1. Markov Random Fields with Efficient Approximations
Yuri Boykov, Olga Veksler, Ramin Zabih
Computer Science Department
CORNELL UNIVERSITY

2. (Maximum Aposteriori Probability estimation of MRF)
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.

3. 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

4. MRF framework in the context of stereo
image pixels (vertices)
neighborhood relationships (n-links)
- disparity at pixel p
- configuration
MRF defining property:
Hammersley-Clifford Theorem:

5. MAP estimation of MRF configuration
Observed data
Bayes rule
Likelihood
function
(sensor noise)
Prior
(MRF model)

6. Energy minimization
Find that minimizes the Posterior Energy Function :
Data term
(sensor noise)
Smoothness term
(MRF prior)

7. Generalized Potts model
Clique potential
Penalty for discontinuity at (p,q)
Energy function

8. Static clues - selecting
Stereo Image: White Rectangle in front of
the black background
Disparity configurations minimizing energy E( f ):

9. Minimization of E(f) via graph cuts
Terminals (possible disparity labels)
Cost of n-link
Cost of t-link
p-vertices
(pixels)

10. Multiway cut C is a multiway cut if terminals are separated in G(C)
vertices V = pixels + terminals
Remove a subset of edges C
edges E = n-links + t-links
Graph G = <V,E>
Graph G(C) = <V, E-C >
C is a multiway cut if terminals are separated in G(C)
A multiway cut C yields some disparity configuration

11. 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

12. 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 >2
(Dahlhaus et al., 1992)
Efficient approximation algorithms that are optimal within a factor of 2

13. 2. Consider connected pixels
Our algorithm
Initialize at arbitrary multiway cut C
1. Choose a pair of terminals
2. Consider connected pixels

14. 2. Consider connected pixels
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

15. 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

16. 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

17. Synthetic example
Correlation
Multiway cut
Image

18. Real imagery with ground truth
Our results

19. Comparison with ground truth

20. Gross errors (> 1 disparity)

21. Comparative results: normalized correlation
Gross errors
Data

22. Statistics

23. Related work (generalized Potts model)
Greig et al., is a special case of our method (two labels)
Two solutions with sensor noise (function g) highly restricted
Ferrari et al., 1995, 1997

24. Linear clique potential model
Penalty for discontinuity at (p,q)
Energy function

25. Minimization of via graph cuts
Cost of t-link
{p,q} part of graph
cut C
Cost of n-link
a cut C yields some
configuration

26. 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.

27. 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

28. Experimental results (linear clique potential model)
Benchmarking on real imagery with dense ground truth
From University of Tsukuba
Image restoration of synthetic data

29. Ground truth stereo image
Generalized Potts model
Linear clique potential model

30. Image restoration Noisy diamond image Generalized Potts model
Linear clique potential model