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Markov Random Fields Probabilistic Models for Images

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Presentation on theme: "Markov Random Fields Probabilistic Models for Images"— Presentation transcript:

1 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

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

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

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

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

6 Markov Random Fields Second order neighborhood

7 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

8 Markov Random Fields What is

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

10 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

11 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

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

13 Ising model pair potential Challenge: auto logistic regression

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

15 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

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

18 Gibbs sampler for Ising model
Challenge: sample from Ising model

19 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

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

21 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

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

23 Maximum likelihood Given Challenge: prove it

24 Stochastic Gradient Given Generate Analysis by synthesis

25 Texture Modeling

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38 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

39 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

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

41 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

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

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