License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.58
URN: urn:nbn:de:0030-drops-138579
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13857/
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Sheehy, Donald R.

A Sparse Delaunay Filtration

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LIPIcs-SoCG-2021-58.pdf (1 MB)

Abstract

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ℝ^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^⌈d/2⌉). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed.

BibTeX - Entry

@InProceedings{sheehy:LIPIcs.SoCG.2021.58,
  author =	{Sheehy, Donald R.},
  title =	{{A Sparse Delaunay Filtration}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13857},
  URN =		{urn:nbn:de:0030-drops-138579},
  doi =		{10.4230/LIPIcs.SoCG.2021.58},
  annote =	{Keywords: Delaunay Triangulation, Persistent Homology, Sparse Filtrations}
}

Keywords: Delaunay Triangulation, Persistent Homology, Sparse Filtrations
Collection: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue Date: 2021
Date of publication: 02.06.2021

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