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12 - Size of the medial axis and stability of Federer's curvature measures

from PART 2 - SURVEYS AND RESEARCH PAPERS

Published online by Cambridge University Press:  05 August 2014

By
Quentin Mérigot
Edited by
Yann Ollivier ,
Hervé Pajot  and
Cedric Villani
Quentin Mérigot
Affiliation:
Université Grenoble Alpes and CNRS
Yann Ollivier
Affiliation:
Université de Paris XI
Hervé Pajot
Affiliation:
Université de Grenoble
Cedric Villani
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
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Summary

Abstract

In this chapter, we study the (d — 1)-volume and the covering numbers of the medial axis of a compact subset of ℝd. In general, this volume is infinite; however, the (d — 1)-volume and covering numbers of a filtered medial axis (the μ-medial axis) that is at distance greater than ε from the compact set can be explicitly bounded. The behavior of the bound we obtain with respect to μ, ε and the covering numbers of K is optimal.

From this result we deduce that the projection function on a compact subset K of ℝd depends continuously on the compact set K, in the L1 sense. This implies in particular that Federer's curvature measures of a compact subset of ℝd with positive reach can be reliably estimated from a Hausdorff approximation of this subset, regardless of any regularity assumption on the approximating subset.

Introduction

We are interested in the following question: given a compact subset K of ℝd with positive reach, and a Hausdorff approximation P of this set, is it possible to approximate Federer's curvature measures of K (see [9] or Section 12.2.2 for a definition) from P only? A positive answer to this question has been given in [8] using convex analysis. In this chapter, we show that such a result can also be deduced from a careful study of the “size” – that is, the covering numbers – of the medial axis.

The notion of medial axis, also known as ambiguous locus in Riemannian geometry, has many applications in computer science. In image analysis and shape recognition, the skeleton of a shape is often used as an idealized version of the shape, which is known to have the same homotopy type as the original shape [14].

Type
Chapter
Information
Optimal Transport
Theory and Applications
 , pp. 288 - 306
Publisher: Cambridge University Press
Print publication year: 2014

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References

[1] N., Amenta and M., Bern, Surface reconstruction by Voronoi filtering, Discrete and Computational Geometry 22 (1999), no. 4, 481-504.Google Scholar
[2] D., Attali, J.-D., Boissonnat, and H., Edelsbrunner, Stability and computation of medial axes: a state of the art report, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, Springer, 2007.
[3] P., Cannarsa, P., Cardaliaguet, and E., Giorgieri, Heilder regularity of the normal distance with an application to a PDE model for growing sandpiles, Transactions of the American Mathematical Society 359 (2007), no. 6, 2741.Google Scholar
[4] P., Cannarsa and C., Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhauser, 2004.
[5] M., Castelpietra and L., Rifford, Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry, ESAIM: Control, Optimisation and Calculus of Variations 16 (2010), no. 3, 695–718.Google Scholar
[6] F., Chazal, D., Cohen-Steiner, and A., Lieutier, A sampling theory for compact sets in Euclidean space, Discrete and Computational Geometry 41 (2009), no. 3, 461–479.Google Scholar
[7] F., Chazal, D., Cohen-Steiner, and A., Lieutier, Normal cone approximation and offset shape isotopy, Computational Geometry: Theory and Applications 42 (2009), no. 6-7, 566-581.Google Scholar
[8] F., Chazal, D., Cohen-Steiner, and Q., Merigot, Boundary measures for geometric inference, Journal of Foundations of Computational Mathematics 10 (2010), 221-240.Google Scholar
[9] H., Federer, Curvature measures, Transactions of the American Mathematical Society 93 (1959), no. 3, 418–491.Google Scholar
[10] J.H.G., Fu, Tubular neighborhoods in Euclidean spaces, Duke Mathematics Journal 52 (1985), no. 4, 1025–1046.Google Scholar
[11] J., Itoh and M., Tanaka, The Lipschitz continuity of the distance function to the cut locus, Transactions ofthe American Mathematical Society 353 (2001), no. 1, 21-40.Google Scholar
[12] H.W.E., Jung, Uber den kleinsten Kreis, der eine ebene Figur einschließt., Journal fur die Reine und Angewandte Mathematik 1910 (1910), no. 137, 310-313.Google Scholar
[13] Y., Li and L., Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Communications on Pure and Applied Mathematics 58 (2005), 0085–0146.Google Scholar
[14] A., Lieutier, Any open bounded subset of Rn has the same homotopy type as its medial axis, Computer Aided Geometric Design 36 (2004), no. 11, 1029-1046.Google Scholar
[15] Q., Merigot, Detection de structure geometrique dans les nuages de points, These de doctorat, Universite de Nice Sophia-Antipolis, 2009.
[16] A., Petrunin, Semiconcave functions in Alexandrov's geometry, Surveys in Differential Geometry. Vol. XI, International Press, Somerville, MA, 2007, pp. 137-201.
[17] C., Villani, Optimal Transport: Old and New, Springer, 2009.
[18] H., Weyl, On the volume of tubes, American Journal of Mathematics 61 (1939), no. 2, 461–472.Google Scholar
[19] T., Zamfirescu, On the cut locus in Alexandrov spaces and applications to convex surfaces, Pacific Journal of Mathematics 217 (2004), 375–386.Google Scholar

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