Introduction p. 1
Bayesian Image Analysis: Introduction
The Bayesian Paradigm p. 9
Warming up for Absolute Beginners p. 10
linages and Observations p. 14
Prior and Posterior Distributions p. 20
Bayes Estimators p. 24
Cleaning Dirty Pictures p. 29
Boundaries and Their Information Content p. 30
Towards Piccewiso Smoothing p. 31
Filters, Smoothers, and Bayes Estimators p. 41
Boundary Extraction p. 48
Dependence on Hyperparameters p. 50
Finite Random Fields p. 55
Markov Random Fields p. 55
Gibbs Fields and Potentials p. 60
Potentials Continued p. 66
The Gibbs Sampler and Simulated Annealing
Markov Chains: Limit Theorems p. 75
Preliminaries p. 75
The Contraction Coefficient p. 81
Homogeneous Markov Chains p. 84
Exact Sampling p. 92
Inhomogeneous Markov Chains p. 102
A Law of Large Numbers for Inhomogeneous Chains p. 106
A Counterexample for the Law of Large Numbers p. 110
Gibbsian Sampling and Annealing p. 113
Sampling p. 113
Simulated Annealing p. 120
Discussion p. 125
Cooling Schedules p. 129
The ICM Algorithm p. 129
Exact MAP Estimation Versus Fast Cooling p. 131
Finite Time Annealing p. 139
Variations of the Gibbs Sampler
Gibbsian Sampling and Annealing Revisited p. 143
A General Gibbs Sampler p. 143
Sampling and Annealing Under Constraints p. 147
Partially Parallel Algorithms p. 153
Synchronous Updating on Independent Sets p. 154
The Swendson-Wang Algorithm p. 156
Synchronous Algorithms p. 159
Invariant Distributions and Convergence p. 159
Support of the Limit Distribution p. 163
Synchronous Algorithms and Reversibility p. 168
Metropolis Algorithms and Spectral Methods
Metropolis Algorithms p. 179
Metropolis Sampling and Annealing p. 179
Convergence Theorems p. 180
Best Constants p. 185
About Visiting Schemes p. 187
Generalizations and Modifications p. 191
The Metropolis Algorithm in Combinatorial Optimization p. 193
The Spectral Gap and Convergence of Markov Chains p. 197
Eigenvalues of Markov Kernels p. 197
Geometric Convergence Rates p. 201
Eigenvalues, Sampling, Variance Reduction p. 203
Samplers and Their Eigenvalues p. 203
Variance Reduction p. 204
Importance Sampling p. 206
Continuous Time Processes p. 209
Discrete State Space p. 210
Continuous State Space p. 211
Texture Analysis
Partitioning p. 217
How to Tell Textures Apart p. 217
Bayesian Texture Segmentation p. 221
Segmentation by a Boundary Model p. 223
Julesz's Conjecture and Two Point Processes p. 225
Random Fields and Texture Models p. 231
Neighbourhood Relations p. 233
Random Field Texture Models p. 235
Texture Synthesis p. 240
Bayesian Texture Classification p. 243
Contextual Classification p. 244
Marginal Posterior Modes Methods p. 246
Parameter Estimation
Maximum Likelihood Estimation p. 251
The Likelihood Function p. 252
Objective Functions p. 257
Consistency of Spatial ML Estimators p. 263
Observation Windows and Specifications p. 263
Pseudolikelihood Methods p. 268
Large Deviations and Full Maximum Likelihood p. 277
Partially Observed Data p. 279
Computation of Pull ML Estimators p. 281
A Naive Algorithm p. 281
Stochastic Optimization for the Full Likelihood p. 285
Main Results p. 286
Error Decomposition p. 291
L2-Estimates p. 295
Supplement
A Glance at Neural Networks p. 301
Boltzmann Machines p. 302
A Learning Rule p. 306
Three Applications p. 313
Motion Analysis p. 313
Tomographic Image Reconstruction p. 317
Biological Shape p. 321
Appendix
Simulation of Random Variables p. 327
Pseudorandom Numbers p. 327
Discrete Random Variables p. 331
Special Distributions p. 334
Analytical Tools p. 343
Concave Functions p. 343
Convergence of Descent Algorithms p. 346
A Discrete Gronwall Lemma p. 347
A Gradient System p. 347
Physical Imaging Systems p. 351
The Software Package AntsInFields p. 355
References p. 357
Symbols p. 379
Index p. 381