The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms

2022/12/02 by Kouřimská, Hana Dal Poz, Lieutier, André, Wintraecken, Mathijs
#68U05 #Computational Geometry (cs.CG) #FOS: Computer and information sciences

paper · doi:10.48550/arxiv.2212.01118

Abstract

We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let S ⊆ ℝ^d be (fixed) closed set (that contains a bounding sphere). 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 some 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.

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