Image Analysis, Random Fields and Markov Chain Monte Carlo Methods (2^nd Ed., 2nd ed. Corr. 3rd printing. 2006) A Mathematical Introduction Stochastic Modelling and Applied Probability Series, Vol. 27

I. Bayesian Image Analysis: Introduction.- 1. The Bayesian Paradigm.- 1.1 Warming up for Absolute Beginners.- 1.2 Images and Observations.- 1.3 Prior and Posterior Distributions.- 1.4 Bayes Estimators.- 2. Cleaning Dirty Pictures.- 2.1 Boundaries and Their Information Content.- 2.2 Towards Piecewise Smoothing.- 2.3 Filters, Smoothers, and Bayes Estimators.- 2.4 Boundary Extraction.- 2.5 Dependence on Hyperparameters.- 3. Finite Random Fields.- 3.1 Markov Random Fields.- 3.2 Gibbs Fields and Potentials.- 3.3 Potentials Continued.- II. The Gibbs Sampler and Simulated Annealing.- 4. Markov Chains: Limit Theorems.- 4.1 Preliminaries.- 4.2 The Contraction Coefficient.- 4.3 Homogeneous Markov Chains.- 4.4 Exact Sampling.- 4.5 Inhomogeneous Markov Chains.- 4.6 A Law of Large Numbers for Inhomogeneous Chains.- 4.7 A Counterexample for the Law of Large Numbers.- 5. Gibbsian Sampling and Annealing.- 5.1 Sampling.- 5.2 Simulated Annealing.- 5.3 Discussion.- 6. Cooling Schedules.- 6.1 The ICM Algorithm.- 6.2 Exact MAP Estimation Versus Fast Cooling.- 6.3 Finite Time Annealing.- III. Variations of the Gibbs Sampler.- 7. Gibbsian Sampling and Annealing Revisited.- 7.1 A General Gibbs Sampler.- 7.2 Sampling and Annealing Under Constraints.- 8. Partially Parallel Algorithms.- 8.1 Synchronous Updating on Independent Sets.- 8.2 The Swendson-Wang Algorithm.- 9. Synchronous Algorithms.- 9.1 Invariant Distributions and Convergence.- 9.2 Support of the Limit Distribution.- 9.3 Synchronous Algorithms and Reversibility.- IV. Metropolis Algorithms and Spectral Methods.- 10. Metropolis Algorithms.- 10.1 Metropolis Sampling and Annealing.- 10.2 Convergence Theorems.- 10.3 Best Constants.- 10.4 About Visiting Schemes.- 10.5 Generalizations and Modifications.- 10.6 The Metropolis Algorithm in Combinatorial Optimization.- 11. The Spectral Gap and Convergence of Markov Chains.- 11.1 Eigenvalues of Markov Kernels.- 11.2 Geometric Convergence Rates.- 12. Eigenvalues, Sampling, Variance Reduction.- 12.1 Samplers and Their Eigenvalues.- 12.2 Variance Reduction.- 12.3 Importance Sampling.- 13. Continuous Time Processes.- 13.1 Discrete State Space.- 13.2 Continuous State Space.- V. Texture Analysis.- 14. Partitioning.- 14.1 How to Tell Textures Apart.- 14.2 Bayesian Texture Segmentation.- 14.3 Segmentation by a Boundary Model.- 14.4 Juleszs Conjecture and Two Point Processes.- 15. Random Fields and Texture Models.- 15.1 Neighbourhood Relations.- 15.2 Random Field Texture Models.- 15.3 Texture Synthesis.- 16. Bayesian Texture Classification.- 16.1 Contextual Classification.- 16.2 Marginal Posterior Modes Methods.- VI. Parameter Estimation.- 17. Maximum Likelihood Estimation.- 17.1 The Likelihood Function.- 17.2 Objective Functions.- 18. Consistency of Spatial ML Estimators.- 18.1 Observation Windows and Specifications.- 18.2 Pseudolikelihood Methods.- 18.3 Large Deviations and Full Maximum Likelihood.- 18.4 Partially Observed Data.- 19. Computation of Full ML Estimators.- 19.1 A Naive Algorithm.- 19.2 Stochastic Optimization for the Full Likelihood.- 19.3 Main Results.- 19.4 Error Decomposition.- 19.5 L2-Estimates.- VII. Supplement.- 20. A Glance at Neural Networks.- 20.1 Boltzmann Machines.- 20.2 A Learning Rule.- 21. Three Applications.- 21.1 Motion Analysis.- 21.2 Tomographic Image Reconstruction.- 21.3 Biological Shape.- VIII. Appendix.- A. Simulation of Random Variables.- A.1 Pseudorandom Numbers.- A.2 Discrete Random Variables.- A.3 Special Distributions.- B. Analytical Tools.- B.1 Concave Functions.- B.2 Convergence of Descent Algorithms.- B.3 A Discrete Gronwall Lemma.- B.4 A Gradient System.- C. Physical Imaging Systems.- D. The Software Package AntslnFields.- References.- Symbols.