efficient simplification of point-sampled geometry

1. efficient simplification of point-sampled geometry Mark Pauly Markus Gross Leif Kobbelt ETH Zurich RWTH Aachen

2. outline • introduction • surface model & local surface analysis • point cloud simplification • hierarchical clustering • iterative simplification • particle simulation • measuring surface error • comparison • conclusions

3. acquisition rendering processing introduction • 3d content creation • many applications require coarser approximations • storage • transmission • editing • rendering • surface simplification for complexity reduction

4. acquisition rendering processing raw scans point cloud triangle mesh registration reconstruction introduction • 3d content creation

5. acquisition rendering processing raw scans point cloud triangle mesh registration reconstruction simplification reduced point cloud introduction • 3d content creation

6. acquisition rendering processing raw scans point cloud registration simplification reduced point cloud introduction • 3d content creation

7. idea: locally approximate surface with polynomial • compute reference plane • compute weighted least-squares fit polynomial surface model • moving least squares (mls) approximation • implicit surface definition using a projection operator • Gaussian weight function  locality

8. surface model • moving least squares (mls) approximation • implicit surface definition using a projection operator • idea: locally approximate surface with polynomial • compute reference plane • compute weighted least-squares fit polynomial • Gaussian weight function  locality

9. local surface analysis • local neighborhood (e.g. k-nearest)

10. covariance matrix centroid • eigenproblem local surface analysis • local neighborhood (e.g. k-nearest)

11. local surface analysis • local neighborhood (e.g. k-nearest) • eigenvectors span covariance ellipsoid • smallest eigenvector is least-squares normal • surface variation • measures deviation from tangent plane  curvature

12. local surface analysis • example original mean curvature variation n=20 variation n=50

13. surface simplification • hierarchical clustering • iterative simplification • particle simulation

14. hierarchical clustering • top-down approach using binary space partition • recursively split the point cloud if: • size is larger than a user-specified threshold or • surface variation is above maximum threshold • split plane defined by centroid and axis of greatest variation • replace clusters by centroid

15. covariance ellipsoid split plane centroid hierarchical clustering • 2d example root

16. hierarchical clustering • 2d example

17. hierarchical clustering • 2d example

18. hierarchical clustering • 2d example

19. hierarchical clustering 43 Clusters 436 Clusters 4,280 Clusters

20. surface simplification • hierarchical clustering • iterative simplification • particle simulation

21. iterative simplification • iteratively contracts point pairs • each contraction reduces the number of points by one • contractions are arranged in priority queue according to quadric error metric • quadric measures cost of contraction and determines optimal position for contracted sample • equivalent to QSlim except for definition of approximating planes

22. compute initial point-pair contraction candidates • compute fundamental quadrics iterative simplification • 2d example • compute edge costs

23. iterative simplification • 2d example priority queue edge cost

24. iterative simplification • 2d example priority queue edge cost

25. iterative simplification • 2d example priority queue edge cost

26. iterative simplification • 2d example priority queue edge cost

27. iterative simplification • 2d example priority queue edge cost

28. iterative simplification • 2d example priority queue edge cost

29. iterative simplification • 2d example priority queue edge cost

30. iterative simplification • 2d example priority queue edge cost

31. iterative simplification • 2d example priority queue edge cost

32. iterative simplification • 2d example priority queue edge cost

33. iterative simplification remaining contraction pairs 296,850 points 2,000 points

34. surface simplification • hierarchical clustering • iterative simplification • particle simulation

35. particle simulation • resample surface by distributing particles on the surface • particles move on surface according to inter-particle repelling forces • particle relaxation terminates when equilibrium is reached (requires damping) • can also be used for up-sampling!

36. mls surface particle simulation • 2d example

37. initialization • randomly spread particles particle simulation • 2d example

38. initialization • randomly spread particles • repulsion • linear repulsion force particle simulation • 2d example

39. projection • project particles onto surface particle simulation • 2d example • initialization • randomly spread particles • repulsion • linear repulsion force

40. particle simulation • 2d example • initialization • randomly spread particles • repulsion • linear repulsion force • projection • project particles onto surface

41. particle simulation original model 296,850 points uniform repulsion 2,000 points adaptive repulsion 3,000 points

42. measuring error • measure distance between two point-sampled surfaces S and S’ using a sampling approach • compute set Q of points on S • maximum error: • two-sided Hausdorff distance • mean error: • area-weighted integral of point-to-surface distances • size of Q determines accuracy of error measure

43. measuring error • d(q,S’) measures the distance of point q to surface S’ using the mls projection operator

44. comparison: surface error • error estimate for Michelangelo’s David simplified from 2,000,000 points to 5,000 points hierarchical clustering iterative simplification particle simulation

45. hierarchical clustering iterative simplification particle simulation comparison: performance • execution time as a function of input model size (simplification to 1% of input model size) time (sec) input size

46. hierarchical clustering iterative simplification particle simulation comparison: performance • execution time as a function of target model size (input: dragon, 435,545 points) time (sec) target size

47. simplification up-sampling smoothing effect

48. point cloud vs. mesh simplification simplification  reconstruction 3.5 sec. 2.45 sec reconstruction  simplification 112.8 sec. 3.5 sec.

49. conclusions • point cloud simplification can be useful to • reduce the complexity of geometric models early in the 3d content creation pipeline • build LOD surface representations • create surface hierarchies • the right method depends on the application • check out: www.pointshop3d.com • acknowledgement: European graduate program on combinatorics, geometry, and computation