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

Abstract

In this paper 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 provide a sharp regularity result for the squared-distance function to any closed nonempty subset $K$ of $\mathbb{R}^n$. Our results exploit properties of the function $C^l_{\lambda}(\mathrm{dist}^2(\cdot;\, K))$ obtained by applying the quadratic lower compensated convex transform of parameter $\lambda$ [K. Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), pp. 743--771] to $\mathrm{dist}^2(\cdot;\, K)$, the Euclidean squared-distance function to $K$. Using a quantitative estimate for the tight approximation of $\mathrm{dist}^2(\cdot;\, K)$ by $C^l_{\lambda}(\mathrm{dist}^2(\cdot;\, K))$, we prove the $C^{1,1}$-regularity of $\mathrm{dist}^2(\cdot;\, K)$ outside a neighborhood of the closure of the medial axis $M_K$ of $K$, which can be viewed as a weak Lusin-type theorem for $\mathrm{dist}^2(\cdot;\, K)$, and give an asymptotic expansion formula for $C^l_{\lambda}(\mathrm{dist}^2(\cdot;\, K))$ in terms of the scaled squared-distance transform to the set and to the convex hull of the set of points that realize the minimum distance to $K$. The multiscale medial axis map, denoted by $M_{\lambda}(\cdot;\, K)$, is a family of nonnegative functions, parametrized by $\lambda>0$, whose limit as $\lambda \to \infty$ exists and is called the multiscale medial axis landscape map, $M_{\infty}(\cdot;\, K)$. We show that $M_{\infty}(\cdot;\, K)$ is strictly positive on the medial axis $M_K$ and zero elsewhere. We give conditions that ensure $M_{\lambda}(\cdot;\, K)$ keeps a constant height along the parts of $M_K$ generated by two-point subsets with the value of the height dependent on the scale of the distance between the generating points, thus providing a hierarchy of heights (hence, the word “multiscale'') between different parts of $M_K$ that enables subsets of $M_K$ to be selected by simple thresholding. Asymptotically, further understanding of the multiscale effect is provided by our exact representation of $M_{\infty}(\cdot;\, K)$. Moreover, given a compact subset $K$ of $\mathbb{R}^n$, while it is well known that $M_K$ is not Hausdorff stable, we prove that in contrast, $M_{\lambda}(\cdot;\, K)$ is stable under the Hausdorff distance, and deduce implications for the localization of the stable parts of $M_K$. Explicitly calculated prototype examples of medial axis maps are also presented and used to illustrate the theoretical findings.