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