Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function - Zhang, Kewei; Crooks, Elaine; Orlando, Antonio (Society for Industrial and Applied Mathematics, 2015)
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 λ, and provide a sharp regularity result for the squared-distance function to any closed nonempty subset K of Rn. Our results exploit properties of the function Clλ (dist2(・; K)) obtained by applying the quadratic lower compensated convex transform of parameter λ [K. Zhang, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 25 (2008), pp. 743–771] to dist2(・; K), the Euclidean squared-distance function to K. Using a quantitative estimate for the tight approximation of dist2(・; K) by Clλ (dist2(・; K)), we prove the C1,1-regularity of dist2(・; K) outside a neighborhood of the closure of the medial axis MK of K, which can be viewed as a weak Lusin-type theorem for dist2(・; K), and give an asymptotic expansion formula for Clλ (dist2(・; 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 Mλ(・; K), is a family of nonnegative functions, parametrized by λ > 0, whose limit as λ→∞exists and is called the multiscale medial axis landscape map, M∞(・; K). We show that M∞(・; K) is strictly positive on the medial axis MK and zero elsewhere. We give conditions that ensure Mλ(・; K) keeps a constant height along the parts of MK 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 MK that enables subsets of MK to be selected by simple thresholding. Asymptotically, further understanding of the multiscale effect is provided by our exact representation of M∞(・; K). Moreover, given a compact subset K of Rn, while it is well known that MK is not Hausdorff stable, we prove that in contrast, Mλ(・; K) is stable under the Hausdorff distance, and deduce implications for the localization of the stable parts of MK. Explicitly calculated prototype examples of medial axis maps are also presented and used to illustrate the theoretical findings.