Jan. 21, 2005

Markov Random Fields &Texture Analysis

Vincent CheungProbabilistic and Statistical Inference GroupElectrical & Computer EngineeringUniversity of Toronto

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Outline

W. Freeman, E. Pasztor, andO. Carmichael, “Learning Low-Level Vision”– Markov random fields– Belief propagation

S. Zhu, Y. Wu, and D. Mumford, “MinimaxEntropy Principle and its Application toTexture Modeling”– Minimax entropy principle– Feature pursuit

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Markov Random Field

An undirected graph– Nodes Variables– Edges Functions

Model the joint probability Perform inference

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Markov Network for Vision

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A Simple Markov Network

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Belief Propagation

MAP estimates can be computed for allthe xi simultaneously using “message-passing”

Factorization structure makes thispossible

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Belief Propagation Example

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Belief Propagation Summary

What a node thinks another should be

What a node thinks itself is

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Scene Candidate Patches

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Applications (1)

Super-Resolution Shading and reflectance Motion estimation

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Applications (2)

Transparency (A. Levin, A. Zomet andY. Weiss)

= +

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Minimax Entropy Principle

Let I be an image Assume that observed images, Ii,

i=1,..,m, are random samples from aprobability distribution f(I)

Goal is to estimate f(I) based on theobserved images

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Maximum Entropy Principle (1)

Maximize entropy to obtain the purestand simplest fusion of the observedfeatures and their statistics

Simplest model as possible

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Maximum Entropy Principle (2)

s.t.

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Minimum Entropy Principle

Minimize entropy to increase modelcomplexity by choosing a set of features

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Feature Pursuit

Inefficient to just try all possible set of Kfeatures

Use greedy approach– Add one feature to the model at a time– Choose feature that maximally decreases the

entropy– Find feature that is most poorly modeled by the

current model and add it to it’s repertoire so thatthe model can better represent this feature.

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Texture Modeling

Histogram of filters as features FRAME - Filter, Random field, And

Minimax Entropy Learn the filters Synthesize texture from the histogram

of the filter response

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Texture Modeling Example (1)

Obs 0 1

2 3 6

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Texture Modeling Example (2)