文章基本信息

标题：Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex
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作者：Jisu Kim ; Jaehyeok Shin ; Frdric Chazal 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：164
页码：54:1-54:19
DOI：10.4230/LIPIcs.SoCG.2020.54
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We derive conditions under which the reconstruction of a target space is topologically correct via the ÄOech 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 ÄOech complex. Second, we demonstrate the homotopy equivalence of a positive Î¼-reach set and its offsets. Applying these results to the restricted ÄOech complex and using the interleaving relations with the ÄOech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the ÄOech complex (or the Vietoris-Rips complex), in terms of the Î¼-reach. Our results sharpen existing results.
关键词：Computational topology; Homotopy reconstruction; Homotopy Equivalence; Vietoris-Rips complex; ÄOech complex; Reach; Î¼-reach; Nerve Theorem; Offset; Double offset; Consistency