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

TL;DR

The paper proves that the medial axis of any closed, bounded subset of $\mathbb{R}^d$ is Lipschitz stable under ambient $C^{1,1}$ diffeomorphisms that preserve a bounding sphere, with explicit bounds on the Hausdorff distance to the medial axis of the transformed set. The authors harness Federer's reach theory, weakly tangent balls, and the closest-point projection to establish a main stability theorem, and then reformulate the result in a Banach-space setting to obtain clear, epsilon-dependent bounds. By extending prior work of Chazal and Soufflet beyond smooth manifolds to general closed sets, the work underpins robust geometric processing in imaging, vision, and astrophysics, and informs computability under realistic computational models. Future directions include extending the stability framework to compact Riemannian manifolds with bounded curvature and relaxing the bounding-sphere assumption to broaden applicability in geometric analysis.

Abstract

We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let $\mathcal{S} \subseteq \mathbb{R}^d$ be (fixed) closed set (that contains a bounding sphere). Consider the space of $C^{1,1}$ diffeomorphisms of $\mathbb{R}^d$ 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 $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F(\mathcal{S})$, is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C^2$ manifolds under $C^2$ ambient diffeomorphisms.

Figures (6)

• Figure 1: The closest point projection to the set ${\mathcal{S}}$ of four points in $\mathbb{R}^2$. When a point lies on the medial axis $\mathop{\mathrm{ax}}\nolimits({\mathcal{S}})$, the closest point projection consists of more points.
• Figure 2: In black the set ${\mathcal{S}}$ and its medial axis, in light blue the perturbed set and its medial axis. Since the lines $\mathop{\mathrm{ax}}\nolimits({\mathcal{S}})$ and $\mathop{\mathrm{ax}}\nolimits(F({\mathcal{S}}))$ are non-parallel, the Hausdorff distance between them is infinite. Hence it is impossible to give a bound on the distance between the medial axes without localizing.
• Figure 3: The projection range $d(p_1,w,\pi_{\mathcal{S}})$ equals infinity if $w\in \operatorname{UBP}(p_1,{\mathcal{S}})$ and zero otherwise. For $p_2, d(p_2,-v,\pi_{\mathcal{S}})$ is infinite, $d(p_2,v,\pi_{\mathcal{S}})$ is finite and non-zero, and $d(p_2,w,\pi_{\mathcal{S}}) = 0$ for $w\neq \pm v$. Hence, the set of unit projection vectors equals $\operatorname{UBP}(p_2,{\mathcal{S}}) = \{v, -v\}$.
• Figure 4: The (affine) generalized tangent and normal spaces of four points in the set ${\mathcal{S}}\subset \mathbb{R}^2$, in light blue and violet, respectively.
• Figure 5: Two families of balls weakly tangent to the set ${\mathcal{S}}\subset \mathbb{R}^2$ (in blue). Each family contains a unique maximal empty ball (in purple). Notice that the centre of the maximal empty ball weakly tangent at the point $p_1$ lies at the medial axis $\mathop{\mathrm{ax}}\nolimits({\mathcal{S}})$, while the centre of the maximal empty ball weakly tangent at the point $p_2$ only lies at its closure, $\overline{\mathop{\mathrm{ax}}\nolimits ( {\mathcal{S}})}$.

Theorems & Definitions (22)

• Remark 1
• Definition 2: Projection range
• Lemma 3: Theorem 4.8 (6) of Federer
• Definition 4: Unit back projection vectors
• Definition 5: Definitions 4.3 and 4.4 of Federer
• Theorem 6: Stability of the reach under ambient diffeomorphisms, Theorem 4.19 of Federer
• Definition 7: Weakly tangent sphere and ball
• Remark 8
• Lemma 9
• Lemma 10