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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.
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https://hal.inria.fr/hal-02425686
Contributor : Jisu Kim <>
Submitted on : Tuesday, May 12, 2020 - 3:27:03 PM
Last modification on : Friday, April 30, 2021 - 10:04:19 AM

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  • HAL Id : hal-02425686, version 2
  • ARXIV : 1903.06955

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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⟩

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