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Title: 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 CC-BY 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 α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an ϵ-sample on the bounding surface, with a weak σ-sparsity 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 » a bounded number of sites in its interior. Compared to state-of-the-art 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.« less

Authors:
 [1];  [2];  [3];  [4];  [3];  [5];  [4]
  1. Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science
  2. Univ. of Texas, Austin, TX (United States). Dept. of Computer Science
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  4. Univ. of California, Davis, CA (United States). Dept. of Electrical and Computer Engineering
  5. Sandia National Lab. (SNL-NM), 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. (SNL-NM), 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):
SAND-2018-3812J
Journal ID: ISSN 1868-8969; 670144
Grant/Contract Number:  
AC04-94AL85000; AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
LIPIcs-Leibniz International Proceedings in Informatics
Additional Journal Information:
Journal Volume: 99; Conference: 34. International Symposium on Computational Geometry (SoCG 2018), Budapest, Hungary, 11-14 Jun 2018; Journal ID: ISSN 1868-8969
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 CC-BY 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 α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an ϵ-sample on the bounding surface, with a weak σ-sparsity 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 state-of-the-art 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 = {LIPIcs-Leibniz International Proceedings in Informatics},
number = ,
volume = 99,
place = {United States},
year = {2018},
month = {6}
}