New York: Springer, 2009. - xiii, 320 p.: ill. - (Applied mathematical sciences; Vol.167). - Incl. bibl. ref. - Ind.: p.315-320. - ISBN 978-0-387-30931-6
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Part I Fundamentals of Imaging
1 Case Examples of Imaging ..................................... 3
1.1 Denoising ............................................... 3
1.2 Chopping and Nodding .................................... 6
1.3 Image Inpainting ........................................ 8
1.4 X-ray-Based Computerized Tomography .................... 10
1.5 Thermoacoustic Computerized Tomography ................. 13
1.6 Schlieren Tomography ................................... 24
2 Image and Noise Models ...................................... 27
2.1 Basic Concepts of Statistics ........................... 27
2.2 Digitized (Discrete) Images ............................ 31
2.3 Noise Models ........................................... 33
2.4 Priors for Images ...................................... 36
2.5 Maximum A Posteriori Estimation ........................ 43
2.6 MAP Estimation for Noisy Images ........................ 46
Part II Regularization
3 Variational Regularization Methods for the Solution
of Inverse Problems ......................................... 53
3.1 Quadratic Tikhonov Regularization in Hilbert Spaces .... 54
3.2 Variational Regularization Methods in Banach Spaces .... 60
3.3 Regularization with Sparsity Constraints ............... 79
3.4 Linear Inverse Problems with Convex Constraints ........ 89
3.5 Schlieren Tomography .................................. 109
3.6 Further Literature on Regularization Methods for
Inverse Problems ...................................... 112
4 Convex Regularization Methods for Denoising ................ 115
4.1 The *-Number .......................................... 220
4.2 Characterization of Minimizers ........................ 125
4.3 One-dimensional Results ............................... 131
4.4 Taut String Algorithm ................................. 137
4.5 Mumford-Shah Regularization ........................... 151
4.6 Recent Topics on Denoising with Variational Methods ... 155
5 Variational Calculus for Non-convex Regularization ......... 159
5.1 Direct Methods ........................................ 159
5.2 Relaxation on Sobolev Spaces .......................... 162
5.3 Relaxation on BV ...................................... 167
5.4 Applications in Non-convex Regularization ............. 172
5.5 One-dimensional Results ............................... 178
5.6 Examples .............................................. 180
6 Semi-group Theory and Scale Spaces ......................... 185
6.1 Linear Semi-group Theory .............................. 186
6.2 Non-linear Semi-groups in Hilbert Spaces .............. 190
6.3 Non-linear Semi-groups m Banach Spaces ................ 193
6.4 Axiomatic Approach to Scale Spaces .................... 197
6.5 Evolution by Non-convex Energy Functionals ............ 200
6.6 Enhancing ............................................. 202
7 Inverse Scale Spaces ....................................... 205
7.1 Iterative Tikhonov Regularization ..................... 206
7.2 Iterative Regularization with Bregman Distances ....... 209
7.3 Recent Topics on Evolutionary Equations for Inverse
Problems .............................................. 217
Part III Mathematical Foundations
8 Functional Analysis ........................................ 221
8.1 General Topology ...................................... 221
8.2 Locally Convex Spaces ................................. 224
8.3 Bounded Linear Operators and Functionals .............. 227
8.4 Linear Operators in Hilbert Spaces .................... 231
8.5 Weak and Weak* Topologies ............................. 234
8.6 Spaces of Differentiable Functions .................... 237
9 Weakly Differentiable Functions ............................ 239
9.1 Measure and Integration Theory ........................ 239
9.2 Distributions and Distributional Derivatives .......... 248
9.3 Geometrical Properties of Functions and Domains ....... 250
9.4 Sobolev Spaces ........................................ 254
9.5 Convolution ........................................... 261
9.6 Sobolev Spaces of Fractional Order .................... 262
9.7 Bochner Spaces ........................................ 263
9.8 Functions of Bounded Variation ........................ 265
10 Convex Analysis and Calculus of Variations ................. 273
10.1 Convex and Lower Semi-continuous Functionals .......... 274
10.2 Fenchel Duality and Subdifferentiability .............. 276
10.3 Duality Mappings ...................................... 280
10.4 Differentiability of Functionals and Operators ........ 281
10.5 Derivatives of Integral Functionals on Lp(Ω) .......... 284
References .................................................... 287
Nomenclature .................................................. 309
Index ......................................................... 315
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