Projecting points onto a point cloud with noise

1. Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008

2. Data Acquisition

3. Point clouds 25893

4. Point clouds 56194

5. topological Unorganized, connectivity-free

6. Surface Reconstruction

7. Noise

8. Definition of “onto” Close? Which?

9. Applications • Rendering • Parameterization • Simplification • Reconstruction • Area computation

10. References An extension on robust directed projection of points onto point clouds Ming-Cui Du, Yu-Shen Liu(CAD, In press) Parameterization-free Projection for Geometry Reconstruction Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer (SIGGRAPH ’07)

11. An extension on robust directed projection of points onto point clouds Ming-Cui Du, Yu-Shen Liu CAD, In press

12. About the author (刘玉身) • Postdoctor of Purdue University, Ph.D. in Tsinghua University. • 3 CAD, 1 The Visual Computer. • CAD, DGP .

14. Result

15. Previous work Parameterization of clouds of unorganized points using dynamic base surfaces (CAD, 04) Drawing curves onto a cloud of points for point-based Modeling (CAD, 05) Automatic least-squares projection of points onto point clouds with applications in reverse engineering (CAD, 06)

16. Weighted squared distances error

17. Weighted squared distances error

18. Proposition Terminating criterion: Simple, direct

19. Error analysis (Robustness) True location Independent of the cloud of points

20. Improved weight distance between pm and the axis stability

21. Improved weight

22. Reduce cloud • Setting the threshold: • 1.

24. Reduce cloud • Setting the threshold: • 1. • 2. Sort the weights in a decreasing order, then choose the nth weight as threshold. (n=N/100).

25. References • Robust diagnostic regression analysis. Atkinson A, Riani M. (Springer;2000) • Robust Moving Least-squares Fitting with Sharp Features Shachar Fleishman, Daniel Cohen-Or, Claudio T. Silva (SIGGRAPH ’05)

26. Forward vs. backward • Backward: Start from the entire sample set, then delete bad samples. • Forward: Begins witha small outlier-free subset, then refining byadding one goodsample at a time. (robust) • Adding of multiple points.

27. Algorithm • 1. Choose a small outlier-free subset Q. • 2. The solution is computed to the current subset Q. • 3. The point with the lowest residual in the remaining points is added into Q. (Forward) • 4. Repeat steps 2 and 3 until the error is larger than a predefined threshold. • 5. Compute the projection position for the final Q.

28. Least median of squares

29. LMS algorithm

30. Random sampling algorithm

31. Robustness • P: Probability of success. • g: Probability of selecting good sample. • k: Number of points are selected at random. (k = p) • T: Number of iteration. (T = 1000)

32. Forward search

36. Disturbing points

37. Disturbing points

38. Limitations

39. Limitations Use the first quartile (25%) instead of the median (50%)

40. Parameterization-free Projection for Geometry Reconstruction Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer (SIGGRAPH ’07)

41. About the author (Yaron Lipman) • Ph.D. student at Tel-Aviv University. His supervisors are Prof. David Levin and Prof. Daniel Cohen-Or. • SIGGRAPH, TOG, EG, SGP

42. About the author (Daniel Cohen-Or) • Professor at the School of Computer Science, Tel Aviv University. • Outstanding Technical Contributions Award 2005(EG) • TOG(19), CGF,TVCG, SGP, VC

43. About the author (David Levin) • Professor of Applied Mathematics, Tel-Aviv University. • Major interests: • Subdivision • Moving Least Squares • Numerical Integration • CAGD • Computer Graphics

44. About the author (David Levin) • Professor of Applied Mathematics, Tel-Aviv University. • Major interests: • Subdivision • Moving Least Squares • Numerical Integration • CAGD • Computer Graphics

45. Results

46. Locally Optimal Projection (LOP) θ(r), η(r)are fast decreasing functions.

47. Regularization

48. Multivariate L1 median

49. Optimization

50. Optimization