Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2020

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

Abstract

We derive conditions under which the reconstruction of a target space is topologically correct via the $\check{C}$ech 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 $\check{C}$ech complex. Second, we demonstrate the homotopy equivalence of a positive $\mu$-reach set and its offsets. Applying these results to the restricted $\check{C}$ech complex and using the interleaving relations with the $\check{C}$ech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the $\check{C}$ech complex (or the Vietoris-Rips complex), in terms of the $\mu$-reach. Our results sharpen existing results.
Fichier principal
Vignette du fichier
lipics-v2019-sample-article.pdf (1.02 Mo) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-02425686 , version 1 (31-12-2019)
hal-02425686 , version 2 (12-05-2020)

Identifiers

Cite

Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. SoCG 2020 - 36th International Symposium on Computational Geometry, Jun 2020, Zurich, Switzerland. ⟨hal-02425686v2⟩
247 View
321 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More