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

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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 prove a sharp regularity result for the squared-distance function to any closed non-empty 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 to dist²(·; K), the Euclidean squared-distance function to K. Using an estimate for the tight approximation of dist²(·; K) by Cˡλ(dist²(·; K)), we prove C¹,¹-regularity of dist²(·; K) outside a neighbourhood of the closure of the medial axis Mₖ of K, and give an asymptotic formula for Cˡλ(dist²(·; 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, Mλ(·; K), is a family of non-negative functions whose limit as λ→ ∞ exists and is called the multiscale medial axis landscape map, M∞(·; K). We show M∞(·; K) is strictly positive on the medial axis Mₖ and zero elsewhere. We give conditions to ensure Mλ(·; K) keeps a constant height along parts of Mₖ generated by two-point subsets with the height dependent on the distance between the generating points, so giving a hierarchy between different parts of Mₖ that enables subsets of Mₖ to be selected by thresholding. Given a compact subset K of Rⁿ, while it is well known that Mₖ is not Hausdorff stable, we prove Mλ(·; K) is stable under Hausdorff distance, and deduce implications for localization of the stable parts of Mₖ. Examples are included.