Markov Random Fields Probabilistic Models for Images

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Presentation transcript:

Markov Random Fields Probabilistic Models for Images Applications in Image Segmentation and Texture Modeling Ying Nian Wu UCLA Department of Statistics IPAM July 22, 2013

Outline Basic concepts, properties, examples Markov chain Monte Carlo sampling Modeling textures and objects Application in image segmentation

Markov Chains Pr(future|present, past) = Pr(future|present) future past | present Markov property: conditional independence limited dependence Makes modeling and learning possible

Markov Chains (higher order) Temporal: a natural ordering Spatial: 2D image, no natural ordering

Markov Random Fields Markov Property all the other pixels Nearest neighborhood, first order neighborhood From Slides by S. Seitz - University of Washington

Markov Random Fields Second order neighborhood

Markov Random Fields Can be generalized to any undirected graphs (nodes, edges) Neighborhood system: each node is connected to its neighbors neighbors are reciprocal Markov property: each node only depends on its neighbors Note: the black lines on the left graph are illustrating the 2D grid for the image pixels they are not edges in the graph as the blue lines on the right

Markov Random Fields What is

Hammersley-Clifford Theorem normalizing constant, partition function potential functions of cliques Cliques for this neighborhood From Slides by S. Seitz - University of Washington

Hammersley-Clifford Theorem Gibbs distribution a clique: a set of pixels, each member is the neighbor of any other member Cliques for this neighborhood From Slides by S. Seitz - University of Washington

Hammersley-Clifford Theorem Gibbs distribution a clique: a set of pixels, each member is the neighbor of any other member Cliques for this neighborhood ……etc, note: the black lines are for illustrating 2D grids, they are not edges in the graph

Ising model Cliques for this neighborhood From Slides by S. Seitz - University of Washington

Ising model pair potential Challenge: auto logistic regression

Gaussian MRF model Challenge: auto regression continuous pair potential Challenge: auto regression

Sampling from MRF Models Markov Chain Monte Carlo (MCMC) Gibbs sampler (Geman & Geman 84) Metropolis algorithm (Metropolis et al. 53) Swedeson & Wang (87) Hybrid (Hamiltonian) Monte Carlo

Gibbs Sampler Repeat: Randomly pick a pixel Simple one-dimension distribution Repeat: Randomly pick a pixel Sample given the current values of

Gibbs sampler for Ising model Challenge: sample from Ising model

Metropolis Algorithm Repeat: energy function Repeat: Proposal: Perturb I to J by sample from K(I, J) = K(J, I) If change I to J otherwise change I to J with prob

Metropolis for Ising model Ising model: proposal --- randomly pick a pixel and flip it Challenge: sample from Ising model

Modeling Images by MRF Ising model Hidden variables, layers, RBM Exponential family model, log-linear model maximum entropy model unknown parameters features (may also need to be learned) reference distribution

Modeling Images by MRF Given How to estimate Maximum likelihood Pseudo-likelihood (Besag 1973) Contrastive divergence (Hinton)

Maximum likelihood Given Challenge: prove it

Stochastic Gradient Given Generate Analysis by synthesis

Texture Modeling

Modeling image pixel labels as MRF (Ising) MRF for Image Segmentation Modeling image pixel labels as MRF (Ising) Bayesian posterior 1 real image label image Slides by R. Huang – Rutgers University

Model joint probability region labels image pixels model param. image-label compatibility Function enforcing Data Constraint label-label compatibility Function enforcing Smoothness constraint label image local Observations neighboring label nodes Slides by R. Huang – Rutgers University

MRF for Image Segmentation Slides by R. Huang – Rutgers University

Inference in MRFs Classical Gibbs sampling, simulated annealing Iterated conditional modes State of the Art Graph cuts Belief propagation Linear Programming Tree-reweighted message passing Slides by R. Huang – Rutgers University

Summary MRF, Gibbs distribution Gibbs sampler, Metropolis algorithm Exponential family model