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Energy functions

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. Example:. Left image. Right image. Stereo. Potts model for stereo. Multiway cut problem.

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Energy functions

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  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. Example: Left image Right image Stereo

  3. Potts model for stereo

  4. Multiway cut problem

  5. Multiway cuts correspond to labelings for Potts model

  6. t-link n-link

  7. Ideal results

  8. Green expansion move Expansion moves

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

  10. Binary image Binary sub-problem Input labeling Expansion move

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

  15. Original (non-binary) smoothness energy: Sum this function over neighboring pixels p,q 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. Potts model Linear model Truncated linear model Metric choices of V

  20. Potts model Linear model Quadratic model Truncated linear model Robust Not robust

  21. truncated linear V linear V Robustness matters!

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

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