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/
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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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 | |
| Seminar: | 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. |
