1. Lena Gorelick Joint work with Frank Schmidt and Yuri Boykov Rochester Institute of Technology, Center of Imaging Science January 2013 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A AA 1

2. 2 S

3. 3 Fg Bg Intensity Probability Distribution Target Appearance Resulting Appearance

4. 4 Non-linear harder to optimize regional term Non-linear harder to optimize regional term

5.  complex appearance models  shape non-linear regional term 5

6.  Can be optimized with gradient descent  First order (linear) approximation models  We use more accurate non-linear approximation models based on trust region 6 Ben Ayed et al. Image Processing 2008, Foulonneau et al., PAMI 2006 Foulonneau et al., IJCV 2009 Ben Ayed et al. Image Processing 2008, Foulonneau et al., PAMI 2006 Foulonneau et al., IJCV 2009

7.  General class of non-linear regional functionals  Optimization algorithm based on trust region framework – Fast Trust Region 7

8.  Non-linear Regional Functionals  Overview of Trust Region Framework  Trust region sub-problem  Lagrangian Formulation for the sub-problem  Fast Trust Region method  Results 8

9. Volume Constraint 9

10. Bin Count Constraint 10

11.  Histogram Constraint 11

12.  Histogram Constraint 12

13.  Histogram Constraint 13

14.  Histogram Constraint 14

15.  Volume Constraint is a very crude shape prior  Can be encoded using a set of shape moments m pq 15 p+q is the order

16.  Volume Constraint is a very crude shape prior 16

17. 17

18.  Shape Prior Constraint Dist(), 18

19. 19

20.  Gradient Descent  First Order Taylor Approximation for R(S)  First Order approximation for B(S) (“curvature flow”)  Only robust with tiny steps  Slow  Sensitive to initialization 20 http://en.wikipedia.org/wiki/File:Level_set_method.jpg Ben Ayed et al. CVPR 2010, Freedman et al. tPAMI 2004 Ben Ayed et al. CVPR 2010, Freedman et al. tPAMI 2004

21.  Speedup via energy- specific methods  Bhattacharyya Distance  Volume Constraint  In contrast:  Fast optimization algorithm for general high-order energies  Based on more accurate non-linear approximation models Ben Ayed et al. CVPR 2010, Werner, CVPR2008 Woodford, ICCV2009 Ben Ayed et al. CVPR 2010, Werner, CVPR2008 Woodford, ICCV2009 21

22.  The goal is to optimize Trust region Trust Region Sub-Problem First Order Taylor for R(S) Keep quadratic B(S) First Order Taylor for R(S) Keep quadratic B(S) 22

23.  The goal is to optimize Trust Region Sub-Problem 23

24.  Constrained optimization minimize  Unconstrained Lagrangian Formulation minimize  Can be optimized globally using graph-cut Can be approximated with unary terms Boykov et al. ECCV 2006 Can be approximated with unary terms Boykov et al. ECCV 2006 24

25. 25 Newton step “Gradient Descent” Exact Line Search (ECCV12) Newton step “Gradient Descent” Exact Line Search (ECCV12)

26. 0 26

27. 0 Smallest discrete step that decreases the energy 27

28.  Repeat  Solve Trust Region Sub-problem around S 0 with radius d  Update solution S 0  Update Trust Region Size d  Until Convergence 28

29.  General Trust Region  Control of the distance constraint d  Lagrangian Formulation  Control of the Lagrange multiplier λ 29 λ

30. λ 30

31.  Simulated Gradient Descent  Exact Line-Search (ECCV 12)  Newton step  Fast Trust Region (CVPR 13) 31

32.  Repeat  Compute all break point solutions S λ  Evaluate actual energy E(S λ )  Choose the best S λ  Until convergence 32

33.  Repeat  Compute all break point solutions S λ  Evaluate actual energy E(S λ )  Choose the best S λ  Until convergence 0 33

34.  Repeat  Solve Trust Region Sub-Problem around S 0 with multiplier λ max  Update solution S 0  Until Convergence 0 34

35. Log-Lik. + length + volume Fast Trust Region Initializations Log-Lik. + length 35

36. Fast Trust Region 36 Log-Likelihoods No Shape Prior Second order geometric moments computed for the user provided initial ellipse

37. Init Fast Trust Region “Gradient Descent” Exact Line Search 37 Appearance model is obtained from the ground truth

38. 38 “ “ Fast Trust Region “Gradient Descent” Exact Line Search Appearance model is obtained from the ground truth

39. 39 Log-Lik. + length + Shape Prior Fast Trust Region Second order Tchebyshev moments computed for the user scribble

40. 40 BHA. + length Fast Trust Region Ground Truth Appearance model is obtained from the ground truth

41.  Multi-label Fast Trust Region  Binary shape prior:  affine-invariant Legendre/Tchebyshev moments  Learning class specific distribution of moments  Multi-label shape prior  moments of multi-label atlas map  Experimental evaluation and comparison between level-sets and FTR. 41

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