A New Voronoi-based Reconstruction Algorithm

1. A New Voronoi-based Reconstruction Algorithm CS 598 MJG Presented by: Ivan Lee N. Amenta, M. Bern, and M. Kamvysselis. In Proceedings of SIGGRAPH 98, pp. 415-422, July 1998.

2. What is Surface Reconstruction? • Set of points in 3-d space • Generate a mesh from the points http://web.mit.edu/manoli/www/crust/crust.html

3. What to talk about • Previous Work • Definitions • The Crust Algorithm • Comparison to Previous Work • Further Research

4. Previous work • Alpha shapes • Zero-set • Delaunay Sculpting

5. Alpha Shapes • Given a parameter, α, connect vertices within α units • Subset of Delaunay triangulation • Generalized convex hull Dey et al. [5]

6. Zero sets • Using input points, define implicit signed distance function • Distance function is interpolated and polygonized using marching cubes • Approximation rather than interpolation • e.g. Curless and Levoy paper

7. Delaunay Sculpting • Remove tetrahedra from Delaunay triangulation • Associate values to tetrahedra and eliminate largest valued ones

8. First, some definitions • Voronoi cell • A cell where all points in the cell are closer to a given sample point than any other point • Voronoi diagram • A space partitioned into Voronoi cells • Voronoi vertex • A point equidistant to d+1 sample points in Rd Amenta et al. [1]

9. Some more definitions • Delaunay triangulation • Dual of Voronoi diagram • Each triangle’s circumcircle contains no other vertices Amenta et al. [1] • Medial axis • Set of points with more than one closest point Amenta et al. [1]

10. And finally… • Poles • Farthest Voronoi vertices for a sample point that are on opposite sides • Crust • Shell created to represent the surface Amenta et al. [1]

11. On to the algorithm • Compute the Voronoi diagram of S, where S is the set of sample points • For each sample point, find the poles on opposite sides of the sample point • Compute Delaunay triangulation of S U P, where P is the set of all poles • Keep all triangles in which all three vertices are sample points

12. On to the algorithm • Delete triangles whose normals differ too much from the direction vectors from the triangle vertices to their poles • Orient triangles consistently with its neighbors and remove sharp dihedral edges to create a manifold

13. Advantages • No need for experimental parameters in basic algorithm • Not sensitive to distribution of points

14. Disadvantages • Sampling of points needs to be dense • Undersampling causes holes • Does not handle sharp edges • Can be fixed by picking two farthest vertices as poles, regardless of being on opposite sides • Boundaries cause problems • But not always

15. Comparison to Previous Work • Alpha Shapes • No need for experimental values • Zero set • Essentially low-pass filtering, lose information • Delaunay sculpting • Very similar to this algorithm

16. Hull • Command line implementation of Voronoi regions in C • Downloadable at: http://cm.bell-labs.com/netlib/voronoi/hull.html

17. Proposed Future Research in 1998 • Fixing problems with boundaries and sharp edges • Using sample points with normals • Allows for sparser samplings • Lossless mesh compression

18. What’s happened since then? • Co-cones (Amenta et al. [2]) • Cone with apex at sample point and aligned with poles • Algorithm only requires one Voronoi diagram computation • Eliminates normal trimming step • Still does not support sharp edges

19. What’s happened since then? • The power crust (Amenta et al. [3]) • Use polar balls and power diagrams to separate the inside and outside of the surface • Approximates medial axis

20. What’s happened since then? • Detecting Undersampling (Dey and Giesen [4]) • Fat Voronoi cells or dissimilarly oriented neighboring Voronoi cells imply undersampling. Add sample points to accommodate • This accounts for sharp edges and boundaries • Tight Co-cone • After detecting undersampling, stitch up holes

21. Summary • “New” Crust Algorithm • Advantages over previous algorithms • Advancements to fix original crust algorithm’s flaws

22. Thank you

23. References • [0] N. Amenta and M. Bern. Surface Reconstruction by Voronoi Filtering. Annual Symposium on Computational Geometry, pp. 39-48, 1998. • [1] N. Amenta, M. Bern, and M. Kamvysselis. A New Voronoi-Based Surface Reconstruction Algorithm. In Proceedings of SIGGRAPH 98, pp. 415-422, July 1998. • [2] N. Amenta, S. Choi, T. Dey, and N. Leekha. A Simple Algorithm for Homeomorphic Surface Reconstruction. Internation Journal of Computational Geometry and its Applications, vol. 12 (1-2), pp. 125-141, 2002. • [3] N. Amenta, S. Choi, and R. Kolluri. The Power Crust. ACM Symposium on Solid Modeling and Applications, pp 249-266, 2001.

24. References • [4] T. Dey and J. Giesen. Detecting Undersampling in Surface Reconstruction. In proceedings for 17th ACM Annual Symposium for Computational Geometry, pp. 257-263, 2001. • [5] T. Dey and S. Goswami. Tight Cocone: A Water-Tight Surface Reconstructor. In Proceedings for 8th ACM Symposium for Solid Modeling Applications, pp. 127-134, 2003. • [6] T. Dey, J. Giesen, and M. John. Alpha-Shapes and Flow Shapes are Homotopy Equivalent. STOC ’03, 2003. • [7] H. Edelsbrunner and E. Mücke. Three-dimensional Alpha Shapes. ACM Transactions on Graphics, 13(1):43-72, 1994.

25. Questions and Discussion