License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.54
URN: urn:nbn:de:0030-drops-122129
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12212/
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Kim, Jisu ; Shin, Jaehyeok ; Chazal, Frédéric ; Rinaldo, Alessandro ; Wasserman, Larry

Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

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LIPIcs-SoCG-2020-54.pdf (0.7 MB)


Abstract

We derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results.

BibTeX - Entry

@InProceedings{kim_et_al:LIPIcs:2020:12212,
  author =	{Jisu Kim and Jaehyeok Shin and Fr{\'e}d{\'e}ric Chazal and Alessandro Rinaldo and Larry Wasserman},
  title =	{{Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{54:1--54:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12212},
  URN =		{urn:nbn:de:0030-drops-122129},
  doi =		{10.4230/LIPIcs.SoCG.2020.54},
  annote =	{Keywords: Computational topology, Homotopy reconstruction, Homotopy Equivalence, Vietoris-Rips complex, Čech complex, Reach, μ-reach, Nerve Theorem, Offset, Double offset, Consistency}
}

Keywords: Computational topology, Homotopy reconstruction, Homotopy Equivalence, Vietoris-Rips complex, Čech complex, Reach, μ-reach, Nerve Theorem, Offset, Double offset, Consistency
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020
Supplementary Material: The code is available at https://github.com/jisuk1/nerveshape.


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