{"date_updated":"2023-05-22T08:15:19Z","article_processing_charge":"No","month":"06","day":"02","type":"conference","abstract":[{"text":"In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.","lang":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/2303.04014","open_access":"1"}],"department":[{"_id":"HeEd"}],"publication_status":"published","date_published":"2023-06-02T00:00:00Z","publisher":"Association for Computing Machinery","date_created":"2023-05-22T08:02:02Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","oa":1,"status":"public","title":"Hausdorff and Gromov-Hausdorff stable subsets of the medial axis","quality_controlled":"1","publication_identifier":{"isbn":["9781450399135"]},"external_id":{"arxiv":["2303.04014"]},"publication":"Proceedings of the 55th Annual ACM Symposium on Theory of Computing","author":[{"first_name":"André","last_name":"Lieutier","full_name":"Lieutier, André"},{"orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","last_name":"Wintraecken"}],"doi":"10.1145/3564246.3585113","page":"1768-1776","_id":"13048","language":[{"iso":"eng"}],"citation":{"ieee":"A. Lieutier and M. Wintraecken, “Hausdorff and Gromov-Hausdorff stable subsets of the medial axis,” in <i>Proceedings of the 55th Annual ACM Symposium on Theory of Computing</i>, Orlando, FL, United States, 2023, pp. 1768–1776.","mla":"Lieutier, André, and Mathijs Wintraecken. “Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial Axis.” <i>Proceedings of the 55th Annual ACM Symposium on Theory of Computing</i>, Association for Computing Machinery, 2023, pp. 1768–76, doi:<a href=\"https://doi.org/10.1145/3564246.3585113\">10.1145/3564246.3585113</a>.","short":"A. Lieutier, M. Wintraecken, in:, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, 2023, pp. 1768–1776.","ama":"Lieutier A, Wintraecken M. Hausdorff and Gromov-Hausdorff stable subsets of the medial axis. In: <i>Proceedings of the 55th Annual ACM Symposium on Theory of Computing</i>. Association for Computing Machinery; 2023:1768-1776. doi:<a href=\"https://doi.org/10.1145/3564246.3585113\">10.1145/3564246.3585113</a>","ista":"Lieutier A, Wintraecken M. 2023. Hausdorff and Gromov-Hausdorff stable subsets of the medial axis. Proceedings of the 55th Annual ACM Symposium on Theory of Computing. STOC: Symposium on Theory of Computing, 1768–1776.","apa":"Lieutier, A., &#38; Wintraecken, M. (2023). Hausdorff and Gromov-Hausdorff stable subsets of the medial axis. In <i>Proceedings of the 55th Annual ACM Symposium on Theory of Computing</i> (pp. 1768–1776). Orlando, FL, United States: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3564246.3585113\">https://doi.org/10.1145/3564246.3585113</a>","chicago":"Lieutier, André, and Mathijs Wintraecken. “Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial Axis.” In <i>Proceedings of the 55th Annual ACM Symposium on Theory of Computing</i>, 1768–76. Association for Computing Machinery, 2023. <a href=\"https://doi.org/10.1145/3564246.3585113\">https://doi.org/10.1145/3564246.3585113</a>."},"conference":{"end_date":"2023-06-23","location":"Orlando, FL, United States","name":"STOC: Symposium on Theory of Computing","start_date":"2023-06-20"},"ec_funded":1,"year":"2023","project":[{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"},{"grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses"}],"acknowledgement":"We are greatly indebted to Erin Chambers for posing a number of questions that eventually led to this paper. We would also like to thank the other organizers of the workshop on ‘Algorithms\r\nfor the medial axis’. We are also indebted to Tatiana Ezubova for helping with the search for and translation of Russian literature. The second author thanks all members of the Edelsbrunner and Datashape groups for the atmosphere in which the research was conducted.\r\nThe research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions). Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411. The Austrian science fund (FWF) M-3073."}