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.

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

What to talk about

• Previous Work

• Definitions

• The Crust Algorithm

• Comparison to Previous Work

• Further Research

Previous work

• Alpha shapes

• Zero-set• Delaunay Sculpting

Alpha Shapes

• Given a parameter, α, connect vertices within α units

• Subset of Delaunay triangulation

• Generalized convex hull

Dey et al. [5]

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

Delaunay Sculpting

• Remove tetrahedra from Delaunay triangulation• Associate values to tetrahedra and eliminate

largest valued ones

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]

Some more definitions

• Delaunay triangulation– Dual of Voronoi diagram– Each triangle’s

circumcircle contains no other vertices

• Medial axis– Set of points with more

than one closest point

Amenta et al. [1]

Amenta et al. [1]

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]

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

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

Advantages

• No need for experimental parameters in basic algorithm

• Not sensitive to distribution of points

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

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

Hull

• Command line implementation of Voronoi regions in C

• Downloadable at:http://cm.bell-labs.com/netlib/voronoi/hull.html

Proposed Future Research in 1998

• Fixing problems with boundaries and sharp edges

• Using sample points with normals– Allows for sparser samplings

• Lossless mesh compression

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

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

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

Summary

• “New” Crust Algorithm

• Advantages over previous algorithms

• Advancements to fix original crust algorithm’s flaws

Thank you

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.

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.

Questions and Discussion