1. Auxiliary Cuts for General Classes of Higher-Order Functionals 1 Ismail Ben Ayed, Lena Gorelick and Yuri Boykov

2. Standard Segmentation Functionals 2 S

3. Historic Data Linear terms are not enough 3 Standard model Learned distributions

4. Linear terms are not enough 3 Segmentation with log likelihoods Learned distributions

5. Linear terms are not enough 3 Standard model Target distributions Segmentation with log likelihoods

6. Linear terms are not enough 3 Standard model Target distributions

7. Segmentation with log likelihoods Linear terms are not enough 3 Standard model Target distributions

8. Segmentation with log likelihoods Linear terms are not enough 3 Standard model Learned distributions Obtained distributions

9. From log-likelihoods to higher-order terms 4 Rother et al. 06, Ben Ayed CVPR 10, Gorelick et al. ECCV 12, Jiang et al. CVPR 12

10. Standard vs. High-order 5 Input High-order Likelihoods High-order Input

11. Regional Functional Examples Volume Constraint 6

12. Bin Count Constraint Regional Functional Examples Volume Constraint 6

13. 7 Contribution: Bound Optimization of General Higher-Order Terms Non-Linear Combination of Linear Terms

14. Optimization Higher-order Pairwise Sub-modular 8

15. 9 Prior Art: General-Purpose Techniques Based on Functional Derivatives

16. 9 -- Level Sets: Ben Ayed et al. CVPR 2008 -- Line search: Gorelick et al. ECCV 2012 Can be slow

17. 9 -- Level Sets: Ben Ayed et al. CVPR 2008 -- Line search: Gorelick et al. ECCV 2012 F differentiable Prior Art: General-Purpose Techniques Based on Functional Derivatives

18. 9 -- Level Sets: Ben Ayed et al. CVPR 2008 -- Line search: Gorelick et al. ECCV 2012 Parameters? Prior Art: General-Purpose Techniques Based on Functional Derivatives

19. Prior Art: Specialized Techniques 9  Volume constraint: Werner, CVPR 2008  Norms between bin counts: Mukherjee et al. CVPR 2009, Jiang et al. CVPR 2012  Bhattacharyya : Ben Ayed et al. CVPR 2010, Punithakumar et al. SIAM 2012 Only particular cases

20. Auxiliary Function Optimization 10

21. Auxiliary Function Optimization 10

22. Auxiliary Function Optimization 10

23. Standard Tricks for Deriving Auxiliary Functions 12 Cauchy-Schwarz inequality Quadratic bound principle First-order expansion Jensen’s inequality  E.g.: EM is based on this approach

24. Jensen’s Inequality bound 11

25. Unary Terms Jensen’s Inequality bound

26. 11 Jensen’s Inequality bound

27. Auxiliary Function Derivation 13

28. Auxiliary Function Derivation 13

29. Auxiliary Function Derivation 13

30. Auxiliary Function Derivation 13 Constant

31. Auxiliary Function Derivation 13 Sum to 1

32. Auxiliary Function Derivation 13 Jensen’sLinear auxiliary function

33. Difference with other methods: the volume constraint case 14

34. Difference with other methods: the volume constraint case 14 Gradient Descent

35. Difference with other methods: the volume constraint case 14 Trust Region: Gorelick et al. CVPR 13

36. Difference with other methods: the volume constraint case 14 Auxiliary Cuts

37. General Form of the Functionals 15 Higher-order Sub-modular

38. General Form of the Functionals 15 Linear bound Sub-modular Higher-order Sub-modular

39. General Form of the Functionals 15 Graph Cut Higher-order Sub-modular Linear bound Sub-modular

40. Experimental examples

41. L2 Bin Count (Aux. Cuts vs. Level Sets) Level-Set, dt=1 Level-Set, dt=50 Level-Set, dt=1000 Init Aux. Cuts 16

42. User input Result User input Iter 2 User input ResultB-J Initial segment Iter. 3Iter. 2 17 L1 Bin Count

43. 18 inputs Input L2 Volume Constraint User input B-J B-J and Volume

44. Conclusions 19 Advantages: Derivative-free No optimization parameters, e.g., step size Easy to implement Never worsen the energy at each iteration

45. Conclusions 19 Limitations: The form of F should verify some conditions Limited to nested evolutions of segments

46. Conclusions 19 Extensions: More general forms of F Arbitrary evolutions of segments

47. 19 inputs Input Thanks