Markov Random Fields with Efficient Approximations

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Markov Random Fields with Efficient
Approximations

• Yuri Boykov, Olga Veksler, Ramin Zabih
• Computer Science Department
• CORNELL UNIVERSITY

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Introduction

• MAP-MRF approach
• (Maximum Aposteriori Probability estimation of
  MRF)
• Bayesian framework suitable for problems in
  Computer Vision (Geman and Geman, 1984)
• Problem High computational cost. Standard
  methods (simulated annealing) are very slow.

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Outline of the talk

• Models where MAP-MRF estimation is equivalent to
  min-cut problem on a graph
• generalized Potts model
• linear clique potential model
• Efficient methods for solving the corresponding
  graph problems
• Experimental results
• stereo, image restoration

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MRF framework in the context of stereo

• image pixels (vertices)

• neighborhood relationships (n-links)

MRF defining property
Hammersley-Clifford Theorem
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MAP estimation of MRF configuration
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Energy minimization
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Generalized Potts model
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Static clues - selecting
Stereo Image White Rectangle in front of
the black background
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Minimization of E(f) via graph cuts
p-vertices (pixels)
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Multiway cut
vertices V pixels terminals
Remove a subset of edges C
edges E n-links t-links

• C is a multiway cut if terminals are separated
  in G(C)

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Main Result (generalized Potts model)

• Under some technical conditions on
  the multiway min-cut C on G gives___
  that minimizes E( f ) - the posterior energy
  function for the generalized Potts model.

• Multiway cut Problem find minimum cost
  multiway cut C graph G

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Solving multiway cut problem

• Case of two terminals
• max-flow algorithm (Ford, Fulkerson 1964)
• polinomial time (almost linear in practice).
• NP-complete if the number of labels gt2
• (Dahlhaus et al., 1992)
• Efficient approximation algorithms that are
  optimal within a factor of 2

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Our algorithm
Initialize at arbitrary multiway cut C
1. Choose a pair of terminals
2. Consider connected pixels
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Our algorithm
Initialize at arbitrary multiway cut C
1. Choose a pair of terminals
2. Consider connected pixels
3. Reallocate pixels between two terminals
by running max-flow algorithm
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Our algorithm
Initialize at arbitrary multiway cut C
1. Choose a pair of terminals
2. Consider connected pixels
3. Reallocate pixels between two terminals
by running max-flow algorithm
4. New multiway cut C is obtained
Iterate until no pair of terminals improves the
cost of the cut
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Experimental results (generalized Potts model)

• Extensive benchmarking on synthetic images and on
  real imagery with dense ground truth
• From University of Tsukuba
• Comparisons with other algorithms

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Synthetic example
Image
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Real imagery with ground truth
Ground truth
Our results
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Comparison with ground truth
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Gross errors (gt 1 disparity)
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Comparative results normalized correlation
Data
Gross errors
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Statistics
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Related work (generalized Potts model)

• Greig et al., 1986 is a special case of our
  method (two labels)
• Two solutions with sensor noise (function g)
  highly restricted
• Ferrari et al., 1995, 1997

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Linear clique potential model
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Minimization of via graph cuts
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Main Result (linear clique potential model)

• Under some technical conditions on
  the min-cut C on gives that
  minimizes - the posterior energy
  function for the linear clique potential model.

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Related work (linear clique potential model)

• Ishikawa and Geiger, 1998
• earlier independently obtained a very similar
  result on a directed graph
• Roy and Cox, 1998
• undirected graph with the same structure
• no optimality properties since edge weights are
  not theoretically justified

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Experimental results (linear clique potential
model)

• Benchmarking on real imagery with dense ground
  truth
• From University of Tsukuba
• Image restoration of synthetic data

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Ground truth stereo image
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Image restoration