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

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• Elaine Crooks (Swansea University)
• Friday 15 December 2017, 10:00-11:00
• Seminar Room 1, Newton Institute.

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VMVW03 - Flows, mappings and shapes

Co-authors: Kewei Zhang (University of Nottingham, UK), Antonio Orlando (Universidad Nacional de Tucuman, Argentina)

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 resu lt 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 (Tucuman). 

This talk is part of the Isaac Newton Institute Seminar Series series.

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