Kernel Methods for Implicit Surface Modeling B. Scholkopf, J. Giesen, S. Spalinger

1. Kernel Methods for Implicit Surface ModelingB. Scholkopf, J. Giesen, S. Spalinger r92922120 黃邦洪 r93922020 楊惠菁 r93922038 柯政宏 r93944009 劉弘偉

2. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

3. Introduction • Step 1: • Given a finite sample points belong to some hypersurface in Euclidean space R3 • The sample can be very large, noisy or sparse

4. Introduction • Step 2: • Use SVM to find an implicit surface • f is the decision function in SVM • The zero set of f is a surface that approximates the sample points • Single-Class SVM v.s. Slab SVM

5. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

6. Single-Class SVM • Primary Object function • Concept

7. Single-Class SVM • Dual Problem • Decision function

8. Single-Class SVM • SVM TOY example

9. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

10. Slab SVM • Primary object function • Concept

11. Slab SVM • Lagrangian dual optimization problem • Decision function

12. Slab SVM • SVM TOY example

13. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

14. Epsilon-SVR • Primary object function • Concept

15. Epsilon-SVR • Lagrangian dual optimization problem • The approximate function

16. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

17. Implement • Single-Class SVM • Use LIBSVM directly • Slab-SVM • Modify epsilon-SVR in LIBSVM to slab-SVM • The result is parameter-sensitive • Modification: • decision function • The result is good but we don’t really know how to explain

18. Outline • Introduction • Single-Class SVMs • Slab SVMs • Epsilon-SVR • Implement • Result

19. Result(1) - One-Class SVM • Bunny • Original: 35947 • One-Class: 2810

20. Result(1) - Slab SVM • Bunny • Left: 8307。Right: 1013

21. Result(1) - Slab SVM • Bunny • Predict accuracy: 97.74%

22. Result(2) - One-Class SVM • Cactus • Original: 3337 • One-Class: 1617

23. Result(2) - Slab SVM • Cactus • Left: 996 • Right: 817

24. Result(2) - Slab SVM • Cactus • Predict accuracy: 96.9434%

25. Result(3) - One-Class SVM • Dinosaur • Original: 56194 • One-Class: 44963

26. Result(3) - Slab SVM • Dinosaur • Left: 13005。Right: 1018

27. Result(3) - Slab SVM • Dinosaur • Predict accuracy: 97.43%

28. Result(4) - One-Class SVM • Knot • Original: 10000 • One-Class: 8807

29. Result(4) - Slab SVM • Knot • Left: 1176。Right: 1036

30. Result(4) - Slab SVM • Knot • Predict accuracy: 96.9%

31. Result(5) - One-Class SVM • Screwdriver • Original: 27152 • One-Class:21728

32. Result(5) - Slab SVM • Screwdriver • Left: 6574 • Right: 884

33. Result(5) - Slab SVM • Screwdriver • Predict accuracy: 97.1752%

34. Result(6) - One-Class SVM • Rockerarm • Original: 40177 • One-Class: 4868

35. Result(6) - Slab SVM • Rockerarm • Left: 2090 • Right: 1029

36. Result(6) - Slab SVM • Rocker arm • Predict accuracy: 96.18%

37. Result(7) - One-Class SVM • Hole • Original: 4000 • One-Class: 3224

38. Result(7) - Slab SVM • Hole • Left: 648 • Right:1039

39. Result(7) - Slab SVM • Hole • Predict accuracy: 93.475%

40. Q&A Thank You