{"id":258,"date":"2016-06-17T07:15:41","date_gmt":"2016-06-17T07:15:41","guid":{"rendered":"http:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/?p=258"},"modified":"2016-06-17T07:15:41","modified_gmt":"2016-06-17T07:15:41","slug":"compensated-convexity-multiscale-medial-axis-maps-and-sharp-regularity-of-the-squared-distance-function","status":"publish","type":"post","link":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/2016\/06\/17\/compensated-convexity-multiscale-medial-axis-maps-and-sharp-regularity-of-the-squared-distance-function\/","title":{"rendered":"Compensated convexity, multiscale medial axis maps, and sharp regularity of the squared distance function"},"content":{"rendered":"<p><em>Elaine\u00a0Crooks (Swansea, UK)<\/em><\/p>\n<p>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.<\/p>\n<p>This is joint work with Kewei Zhang, Nottingham, and Antonio Orlando, Tucum\u00e1n.<\/p>\n<p><a href=\"http:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/programme\/\">Back to Programme<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Elaine\u00a0Crooks (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, &hellip; <a href=\"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/2016\/06\/17\/compensated-convexity-multiscale-medial-axis-maps-and-sharp-regularity-of-the-squared-distance-function\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Compensated convexity, multiscale medial axis maps, and sharp regularity of the squared distance function<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-258","post","type-post","status-publish","format-standard","hentry","category-abstracts"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts\/258","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/comments?post=258"}],"version-history":[{"count":0,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts\/258\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/media?parent=258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/categories?post=258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/tags?post=258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}