1. Energy functions f(p)  {0,1} : Ising model Can solve fast with graph cuts V( ,  ) = T[  ] : Potts model NP-hard Closely related to Multiway Cut Problem Local minimum via expansion move algorithm

2. Stereo Left imageRight image Example:

3. Potts model for stereo

4. Multiway cut problem

5. Multiway cuts correspond to labelings for Potts model

6. n-link t-link

7. Ideal results

8. Expansion moves Green expansion move

9. Expansion moves in action initial solution -expansion For each move we choose expansion that gives the largest decrease in the energy: binary energy minimization subproblem

10. Binary sub-problem Input labelingExpansion move Binary image

11. Expansion move energy Goal: find the binary image with lowest energy Binary image energy depends on f, 

12. Original energy function

13. Binary image notation Also depends on f,  !

14. Binary data energy (given f,  ) Sum this function over pixels p Original (non-binary) data energy:

15. Binary smoothness energy Sum this function over neighboring pixels p,q Original (non-binary) smoothness energy:

16. Binary energy minimization Finding the cheapest expansion move requires minimizing Can be done efficiently by graph cuts!

17. Graph cuts solution This can be done as long as V has a specific form (works for arbitrary D ) Regularity constraint: for f,  we need

18. Regular choices of V Suppose that V is a metric Then what?

19. Metric choices of V Potts model Truncated linear model Linear model

20. Potts model Truncated linear model Linear model Quadratic model RobustNot robust

21. Robustness matters! linear V truncated linear V

22. Potts regularity (the hard way) Case f(p)=f(q)=  : Case f(p)= , f(q)  : Case f(p) , f(q)  : √ √ √