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Documents authored by Mahmoud, Ahmed H.

Document
DOI: 10.4230/LIPIcs.SoCG.2018.1
Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

Authors: Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
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 alpha-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 epsilon-sample on the bounding surface, with a weak sigma-sparsity condition, we work with the balls of radius delta 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 epsilon, sigma and delta, 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.
Cite as

Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi. Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abdelkader_et_al:LIPIcs.SoCG.2018.1,
  author =	{Abdelkader, Ahmed and Bajaj, Chandrajit L. and Ebeida, Mohamed S. and Mahmoud, Ahmed H. and Mitchell, Scott A. and Owens, John D. and Rushdi, Ahmad A.},
  title =	{{Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.1},
  URN =		{urn:nbn:de:0030-drops-87147},
  doi =		{10.4230/LIPIcs.SoCG.2018.1},
  annote =	{Keywords: sampling conditions, surface reconstruction, polyhedral meshing, Voronoi}
}
Document
Multimedia Exposition
DOI: 10.4230/LIPIcs.SoCG.2018.77
VoroCrust Illustrated: Theory and Challenges (Multimedia Exposition)

Authors: Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.
Cite as

Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi. VoroCrust Illustrated: Theory and Challenges (Multimedia Exposition). In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 77:1-77:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{abdelkader_et_al:LIPIcs.SoCG.2018.77,
  author =	{Abdelkader, Ahmed and Bajaj, Chandrajit L. and Ebeida, Mohamed S. and Mahmoud, Ahmed H. and Mitchell, Scott A. and Owens, John D. and Rushdi, Ahmad A.},
  title =	{{VoroCrust Illustrated: Theory and Challenges}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{77:1--77:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.77},
  URN =		{urn:nbn:de:0030-drops-87903},
  doi =		{10.4230/LIPIcs.SoCG.2018.77},
  annote =	{Keywords: sampling, surface reconstruction, polyhedral meshing, Voronoi}
}
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