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Uncertainty and Variability in Point Cloud Surface Data

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. Point Cloud Data (PCD). To model some underlying curve/surface. Sources of Uncertainty. Discrete sampling of a manifold

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Uncertainty and Variability in Point Cloud Surface Data

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  1. Uncertainty and Variability in Point Cloud Surface Data Mark Pauly1,2, Niloy J. Mitra1, Leonidas J. Guibas1 1 Stanford University 2 ETH, Zurich

  2. Point Cloud Data (PCD) To model some underlying curve/surface

  3. Sources of Uncertainty • Discrete sampling of a manifold • Sampling density • Features of the underlying curve/surface • Noise • Noise characteristics

  4. Uncertainty in PCD Reconstruction algorithm PCD curve/ surface But is this unique?

  5. A possible reconstruction Motivation

  6. or this one, Motivation

  7. or this ….. Motivation

  8. So look for probabilistic answers. Motivation priors !

  9. 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. 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. Likelihood that a surface interpolating P passes though a point x in space Set of all interpolating surfaces for PCD P Prior for a surface S in MP Notations

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

  13. How to get FP(x) ? • input : set of points P • implicitly assume some priors (geometric) • General idea: • Each point piP gives a local vote of likelihood • 1.Local likelihood depends on how well neighborhood of piagrees with x. • 2. Weight of vote depends on distance of pi from x.

  14. x x Estimates for x Interpolating curve more likely to pass through x Prior : preference to linear interpolation

  15. x qi(x) qi(x) x pi pj pi pj Estimates for x

  16. Likelihood Estimate by pi Distance weighing High if x agrees with neighbors of pi

  17. Likelihood Estimates Normalization constant

  18. Finally… O(N) O(1) Covariance matrix (independent of x !)

  19. Likelihood Map: Fi(x) likelihood Estimates by point pi

  20. Likelihood Map: Fi(x) Pinch point is pi High likelihood Estimates by point pi

  21. Likelihood Map: Fi(x) Distance weighting

  22. Likelihood Map: FP(x) likelihood O(N)

  23. Confidence Map • How much do we trust the local estimates? • Eigenvalue based approach • Likelihood estimates based on covariance matrices Ci • Tangency information implicitly coded in Ci

  24. Confidence Map denote the eigenvalues of Ci. Low value denotes high confidence (similar to sampling criteria proposed by Alexa et al. )

  25. Confidence Map confidence Red indicates regions with bad normal estimates

  26. Maps in 2d Likelihood Map Confidence Map

  27. Confidence Map Likelihood Map Maps in 3d

  28. Noise Model • Each point pi corrupted with additive noise i • zero mean • noise distribution gi • noise covariance matrix i • Noise distributions gi-s are assumed to be independent

  29. Noise Expected likelihood map simplifies to a convolution. Modified covariance matrix convolution

  30. Likelihood Map for Noisy PCD gi No noise With noise

  31. Scale Space Proportional to local sampling density

  32. Scale Space Good separation Bad estimates in noisy section

  33. Scale Space Cannot detect separation Better estimates in noisy section

  34. Application 1: Most Likely Surface Noisy PCD Likelihood Map

  35. Application 1: Most Likely Surface Active Contour Sharp features missed?

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

  37. Application 2: Re-sampling Add points in low confidence areas

  38. Application 2: Re-sampling

  39. Application 3: Weighted PCD PCD 1 PCD 2

  40. Application 3: Weighted PCD Merged PCD

  41. Application 3: Weighted PCD Merged PCD Too noisy Too smooth

  42. Application 3: Weighted PCD Confidence Map Likelihood Map

  43. Application 3: Weighted PCD Weighted PCD

  44. Application 3: Weighted PCD Weighted PCD Merged PCD

  45. 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. Questions ?

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