article

Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes

Authors:

Dominique Attali,

André Lieutier,

David SalinasAuthors Info & Claims

Computational Geometry: Theory and Applications, Volume 46, Issue 4

Pages 448 - 465

https://doi.org/10.1016/j.comgeo.2012.02.009

Published: 01 May 2013 Publication History

Abstract

Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of R^n two real-valued functions c"X and h"X defined on R"+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on c"P(t) entails that the Rips complex of P at scale r collapses to the Cech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on h"X over some interval guarantees a topologically correct reconstruction of the shape X either with a Cech complex of P or with a Rips complex of P. Regarding the reconstruction with Cech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive @m-reach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions.

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Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
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cover image Computational Geometry: Theory and Applications

Computational Geometry: Theory and Applications Volume 46, Issue 4

May, 2013

91 pages

ISSN:0925-7721

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2012.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 May 2013

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Cited By

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Cohen-Steiner DLieutier AVuillamy J(2022)Lexicographic Optimal Homologous Chains and Applications to Point Cloud TriangulationsDiscrete & Computational Geometry10.1007/s00454-022-00432-668:4(1155-1174)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s00454-022-00432-6
Berenfeld CHarvey JHoffmann MShankar K(2022)Estimating the Reach of a Manifold via its Convexity Defect FunctionDiscrete & Computational Geometry10.1007/s00454-021-00290-867:2(403-438)Online publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1007/s00454-021-00290-8
Watanabe KMaeda KOgawa THaseyama M(2022)Summarizing Data Structures with Gaussian Process and Robust Neighborhood PreservationMachine Learning and Knowledge Discovery in Databases10.1007/978-3-031-26419-1_10(157-173)Online publication date: 19-Sep-2022
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Adamaszek MFrick FVakili A(2017)On Homotopy Types of Euclidean Rips ComplexesDiscrete & Computational Geometry10.1007/s00454-017-9916-558:3(526-542)Online publication date: 1-Oct-2017
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Attali DBauer UDevillers OGlisse MLieutier A(2015)Homological reconstruction and simplification in R 3Computational Geometry: Theory and Applications10.1016/j.comgeo.2014.08.01048:8(606-621)Online publication date: 1-Sep-2015
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