Article

A sampling theory for compact sets in Euclidean space

Authors:

Frédéric Chazal,

David Cohen-Steiner,

André LieutierAuthors Info & Claims

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

Pages 319 - 326

https://doi.org/10.1145/1137856.1137904

Published: 05 June 2006 Publication History

Abstract

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points and critical values of distance functions.

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Cited By

Aamari EBerenfeld CLevrard C(2023)Optimal reach estimation and metric learningThe Annals of Statistics10.1214/23-AOS228151:3Online publication date: 1-Jun-2023
https://doi.org/10.1214/23-AOS2281
Bürgisser PCucker FTonelli-Cueto J(2019)Computing the Homology of Semialgebraic Sets. I: Lax FormulasFoundations of Computational Mathematics10.1007/s10208-019-09418-yOnline publication date: 6-May-2019
https://doi.org/10.1007/s10208-019-09418-y
Aamari ELevrard C(2018)Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold ReconstructionDiscrete & Computational Geometry10.1007/s00454-017-9962-z59:4(923-971)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00454-017-9962-z
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Index Terms

A sampling theory for compact sets in Euclidean space
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

June 2006

500 pages

ISBN:1595933409

DOI:10.1145/1137856

Program Chairs:
Nina Amenta
University of California at Davis
,
Otfried Cheong
Korea Advanced Institute of Science and Technology

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 05 June 2006

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SoCG06

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SoCG06: 22nd Annual Symposium on Computational Geometry

June 5 - 7, 2006

Arizona, Sedona, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Aamari EBerenfeld CLevrard C(2023)Optimal reach estimation and metric learningThe Annals of Statistics10.1214/23-AOS228151:3Online publication date: 1-Jun-2023
https://doi.org/10.1214/23-AOS2281
Bürgisser PCucker FTonelli-Cueto J(2019)Computing the Homology of Semialgebraic Sets. I: Lax FormulasFoundations of Computational Mathematics10.1007/s10208-019-09418-yOnline publication date: 6-May-2019
https://doi.org/10.1007/s10208-019-09418-y
Aamari ELevrard C(2018)Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold ReconstructionDiscrete & Computational Geometry10.1007/s00454-017-9962-z59:4(923-971)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00454-017-9962-z
Cazals FCohen-Steiner D(2012)Reconstructing 3D compact setsComputational Geometry: Theory and Applications10.1016/j.comgeo.2011.07.00545:1-2(1-13)Online publication date: 1-Jan-2012
https://dl.acm.org/doi/10.1016/j.comgeo.2011.07.005
Chazal FGuibas LOudot SSkraba P(2011)Scalar Field Analysis over Point Cloud DataDiscrete & Computational Geometry10.5555/3116672.311713346:4(743-775)Online publication date: 1-Dec-2011
https://dl.acm.org/doi/10.5555/3116672.3117133
Reem DHurtado Fvan Kreveld M(2011)The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the SitesProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998234(254-263)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998234
Cheng SDey TRamos E(2010)Delaunay Refinement for Piecewise Smooth ComplexesDiscrete & Computational Geometry10.5555/3116259.311638143:1(121-166)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/3116259.3116381
Chazal FCohen-Steiner DMérigot Q(2010)Boundary Measures for Geometric InferenceFoundations of Computational Mathematics10.5555/3115493.311588210:2(221-240)Online publication date: 1-Apr-2010
https://dl.acm.org/doi/10.5555/3115493.3115882
Chen DGuibas LHershberger JSun JCharikar M(2010)Road network reconstruction for organizing pathsProceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms10.5555/1873601.1873706(1309-1320)Online publication date: 17-Jan-2010
https://dl.acm.org/doi/10.5555/1873601.1873706
Boissonnat JGuibas LOudot S(2009)Manifold Reconstruction in Arbitrary Dimensions Using Witness ComplexesDiscrete & Computational Geometry10.5555/3116258.311637042:1(37-70)Online publication date: 1-Jul-2009
https://dl.acm.org/doi/10.5555/3116258.3116370
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