โ Scribed by Gerhard Winkler
No coin nor oath required. For personal study only.
This second edition of G. Winkler's successful book on random field approaches to image analysis, related Markov Chain Monte Carlo methods, and statistical inference with emphasis on Bayesian image analysis concentrates more on general principles and models and less on details of concrete applications. Addressed to students and scientists from mathematics, statistics, physics, engineering, and computer science, it will serve as an introduction to the mathematical aspects rather than a survey. Basically no prior knowledge of mathematics or statistics is required.
The second edition is in many parts completely rewritten and improved, and most figures are new. The topics of exact sampling and global optimization of likelihood functions have been added.
Cover
Applications of Mathematics, vol. 27
Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction (Second Edition)
Copyright
Dedication
Preface to the Second Edition
Preface to the First Edition
Contents
Introduction
Part 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
Part 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 ยฑ
6. Cooling Schedules
6.1 The ICM Algorithm
6.2 Exact MAP Estimation Versus Fast Cooling
6.3 Finite Time Annealing
Part 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
Part 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
Part 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 Julesz's 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
Part 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
Part 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
Part 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 AntsInFields
References
Symbols
Index
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