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

Authors: Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


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.

Cite as

Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kim_et_al:LIPIcs.SoCG.2020.54,
  author =	{Kim, Jisu and Shin, Jaehyeok and Chazal, Fr\'{e}d\'{e}ric and Rinaldo, Alessandro and Wasserman, Larry},
  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 =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.54},
  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, \v{C}ech complex, Reach, \mu-reach, Nerve Theorem, Offset, Double offset, Consistency}
}
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