1. Sample Shuffling for Quality Hierarchic Surface Meshing

2. Surface Meshing

3. Sample Decimation

4. Surface Reconstruction

5. Foot data No decimation 

6. Medial axis Local feature size f(p)  -sampling   d(p)/f(p) Local feature size and sampling Amenta-Bern-Eppstein

7. Reconstruction Functional approach Tangent plane [HDeDDMS92] Natural Neighbors [BC00] Voronoi/Delaunay filtering Alpha shapes [EM94] Crust [AB98] Cocone [ACDL00]

8. Cocones Compute cocones Filter triangles whose duals intersect cocones Extract manifold Space spanned by vectors making angle   /8 with horizontal

9. Approximating density Need an approximation to Restricted Voronoi on S Need an approximation to local feature sizes

10. Radius and height radius r(p): distance from p to pº. height h(p): min distance to the poles C(p,  space spanned by vectors making angles <=  with horizontal. pº : point at max distance from p

11. Deletion and Insertion Vertex p is deleted if there is nearby sample point  p)/h(p) <  '. Insert p° if deletion of p destroys density r(p)/h(p) > 

12. Shuffling

13. Reconstruction (Dey-Giesen) Cocone (P,  Compute V P ; for each p  P if p  B compute T of triangles with duals intersecting C(p  endif endfor; Extract manifold; end B:= Boundary( P )

14. Main Theorem Theorem 1: For sufficiently small  an  sample P of S can be shuffled to Q s.t. a surface mesh M can be computed from Q with M is homeomorphic to S |M-S| = O(  f(p) for some p on S each triangle has aspect ratio O( 

15. Synthetic data (Parbol) No decimation, 8K pts   pts   pts

16. Synthetic data (Hyperbol) No decimation, 8K pts    

17. Synthetic data (Parcyl) No decimation, 6K pts  1K pts  0.7K pts

18. Experimental Data

19. Rocker data No decimation, 40K pts   pts   pts

20. Rocker data  pts  pts

21. Experimental data

22. Hip data No decim, 265k pts   pts   pts  126K pts   pts

23. Conclusions Introduced sample shuffling Achieves sample decimation retaining features Achieves quality meshing What about both coarsening and refining? How to take care of the boundaries? How to take care of noise? Softwares: www.cis.ohio-state.edu/~tamaldey/cocone.html