Compensated Convexity, Multiscale Medial Axis Maps and Sharp Regularity of the Squared Distance Function

TL;DR

This paper introduces a new mathematical model for medial axis detection using quadratic multiscale transforms, proving regularity results and stability properties that improve understanding and localization of medial axes in geometric objects.

Contribution

The paper develops a stable multiscale medial axis map based on compensated convexity, providing sharp regularity results and stability under Hausdorff distance for geometric analysis.

Findings

Proves $C^{1,1}$-regularity of squared-distance functions outside medial axis neighborhoods.

Introduces the multiscale medial axis map $M_ ext{lambda}$ and its limit $M_ ext{infinity}$, which characterizes the medial axis.

Establishes stability of the multiscale medial axis map under Hausdorff distance, aiding in robust medial axis localization.

Abstract

We introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale $\lambda$, and prove a sharp regularity result for the squared-distance function to any closed non-empty subset $K$ of $\mathbb{R}^n$. Our results exploit properties of the function $C^l_{\lambda}(dist^2(\cdot;\, K))$ obtained by applying the quadratic lower compensated convex transform of parameter $\lambda$ to $dist^2(\cdot;\, K)$, the Euclidean squared-distance function to $K$. Using an estimate for the tight approximation of $dist^2(\cdot;\, K)$ by $C^l_{\lambda}(dist^2(\cdot;\, K))$, we prove $C^{1,1}$-regularity of $dist^2(\cdot;\, K)$ outside a neighbourhood of the closure of the medial axis $M_K$ of $K$, and give an asymptotic formula for $C^l_{\lambda}(dist^2(\cdot;\, K))$ in terms of the scaled squared distance to $K$ and to the convex hull of the set of points that realize the minimum distance to $K$. The multiscale medial axis map, $M_{\lambda}(\cdot;\, K)$, is a family of non-negative functions whose limit as $\lambda \to \infty$ exists and is called the multiscale medial axis landscape map, $M_{\infty}(\cdot;\, K)$. We show $M_{\infty}(\cdot;\, K)$ is strictly positive on the medial axis $M_K$ and zero elsewhere. We give conditions to ensure $M_{\lambda}(\cdot;\, K)$ keeps a constant height along parts of $M_K$ generated by two-point subsets with the height dependent on the distance between the generating points, so giving a hierarchy between different parts of $M_K$ that enables subsets of $M_K$ to be selected by thresholding. Given a compact subset $K$ of $\mathbb{R}^n$, while it is well known that $M_K$ is not Hausdorff stable, we prove $M_{\lambda}(\cdot;\, K)$ is stable under Hausdorff distance, and deduce implications for localization of the stable parts of $M_K$. Examples are included.