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

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 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]

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]

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.

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