Article

Linear onesided stability of MAT for weakly injective 3D domain

Authors:

Hans-Peter SeidelAuthors Info & Claims

SMA '02: Proceedings of the seventh ACM symposium on Solid modeling and applications

Pages 344 - 355

https://doi.org/10.1145/566282.566332

Published: 17 June 2002 Publication History

Abstract

Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. However, it is known that MATs of 2D domains are stable if we view this phenomenon with the one-sided Hausdorff distance. This result depends on the fact that MATs are stable if the differences between them are measured with the recently introduced hyperbolic Hausdorff distance. In this paper, we extend the result for the one-sided stability of the MAT to a class of 3D domains called weakly injective, which contains many important 3D shapes typically appearing in solid modeling. Especially, the weakly injective 3D domains can have sharp features like corners or edges. In fact, by using the stability of the MAT under the hyperbolic Hausdorff distance, we obtain an explicit bound for the one-sided Hausdorff distance of the MAT of a weakly injective 3D domain with respect to that of a perturbed domain, which is linear with respect to the domain perturbation. We discuss some consequences of this result concerning the computation and the approximation of the medial axis transform of 3D objects.

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Cited By

Aaron C(2019)Convergence rate for the $\lambda $-Medial-Axis estimation under regularity conditionsElectronic Journal of Statistics10.1214/19-EJS158113:2Online publication date: 1-Jan-2019
https://doi.org/10.1214/19-EJS1581
Beristain AGraña MGonzalez A(2012)A Pruning Algorithm for Stable Voronoi SkeletonsJournal of Mathematical Imaging and Vision10.1007/s10851-011-0291-142:2-3(225-237)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1007/s10851-011-0291-1
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
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Index Terms

Linear onesided stability of MAT for weakly injective 3D domain
1. Computing methodologies
  1. Artificial intelligence
    1. Computer vision
      1. Computer vision representations
        Hierarchical representations
        Shape representations
  2. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models
2. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Approximation algorithms

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Published In

cover image ACM Conferences

SMA '02: Proceedings of the seventh ACM symposium on Solid modeling and applications

June 2002

424 pages

ISBN:1581135068

DOI:10.1145/566282

Conference Chairs:
Hans-Peter Seidel
Max-Planck-Institut für Informatik, Saarbrücken, Germany
,
Vadim Shapiro
University of Wisconsin, Madison, U.S.A.
,
Program Chairs:
Kunwoo Lee
Seoul National University, Seoul, Korea
,
Nick Patrikalakis
Massachsetts Institute of Technology, Cambridge, U.S.A.

Copyright © 2002 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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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2002

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SM02: ACM Symposium on Solid Modeling and Applications 2002

June 17 - 21, 2002

Saarbrücken, Germany

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SMA '02 Paper Acceptance Rate 43 of 93 submissions, 46%;

Overall Acceptance Rate 86 of 173 submissions, 50%

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Citations

Cited By

Aaron C(2019)Convergence rate for the $\lambda $-Medial-Axis estimation under regularity conditionsElectronic Journal of Statistics10.1214/19-EJS158113:2Online publication date: 1-Jan-2019
https://doi.org/10.1214/19-EJS1581
Beristain AGraña MGonzalez A(2012)A Pruning Algorithm for Stable Voronoi SkeletonsJournal of Mathematical Imaging and Vision10.1007/s10851-011-0291-142:2-3(225-237)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1007/s10851-011-0291-1
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
Attali DBoissonnat JEdelsbrunner H(2009)Stability and Computation of Medial Axes - a State-of-the-Art ReportMathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration10.1007/b106657_6(109-125)Online publication date: 25-Mar-2009
https://doi.org/10.1007/b106657_6
Cornea NSilver DMin P(2007)Curve-Skeleton Properties, Applications, and AlgorithmsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2007.100213:3(530-548)Online publication date: 1-May-2007
https://dl.acm.org/doi/10.1109/TVCG.2007.1002
Ferchichi SWang SGrira S(2007)New Algorithm to Extract Centerline of 2D Objects Based on ClusteringImage Analysis and Recognition10.1007/978-3-540-74260-9_33(364-374)Online publication date: 2007
https://doi.org/10.1007/978-3-540-74260-9_33
Cornea NSilver DMin P(2005)Curve-Skeleton ApplicationsVIS 05. IEEE Visualization, 2005.10.1109/VISUAL.2005.1532783(95-102)Online publication date: 2005
https://doi.org/10.1109/VISUAL.2005.1532783
Jain NKabul IGovindaraju NManocha DLin M(2005)Multi‐resolution collision handling for cloth‐like simulationsComputer Animation and Virtual Worlds10.1002/cav.10616:3-4(141-151)Online publication date: 15-Sep-2005
https://doi.org/10.1002/cav.106
Chazal FSoufflet R(2004)Stability and Finiteness Properties of Medial Axis and SkeletonJournal of Dynamical and Control Systems10.1023/B:JODS.0000024119.38784.ff10:2(149-170)Online publication date: 1-Apr-2004
https://dl.acm.org/doi/10.1023/B%3AJODS.0000024119.38784.ff
Foskey MLin MManocha DTurkiyyah GBrunet PElber GShapiro V(2003)Efficient computation of a simplified medial axisProceedings of the eighth ACM symposium on Solid modeling and applications10.1145/781606.781623(96-107)Online publication date: 16-Jun-2003
https://dl.acm.org/doi/10.1145/781606.781623
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