1. Adaptive Cooperative Systems Chapter 6 Markov Random Fields
6.12 ~ 6.13
Summary by Byoung-Hee Kim
Biointelligence Lab
School of Computer Sci. & Eng.
Seoul National University

2. (C) 2009 SNU CSE Biointelligence Lab
Contents
6.12 Mean-field annealing
Mechanical models, graduated nonconvexity, and mean-field theory
Mean-field annealing
Convex functions and Jensen’s inequality
The mean-field annealing algorithm
6.13 Graduated nonconvexity
Weak continuity constraints
Elimination of the line process
Nonconvexity and convex approximation
(C) 2009 SNU CSE Biointelligence Lab

3. Potentials and Deterministic Approaches
One of our main objectives in studying image-processing applications is to explore ways of constructing useful potential functions
Sources of inspiration: regularization theory, mechanical models
Potentials using mean-field annealing and graduated nonconvexity
Deterministic approximations to the stochastic method of Geman and Geman
Mean field formalism and resulting potentials
Corresponding graduated nonconvexity potential functions
Problem domain: image reconstruction problem
(C) 2009 SNU CSE Biointelligence Lab

4. Image Reconstruction Problem
Goal
Simultaneously remove noise and strengthen and preserve boundaries
(C) 2009 SNU CSE Biointelligence Lab

5. (C) 2009 SNU CSE Biointelligence Lab
Mean Field Annealing
We approximate the posterior hamiltonian by the mean-field hamiltoniam
Illustration with the simple, piecewise constant image model
Starting point: the posterior hamiltonian
Choose a piecewise constant prior potential
(C) 2009 SNU CSE Biointelligence Lab

6. (C) 2009 SNU CSE Biointelligence Lab
For continuous-valued variables
Neglecting independent terms
Mean-field hamiltonian
Form: a Gibbs distribution with a Gaussian normalization
We want to determine the form of the mean-field x
(C) 2009 SNU CSE Biointelligence Lab

7. Convex Functions and Jensen’s Inequality
We want to determine the form of the mean-field x
How to find the approximation?
Self-consistency procedure based on Jensen’s inequality
Upper bound to the free energy
 Minimize this w.r.t x to get Emf that best approximates E
(C) 2009 SNU CSE Biointelligence Lab

8. The Mean-Field Annealing Algorithm
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9. Graduated Nonconvexity
Mechanical approach to solving optimization problems in visual surface reconstruction
Key elements
Having the cooperativity intrinsic to mechanical systems (rods, springs, flexible plates, …)
These systems develop global order from local interactions
Posses all the properties expected of cooperative systems
Examples
Spring-loaded dipole model
Having piecewise continuity
Analogous to piecewise smoothness of curves and surfaces in discrete systems
Having nonconvex optimization  piecewise convex `
(C) 2009 SNU CSE Biointelligence Lab

10. (C) 2009 SNU CSE Biointelligence Lab