Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm
Abstract
© Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens and Ahmad A. Rushdi; licensed under Creative Commons License CCBY 34th Symposium on Computational Geometry (SoCG 2018). We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from αshapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provablycorrect algorithm for this problem. Given an ϵsample on the bounding surface, with a weak σsparsity condition, we work with the balls of radius δ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of ϵ, σ and δ, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generatingmore »
 Authors:

 Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science
 Univ. of Texas, Austin, TX (United States). Dept. of Computer Science
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States). Center for Computing Research
 Univ. of California, Davis, CA (United States). Dept. of Electrical and Computer Engineering
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States). Center for Computing Research; Univ. of California, Davis, CA (United States). Dept. of Electrical and Computer Engineering
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
 OSTI Identifier:
 1502453
 Alternate Identifier(s):
 OSTI ID: 1526526
 Report Number(s):
 SAND20183812J
Journal ID: ISSN 18688969; 670144
 Grant/Contract Number:
 AC0494AL85000; AC0205CH11231
 Resource Type:
 Accepted Manuscript
 Journal Name:
 LIPIcsLeibniz International Proceedings in Informatics
 Additional Journal Information:
 Journal Volume: 99; Conference: 34. International Symposium on Computational Geometry (SoCG 2018), Budapest, Hungary, 1114 Jun 2018; Journal ID: ISSN 18688969
 Publisher:
 Dagstuhl Research Online Publication Server (DROPS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; sampling conditions; surface reconstruction; polyhedral meshing; Voronoi
Citation Formats
Abdelkader, Ahmed, Bajaja, Chandrajit L., Ebeida, Mohamed Salah, Mahmoud, Ahmed H., Mitchell, Scott A., Owens, John D., and Rushdi, Ahmad A. Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm. United States: N. p., 2018.
Web. doi:10.4230/LIPIcs.SoCG.2018.1.
Abdelkader, Ahmed, Bajaja, Chandrajit L., Ebeida, Mohamed Salah, Mahmoud, Ahmed H., Mitchell, Scott A., Owens, John D., & Rushdi, Ahmad A. Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm. United States. https://doi.org/10.4230/LIPIcs.SoCG.2018.1
Abdelkader, Ahmed, Bajaja, Chandrajit L., Ebeida, Mohamed Salah, Mahmoud, Ahmed H., Mitchell, Scott A., Owens, John D., and Rushdi, Ahmad A. Fri .
"Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm". United States. https://doi.org/10.4230/LIPIcs.SoCG.2018.1. https://www.osti.gov/servlets/purl/1502453.
@article{osti_1502453,
title = {Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm},
author = {Abdelkader, Ahmed and Bajaja, Chandrajit L. and Ebeida, Mohamed Salah and Mahmoud, Ahmed H. and Mitchell, Scott A. and Owens, John D. and Rushdi, Ahmad A.},
abstractNote = {© Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens and Ahmad A. Rushdi; licensed under Creative Commons License CCBY 34th Symposium on Computational Geometry (SoCG 2018). We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from αshapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provablycorrect algorithm for this problem. Given an ϵsample on the bounding surface, with a weak σsparsity condition, we work with the balls of radius δ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of ϵ, σ and δ, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to stateoftheart methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.},
doi = {10.4230/LIPIcs.SoCG.2018.1},
journal = {LIPIcsLeibniz International Proceedings in Informatics},
number = ,
volume = 99,
place = {United States},
year = {2018},
month = {6}
}