The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms
Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.
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LIPIcs
Abstract
We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.
Publishing Year
Date Published
2024-06-01
Proceedings Title
40th International Symposium on Computational Geometry
Acknowledgement
This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.
Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Cô d'Azur.
We are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.
Volume
293
Article Number
69
Conference
SoCG: Symposium on Computational Geometry
Conference Location
Athens, Greece
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IST-REx-ID
Cite this
Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.69
Kourimska, H., Lieutier, A., & Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.69
Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.69.
H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293.
Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.
Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” 40th International Symposium on Computational Geometry, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.69.
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