efficient simplification of point-sampled geometry

1. efficient simplification of point-sampled geometry Jeffrey Sukharev CMPS260 Final Project From the paper “Efficient Simplification of Point-Sampled Surfaces” by Mark Pauly, Markus Gross, Leif Kobbelt

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 • 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 used for 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 used for locality

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

10. covariance matrix centroid eigenvalue problem 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 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 • incremental clustering • hierarchical clustering • iterative simplification • particle simulation

14. incremental clustering • Clustering by growing regions • start with a random seed point • successively add nearest points to cluster until cluster reaches desired maximum size • the growth of clusters can also be limited be surface variation and in that way the curvature adaptive clustering is achieved.

15. incremental clustering • Incremental growth leads to some fragmentation. Therefore stray samples need to be added to closest clusters at the end of the run.

16. incremental clustering • each cluster is replaced by its centroid Origina model 34,384 points Simplified model 1,000 pts

17. incremental clustering • Results from my incremental clustering implementation. 1,222 pts 35,000 pts

18. surface simplification • incremental clustering • hierarchical clustering • iterative simplification • particle simulation

19. 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

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

21. hierarchical clustering • 2d example

22. hierarchical clustering • 2d example

23. hierarchical clustering • 2d example

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

25. surface simplification • incremental clustering • hierarchical clustering • iterative simplification • particle simulation

26. 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

27. surface simplification • incremental clustering • hierarchical clustering • iterative simplification • particle simulation

28. particle simulation • Method proposed by Turk G. (for polygonal surfaces) • 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 • can also be used for up-sampling!

29. 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

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

31. conclusions • point cloud simplification can be useful to • reduce the complexity of geometric models early in the 3d content creation pipeline • create surface hierarchies • References • Mark Pauly et al “Efficient Simplification of Point Sampled Surfaces” • Mark Alexa et al “Point Set Surfaces” • Levin D. “Mesh-independent surface interpolation”