Compensated convexity, multiscale medial axis maps, and sharp regularity of the squared distance function

Elaine Crooks (Swansea, UK)

Compensated convex transforms enjoy tight-approximation and locality properties that can be exploited to develop multiscale, parametrised methods for identifying singularities in functions. When applied to the squared distance function to a closed subset of Euclidean space, these ideas yield a new tool for locating and analyzing the medial axis of geometric objects, called the multiscale medial axis map. This consists of a parametrised family of nonnegative functions that provides a Hausdorff-stable multiscale representation of the medial axis, in particular producing a hierarchy of heights between different parts of the medial axis depending on the distance between the generating points of that part of the medial axis. Such a hierarchy enables subsets of the medial axis to be selected by simple thresholding, which tackles the well-known stability issue that small perturbations in an object can produce large variations in the corresponding medial axis. A sharp regularity result for the squared distance function is obtained as a by-product of the analysis of this multiscale medial axis map.

This is joint work with Kewei Zhang, Nottingham, and Antonio Orlando, Tucumán.

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