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

(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

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

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

(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

(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

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

The Mean-Field Annealing Algorithm (C) 2009 SNU CSE Biointelligence Lab

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

(C) 2009 SNU CSE Biointelligence Lab