[{"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","last_name":"Ölsböck","full_name":"Ölsböck, Katharina","first_name":"Katharina"},{"last_name":"Synak","full_name":"Synak, Peter","first_name":"Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics."}],"citation":{"chicago":"Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In <i>Topological Data Analysis</i>, 15:181–218. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>.","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., &#38; Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In <i>Topological Data Analysis</i> (Vol. 15, pp. 181–218). Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>","ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in <i>Topological Data Analysis</i>, 2020, vol. 15, pp. 181–218.","ista":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218.","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” <i>Topological Data Analysis</i>, vol. 15, Springer Nature, 2020, pp. 181–218, doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>.","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: <i>Topological Data Analysis</i>. Vol 15. Springer Nature; 2020:181-218. doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>"},"publication_status":"published","project":[{"grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"grant_number":"638176","_id":"2533E772-B435-11E9-9278-68D0E5697425","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales","call_identifier":"H2020"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"quality_controlled":"1","oa_version":"Submitted Version","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"eissn":["21978549"],"issn":["21932808"],"isbn":["9783030434076"]},"_id":"8135","article_processing_charge":"No","volume":15,"oa":1,"date_updated":"2021-01-12T08:17:06Z","title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","ec_funded":1,"doi":"10.1007/978-3-030-43408-3_8","year":"2020","ddc":["510"],"alternative_title":["Abel Symposia"],"status":"public","intvolume":"        15","type":"conference","day":"22","file_date_updated":"2020-10-08T08:56:14Z","page":"181-218","publication":"Topological Data Analysis","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","date_published":"2020-06-22T00:00:00Z","month":"06","file":[{"access_level":"open_access","date_updated":"2020-10-08T08:56:14Z","checksum":"7b5e0de10675d787a2ddb2091370b8d8","date_created":"2020-10-08T08:56:14Z","file_size":2207071,"file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","file_id":"8628","creator":"dernst","relation":"main_file","content_type":"application/pdf","success":1}],"date_created":"2020-07-19T22:00:59Z","department":[{"_id":"HeEd"}],"has_accepted_license":"1"},{"file_date_updated":"2020-07-14T12:47:58Z","page":"155","status":"public","supervisor":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"type":"dissertation","day":"10","date_created":"2020-02-06T14:56:53Z","file":[{"relation":"main_file","content_type":"application/pdf","creator":"koelsboe","file_id":"7461","file_size":76195184,"file_name":"thesis_ist-final_noack.pdf","checksum":"1df9f8c530b443c0e63a3f2e4fde412e","date_created":"2020-02-06T14:43:54Z","access_level":"open_access","date_updated":"2020-07-14T12:47:58Z"},{"access_level":"closed","date_updated":"2020-07-14T12:47:58Z","checksum":"7a52383c812b0be64d3826546509e5a4","date_created":"2020-02-06T14:52:45Z","file_size":122103715,"file_name":"latex-files.zip","file_id":"7462","creator":"koelsboe","relation":"source_file","description":"latex source files, figures","content_type":"application/x-zip-compressed"}],"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","has_accepted_license":"1","degree_awarded":"PhD","language":[{"iso":"eng"}],"publisher":"Institute of Science and Technology Austria","date_published":"2020-02-10T00:00:00Z","month":"02","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Published Version","_id":"7460","publication_identifier":{"issn":["2663-337X"]},"oa":1,"date_updated":"2023-09-07T13:15:30Z","article_processing_charge":"No","keyword":["shape reconstruction","hole manipulation","ordered complexes","Alpha complex","Wrap complex","computational topology","Bregman geometry"],"author":[{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","last_name":"Ölsböck","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297"}],"abstract":[{"text":"Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.\r\n\r\nFor the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.","lang":"eng"}],"publication_status":"published","citation":{"ista":"Ölsböck K. 2020. The hole system of triangulated shapes. Institute of Science and Technology Austria.","short":"K. Ölsböck, The Hole System of Triangulated Shapes, Institute of Science and Technology Austria, 2020.","ama":"Ölsböck K. The hole system of triangulated shapes. 2020. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7460\">10.15479/AT:ISTA:7460</a>","mla":"Ölsböck, Katharina. <i>The Hole System of Triangulated Shapes</i>. Institute of Science and Technology Austria, 2020, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7460\">10.15479/AT:ISTA:7460</a>.","chicago":"Ölsböck, Katharina. “The Hole System of Triangulated Shapes.” Institute of Science and Technology Austria, 2020. <a href=\"https://doi.org/10.15479/AT:ISTA:7460\">https://doi.org/10.15479/AT:ISTA:7460</a>.","apa":"Ölsböck, K. (2020). <i>The hole system of triangulated shapes</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:7460\">https://doi.org/10.15479/AT:ISTA:7460</a>","ieee":"K. Ölsböck, “The hole system of triangulated shapes,” Institute of Science and Technology Austria, 2020."},"ddc":["514"],"related_material":{"record":[{"id":"6608","relation":"part_of_dissertation","status":"public"}]},"alternative_title":["ISTA Thesis"],"tmp":{"short":"CC BY-NC-SA (4.0)","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode"},"title":"The hole system of triangulated shapes","year":"2020","doi":"10.15479/AT:ISTA:7460"},{"article_processing_charge":"Yes (via OA deal)","date_updated":"2023-08-21T06:13:48Z","oa":1,"volume":64,"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"quality_controlled":"1","oa_version":"Published Version","acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"_id":"7666","citation":{"apa":"Edelsbrunner, H., &#38; Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>","ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” <i>Discrete and Computational Geometry</i>, vol. 64. Springer Nature, pp. 759–775, 2020.","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s00454-020-00188-x\">https://doi.org/10.1007/s00454-020-00188-x</a>.","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. <i>Discrete and Computational Geometry</i>. 2020;64:759-775. doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” <i>Discrete and Computational Geometry</i>, vol. 64, Springer Nature, 2020, pp. 759–75, doi:<a href=\"https://doi.org/10.1007/s00454-020-00188-x\">10.1007/s00454-020-00188-x</a>.","ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775."},"publication_status":"published","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"first_name":"Katharina","last_name":"Ölsböck","full_name":"Ölsböck, Katharina","orcid":"0000-0002-4672-8297","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups."}],"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["510"],"ec_funded":1,"doi":"10.1007/s00454-020-00188-x","year":"2020","external_id":{"isi":["000520918800001"]},"title":"Tri-partitions and bases of an ordered complex","file_date_updated":"2020-11-20T13:22:21Z","page":"759-775","publication":"Discrete and Computational Geometry","type":"journal_article","day":"20","status":"public","intvolume":"        64","department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"date_updated":"2020-11-20T13:22:21Z","access_level":"open_access","file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","file_size":701673,"date_created":"2020-11-20T13:22:21Z","checksum":"f8cc96e497f00c38340b5dafe0cb91d7","content_type":"application/pdf","relation":"main_file","creator":"dernst","file_id":"8786","success":1}],"date_created":"2020-04-19T22:00:56Z","article_type":"original","date_published":"2020-03-20T00:00:00Z","month":"03","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature"},{"type":"journal_article","day":"01","status":"public","intvolume":"        73","file_date_updated":"2020-07-14T12:47:34Z","page":"1-15","publication":"Computer Aided Geometric Design","date_published":"2019-08-01T00:00:00Z","month":"08","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Elsevier","department":[{"_id":"HeEd"}],"has_accepted_license":"1","license":"https://creativecommons.org/licenses/by-nc-nd/4.0/","file":[{"date_updated":"2020-07-14T12:47:34Z","access_level":"open_access","file_name":"Elsevier_2019_Edelsbrunner.pdf","file_size":2665013,"date_created":"2019-07-08T15:24:26Z","checksum":"7c99be505dc7533257d42eb1830cef04","content_type":"application/pdf","relation":"main_file","file_id":"6624","creator":"kschuh"}],"date_created":"2019-07-07T21:59:20Z","citation":{"ama":"Edelsbrunner H, Ölsböck K. Holes and dependences in an ordered complex. <i>Computer Aided Geometric Design</i>. 2019;73:1-15. doi:<a href=\"https://doi.org/10.1016/j.cagd.2019.06.003\">10.1016/j.cagd.2019.06.003</a>","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Holes and Dependences in an Ordered Complex.” <i>Computer Aided Geometric Design</i>, vol. 73, Elsevier, 2019, pp. 1–15, doi:<a href=\"https://doi.org/10.1016/j.cagd.2019.06.003\">10.1016/j.cagd.2019.06.003</a>.","ista":"Edelsbrunner H, Ölsböck K. 2019. Holes and dependences in an ordered complex. Computer Aided Geometric Design. 73, 1–15.","short":"H. Edelsbrunner, K. Ölsböck, Computer Aided Geometric Design 73 (2019) 1–15.","apa":"Edelsbrunner, H., &#38; Ölsböck, K. (2019). Holes and dependences in an ordered complex. <i>Computer Aided Geometric Design</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.cagd.2019.06.003\">https://doi.org/10.1016/j.cagd.2019.06.003</a>","ieee":"H. Edelsbrunner and K. Ölsböck, “Holes and dependences in an ordered complex,” <i>Computer Aided Geometric Design</i>, vol. 73. Elsevier, pp. 1–15, 2019.","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Holes and Dependences in an Ordered Complex.” <i>Computer Aided Geometric Design</i>. Elsevier, 2019. <a href=\"https://doi.org/10.1016/j.cagd.2019.06.003\">https://doi.org/10.1016/j.cagd.2019.06.003</a>."},"publication_status":"published","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Katharina","orcid":"0000-0002-4672-8297","last_name":"Ölsböck","full_name":"Ölsböck, Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram.","lang":"eng"}],"article_processing_charge":"No","volume":73,"date_updated":"2023-09-07T13:15:29Z","oa":1,"project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","oa_version":"Published Version","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"6608","ec_funded":1,"doi":"10.1016/j.cagd.2019.06.003","year":"2019","title":"Holes and dependences in an ordered complex","external_id":{"isi":["000485207800001"]},"isi":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","image":"/images/cc_by_nc_nd.png","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","short":"CC BY-NC-ND (4.0)"},"ddc":["000"],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"7460"}]}}]