1. Uncertainty and Variability in Point Cloud Surface Data Mark Pauly 1,2, Niloy J. Mitra 1, Leonidas J. Guibas 1 1 Stanford University 2 ETH, Zurich

2. Uncertainty and Variability in PCD Point Cloud Data (PCD) To model some underlying curve/surface

3. Uncertainty and Variability in PCD Sources of Uncertainty Discrete sampling of a manifold  Sampling density  Features of the underlying curve/surface Noise  Noise characteristics

4. Uncertainty and Variability in PCD Uncertainty in PCD PCDcurve/ surface Reconstruction algorithm But is this unique?

5. Uncertainty and Variability in PCD Motivation A possible reconstruction

6. Uncertainty and Variability in PCD Motivation or this one,

7. Uncertainty and Variability in PCD Motivation or this …..

8. Uncertainty and Variability in PCD Motivation So look for probabilistic answers. priors !

9. Uncertainty and Variability in PCD What are our Goals? Try to evaluate properties of the set of (interpolating) curves/surfaces. Answers in probabilistic sense. Capture the uncertainty introduced by point representation.

10. Uncertainty and Variability in PCD Related Work Surface reconstruction reconstruct the connectivity get a possible mesh representation PCD for geometric modeling MLS based algorithms Kalaiah and Varshney PCA based statistical model Tensor voting

11. Uncertainty and Variability in PCD Notations Likelihood that a surface interpolating P passes though a point x in space Prior for a surface S in M P Set of all interpolating surfaces for PCD P

12. Uncertainty and Variability in PCD Expected Value Surface prior ? Characteristic function Set of all interpolating surfaces ? Conceptually we can define likelihood as

13. Uncertainty and Variability in PCD How to get F P (x) ? input : set of points P implicitly assume some priors (geometric) General idea: Each point p i  P gives a local vote of likelihood 1. Local likelihood depends on how well neighborhood of p i agrees with x. 2. Weight of vote depends on distance of p i from x.

14. Uncertainty and Variability in PCD Estimates for x x x Interpolating curve more likely to pass through x Prior : preference to linear interpolation

15. Uncertainty and Variability in PCD Estimates for x x x pipi pipi pjpj pjpj qi(x)qi(x) qi(x)qi(x)

16. Uncertainty and Variability in PCD Likelihood Estimate by p i High if x agrees with neighbors of p i Distance weighing

17. Uncertainty and Variability in PCD Likelihood Estimates Normalization constant

18. Uncertainty and Variability in PCD Finally… Covariance matrix (independent of x !) O(N) O(1)

19. Uncertainty and Variability in PCD Likelihood Map: F i (x) Estimates by point p i likelihood

20. Uncertainty and Variability in PCD Likelihood Map: F i (x) Estimates by point p i High likelihood Pinch point is p i

21. Uncertainty and Variability in PCD Likelihood Map: F i (x) Distance weighting

22. Uncertainty and Variability in PCD Likelihood Map: F P (x) likelihood O(N)

23. Uncertainty and Variability in PCD Confidence Map How much do we trust the local estimates?  Eigenvalue based approach Likelihood estimates based on covariance matrices C i Tangency information implicitly coded in C i

24. Uncertainty and Variability in PCD Confidence Map denote the eigenvalues of C i. Low value denotes high confidence (similar to sampling criteria proposed by Alexa et al. )

25. Uncertainty and Variability in PCD Confidence Map confidence Red indicates regions with bad normal estimates

26. Uncertainty and Variability in PCD Maps in 2d Likelihood MapConfidence Map

27. Uncertainty and Variability in PCD Maps in 3d Likelihood Map Confidence Map

28. Uncertainty and Variability in PCD Noise Model Each point p i corrupted with additive noise  i zero mean noise distribution g i noise covariance matrix  i Noise distributions g i -s are assumed to be independent

29. Uncertainty and Variability in PCD Noise Expected likelihood map simplifies to a convolution. Modified covariance matrix convolution

30. Uncertainty and Variability in PCD Likelihood Map for Noisy PCD No noiseWith noise gigi

31. Uncertainty and Variability in PCD Scale Space Proportional to local sampling density

32. Uncertainty and Variability in PCD Scale Space Bad estimates in noisy section Good separation

33. Uncertainty and Variability in PCD Scale Space Better estimates in noisy section Cannot detect separation

34. Uncertainty and Variability in PCD Application 1: Most Likely Surface Noisy PCDLikelihood Map

35. Uncertainty and Variability in PCD Application 1: Most Likely Surface Sharp features missed? Active Contour

36. Uncertainty and Variability in PCD Application 2: Re-sampling Add points in low confidence areas Given the shape !! Confidence map

37. Uncertainty and Variability in PCD Application 2: Re-sampling Add points in low confidence areas

38. Uncertainty and Variability in PCD Application 2: Re-sampling

39. Uncertainty and Variability in PCD Application 3: Weighted PCD PCD 1PCD 2

40. Uncertainty and Variability in PCD Application 3: Weighted PCD Merged PCD

41. Uncertainty and Variability in PCD Application 3: Weighted PCD Too noisyToo smoothMerged PCD

42. Uncertainty and Variability in PCD Application 3: Weighted PCD Likelihood Map Confidence Map

43. Uncertainty and Variability in PCD Application 3: Weighted PCD Weighted PCD

44. Uncertainty and Variability in PCD Application 3: Weighted PCD Merged PCD Weighted PCD

45. Uncertainty and Variability in PCD Future Work Soft classification of medical data Analyze variability in family of shapes Incorporate context information to get better priors Statistical modeling of surface topology

46. Uncertainty and Variability in PCD Questions ?