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 , July 1998.

2. What is Surface Reconstruction?
Set of points in 3-d space
Generate a mesh from the points

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

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 Medial axis
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 Crust
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:

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 , 1998.
[1] N. Amenta, M. Bern, and M. Kamvysselis. A New Voronoi-Based Surface Reconstruction Algorithm. In Proceedings of SIGGRAPH 98, pp , 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 , 2002.
[3] N. Amenta, S. Choi, and R. Kolluri. The Power Crust. ACM Symposium on Solid Modeling and Applications, pp , 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 , 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 , 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