[{"date_updated":"2024-02-26T23:30:04Z","status":"public","degree_awarded":"MS","abstract":[{"text":"We introduce the notion of a Faustian interchange in a 1-parameter family of smooth\r\nfunctions to generalize the medial axis to critical points of index larger than 0.\r\nWe construct and implement a general purpose algorithm for approximating such\r\ngeneralized medial axes.","lang":"eng"}],"day":"24","publication_status":"published","oa_version":"Published Version","page":"43","citation":{"apa":"Stephenson, E. R. (2023). <i>Generalizing medial axes with homology switches</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:14226\">https://doi.org/10.15479/at:ista:14226</a>","ista":"Stephenson ER. 2023. Generalizing medial axes with homology switches. Institute of Science and Technology Austria.","mla":"Stephenson, Elizabeth R. <i>Generalizing Medial Axes with Homology Switches</i>. Institute of Science and Technology Austria, 2023, doi:<a href=\"https://doi.org/10.15479/at:ista:14226\">10.15479/at:ista:14226</a>.","short":"E.R. Stephenson, Generalizing Medial Axes with Homology Switches, Institute of Science and Technology Austria, 2023.","ama":"Stephenson ER. Generalizing medial axes with homology switches. 2023. doi:<a href=\"https://doi.org/10.15479/at:ista:14226\">10.15479/at:ista:14226</a>","chicago":"Stephenson, Elizabeth R. “Generalizing Medial Axes with Homology Switches.” Institute of Science and Technology Austria, 2023. <a href=\"https://doi.org/10.15479/at:ista:14226\">https://doi.org/10.15479/at:ista:14226</a>.","ieee":"E. R. Stephenson, “Generalizing medial axes with homology switches,” Institute of Science and Technology Austria, 2023."},"has_accepted_license":"1","alternative_title":["ISTA Master's Thesis"],"file":[{"creator":"cchlebak","access_level":"closed","relation":"source_file","content_type":"application/x-zip-compressed","file_name":"documents-export-2023-08-24.zip","file_size":15501411,"date_created":"2023-08-24T13:02:49Z","file_id":"14227","checksum":"453caf851d75c3478c10ed09bd242a91","embargo_to":"open_access","date_updated":"2024-02-26T23:30:03Z"},{"access_level":"open_access","content_type":"application/pdf","relation":"main_file","creator":"cchlebak","embargo":"2024-02-25","checksum":"7349d29963d6695e555e171748648d9a","file_id":"14228","date_created":"2023-08-24T13:03:42Z","date_updated":"2024-02-26T23:30:03Z","file_name":"thesis_pdf_a.pdf","file_size":6854783}],"title":"Generalizing medial axes with homology switches","year":"2023","publisher":"Institute of Science and Technology Austria","doi":"10.15479/at:ista:14226","language":[{"iso":"eng"}],"type":"dissertation","_id":"14226","oa":1,"author":[{"first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","orcid":"0000-0002-6862-208X","last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2023-08-24T00:00:00Z","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"date_created":"2023-08-24T13:01:18Z","file_date_updated":"2024-02-26T23:30:03Z","article_processing_charge":"No","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","ddc":["500"],"publication_identifier":{"issn":["2791-4585"]},"month":"08","supervisor":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}]},{"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"date_created":"2021-02-02T14:11:06Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","article_processing_charge":"No","ddc":["006","514","516"],"place":"Klosterneuburg","file_date_updated":"2021-02-03T10:37:28Z","publication_identifier":{"issn":["2663-337X"]},"month":"02","supervisor":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"}],"publisher":"Institute of Science and Technology Austria","year":"2021","language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:9056","type":"dissertation","oa":1,"_id":"9056","author":[{"first_name":"Georg F","full_name":"Osang, Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","orcid":"0000-0002-8882-5116"}],"date_published":"2021-02-01T00:00:00Z","citation":{"ieee":"G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021.","chicago":"Osang, Georg F. “Multi-Cover Persistence and Delaunay Mosaics.” Institute of Science and Technology Austria, 2021. <a href=\"https://doi.org/10.15479/AT:ISTA:9056\">https://doi.org/10.15479/AT:ISTA:9056</a>.","ama":"Osang GF. Multi-cover persistence and Delaunay mosaics. 2021. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9056\">10.15479/AT:ISTA:9056</a>","short":"G.F. Osang, Multi-Cover Persistence and Delaunay Mosaics, Institute of Science and Technology Austria, 2021.","ista":"Osang GF. 2021. Multi-cover persistence and Delaunay mosaics. Klosterneuburg: Institute of Science and Technology Austria.","apa":"Osang, G. F. (2021). <i>Multi-cover persistence and Delaunay mosaics</i>. Institute of Science and Technology Austria, Klosterneuburg. <a href=\"https://doi.org/10.15479/AT:ISTA:9056\">https://doi.org/10.15479/AT:ISTA:9056</a>","mla":"Osang, Georg F. <i>Multi-Cover Persistence and Delaunay Mosaics</i>. Institute of Science and Technology Austria, 2021, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9056\">10.15479/AT:ISTA:9056</a>."},"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"187"},{"status":"public","id":"8703","relation":"part_of_dissertation"}]},"file":[{"file_size":13446994,"file_name":"thesis_source.zip","date_updated":"2021-02-03T10:37:28Z","date_created":"2021-02-02T14:09:25Z","file_id":"9063","checksum":"bcf27986147cab0533b6abadd74e7629","creator":"patrickd","relation":"source_file","content_type":"application/zip","access_level":"closed"},{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","creator":"patrickd","success":1,"date_updated":"2021-02-02T14:09:18Z","checksum":"9cc8af266579a464385bbe2aff6af606","date_created":"2021-02-02T14:09:18Z","file_id":"9064","file_size":5210329,"file_name":"thesis_pdfA2b.pdf"}],"title":"Multi-cover persistence and Delaunay mosaics","date_updated":"2023-09-07T13:29:01Z","status":"public","degree_awarded":"PhD","abstract":[{"lang":"eng","text":"In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,\r\nand thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration\r\nfunction on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets."}],"oa_version":"Published Version","publication_status":"published","page":"134","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"day":"01"},{"language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:7944","publisher":"Institute of Science and Technology Austria","year":"2020","type":"dissertation","author":[{"last_name":"Masárová","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6660-1322","full_name":"Masárová, Zuzana","first_name":"Zuzana"}],"oa":1,"_id":"7944","date_published":"2020-06-09T00:00:00Z","date_created":"2020-06-08T00:49:46Z","department":[{"_id":"HeEd"},{"_id":"UlWa"}],"ddc":["516","514"],"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file_date_updated":"2020-07-14T12:48:05Z","keyword":["reconfiguration","reconfiguration graph","triangulations","flip","constrained triangulations","shellability","piecewise-linear balls","token swapping","trees","coloured weighted token swapping"],"month":"06","publication_identifier":{"isbn":["978-3-99078-005-3"],"issn":["2663-337X"]},"supervisor":[{"full_name":"Wagner, Uli","first_name":"Uli","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner"},{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"}],"date_updated":"2023-09-07T13:17:37Z","status":"public","license":"https://creativecommons.org/licenses/by-sa/4.0/","degree_awarded":"PhD","page":"160","oa_version":"Published Version","publication_status":"published","day":"09","tmp":{"name":"Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)","short":"CC BY-SA (4.0)","image":"/images/cc_by_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode"},"abstract":[{"text":"This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.\r\n\r\nFor triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.\r\n\r\nIn the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.","lang":"eng"}],"citation":{"ieee":"Z. Masárová, “Reconfiguration problems,” Institute of Science and Technology Austria, 2020.","chicago":"Masárová, Zuzana. “Reconfiguration Problems.” Institute of Science and Technology Austria, 2020. <a href=\"https://doi.org/10.15479/AT:ISTA:7944\">https://doi.org/10.15479/AT:ISTA:7944</a>.","ama":"Masárová Z. Reconfiguration problems. 2020. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7944\">10.15479/AT:ISTA:7944</a>","apa":"Masárová, Z. (2020). <i>Reconfiguration problems</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:7944\">https://doi.org/10.15479/AT:ISTA:7944</a>","short":"Z. Masárová, Reconfiguration Problems, Institute of Science and Technology Austria, 2020.","ista":"Masárová Z. 2020. Reconfiguration problems. Institute of Science and Technology Austria.","mla":"Masárová, Zuzana. <i>Reconfiguration Problems</i>. Institute of Science and Technology Austria, 2020, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:7944\">10.15479/AT:ISTA:7944</a>."},"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","related_material":{"record":[{"status":"public","id":"7950","relation":"part_of_dissertation"},{"id":"5986","status":"public","relation":"part_of_dissertation"}]},"file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","creator":"zmasarov","date_created":"2020-06-08T00:34:00Z","file_id":"7945","checksum":"df688bc5a82b50baee0b99d25fc7b7f0","date_updated":"2020-07-14T12:48:05Z","file_name":"THESIS_Zuzka_Masarova.pdf","file_size":13661779},{"creator":"zmasarov","content_type":"application/zip","relation":"source_file","access_level":"closed","file_size":32184006,"file_name":"THESIS_Zuzka_Masarova_SOURCE_FILES.zip","date_updated":"2020-07-14T12:48:05Z","date_created":"2020-06-08T00:35:30Z","checksum":"45341a35b8f5529c74010b7af43ac188","file_id":"7946"}],"title":"Reconfiguration problems"},{"oa":1,"_id":"7460","author":[{"full_name":"Ölsböck, Katharina","first_name":"Katharina","orcid":"0000-0002-4672-8297","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","last_name":"Ölsböck"}],"date_published":"2020-02-10T00:00:00Z","publisher":"Institute of Science and Technology Austria","year":"2020","language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:7460","type":"dissertation","publication_identifier":{"issn":["2663-337X"]},"month":"02","supervisor":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"department":[{"_id":"HeEd"},{"_id":"GradSch"}],"date_created":"2020-02-06T14:56:53Z","keyword":["shape reconstruction","hole manipulation","ordered complexes","Alpha complex","Wrap complex","computational topology","Bregman geometry"],"ddc":["514"],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","article_processing_charge":"No","file_date_updated":"2020-07-14T12:47:58Z","degree_awarded":"PhD","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","oa_version":"Published Version","page":"155","tmp":{"image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","short":"CC BY-NC-SA (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode"},"day":"10","date_updated":"2023-09-07T13:15:30Z","license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","status":"public","related_material":{"record":[{"id":"6608","status":"public","relation":"part_of_dissertation"}]},"title":"The hole system of triangulated shapes","file":[{"access_level":"open_access","content_type":"application/pdf","relation":"main_file","creator":"koelsboe","file_id":"7461","checksum":"1df9f8c530b443c0e63a3f2e4fde412e","date_created":"2020-02-06T14:43:54Z","date_updated":"2020-07-14T12:47:58Z","file_name":"thesis_ist-final_noack.pdf","file_size":76195184},{"file_id":"7462","date_created":"2020-02-06T14:52:45Z","checksum":"7a52383c812b0be64d3826546509e5a4","date_updated":"2020-07-14T12:47:58Z","file_name":"latex-files.zip","file_size":122103715,"access_level":"closed","content_type":"application/x-zip-compressed","description":"latex source files, figures","relation":"source_file","creator":"koelsboe"}],"citation":{"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>","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>","short":"K. Ölsböck, The Hole System of Triangulated Shapes, Institute of Science and Technology Austria, 2020.","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>.","ista":"Ölsböck K. 2020. The hole system of triangulated shapes. Institute of Science and Technology Austria.","ieee":"K. Ölsböck, “The hole system of triangulated shapes,” Institute of Science and Technology Austria, 2020.","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>."},"alternative_title":["ISTA Thesis"],"has_accepted_license":"1"},{"date_published":"2018-06-11T00:00:00Z","oa":1,"_id":"201","author":[{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel","first_name":"Mabel"}],"type":"dissertation","publisher":"Institute of Science and Technology Austria","year":"2018","language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:th_1026","supervisor":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"}],"publist_id":"7712","publication_identifier":{"issn":["2663-337X"]},"month":"06","article_processing_charge":"No","ddc":["514","516"],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file_date_updated":"2020-07-14T12:45:24Z","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:45:10Z","abstract":[{"lang":"eng","text":"We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications."}],"publication_status":"published","page":"171","oa_version":"Published Version","day":"11","degree_awarded":"PhD","status":"public","date_updated":"2023-09-07T12:25:32Z","title":"Multiple covers with balls","file":[{"file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","file_size":11827713,"checksum":"dd699303623e96d1478a6ae07210dd05","date_created":"2019-02-05T07:43:31Z","file_id":"5918","date_updated":"2020-07-14T12:45:24Z","creator":"kschuh","access_level":"closed","relation":"source_file","content_type":"application/zip"},{"creator":"kschuh","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_size":4783846,"file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","date_updated":"2020-07-14T12:45:24Z","date_created":"2019-02-05T07:43:45Z","checksum":"ba163849a190d2b41d66fef0e4983294","file_id":"5919"}],"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","citation":{"ama":"Iglesias Ham M. Multiple covers with balls. 2018. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>","mla":"Iglesias Ham, Mabel. <i>Multiple Covers with Balls</i>. Institute of Science and Technology Austria, 2018, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>.","ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria.","short":"M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018.","apa":"Iglesias Ham, M. (2018). <i>Multiple covers with balls</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>","ieee":"M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018.","chicago":"Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>."},"pubrep_id":"1026"},{"ddc":["514","516","519"],"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file_date_updated":"2020-07-14T12:47:26Z","department":[{"_id":"HeEd"}],"date_created":"2019-04-09T15:04:32Z","supervisor":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"}],"publication_identifier":{"issn":["2663-337X"]},"month":"10","type":"dissertation","publisher":"Institute of Science and Technology Austria","year":"2017","language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:th_873","date_published":"2017-10-27T00:00:00Z","oa":1,"_id":"6287","author":[{"first_name":"Anton","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko"}],"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","citation":{"chicago":"Nikitenko, Anton. “Discrete Morse Theory for Random Complexes .” Institute of Science and Technology Austria, 2017. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>.","ieee":"A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of Science and Technology Austria, 2017.","mla":"Nikitenko, Anton. <i>Discrete Morse Theory for Random Complexes </i>. Institute of Science and Technology Austria, 2017, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>.","apa":"Nikitenko, A. (2017). <i>Discrete Morse theory for random complexes </i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>","short":"A. Nikitenko, Discrete Morse Theory for Random Complexes , Institute of Science and Technology Austria, 2017.","ista":"Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute of Science and Technology Austria.","ama":"Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>"},"pubrep_id":"873","related_material":{"record":[{"relation":"part_of_dissertation","id":"718","status":"public"},{"relation":"part_of_dissertation","id":"5678","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"87"}]},"file":[{"access_level":"open_access","content_type":"application/pdf","relation":"main_file","creator":"dernst","checksum":"ece7e598a2f060b263c2febf7f3fe7f9","file_id":"6289","date_created":"2019-04-09T14:54:51Z","date_updated":"2020-07-14T12:47:26Z","file_name":"2017_Thesis_Nikitenko.pdf","file_size":2324870},{"file_size":2863219,"file_name":"2017_Thesis_Nikitenko_source.zip","date_updated":"2020-07-14T12:47:26Z","date_created":"2019-04-09T14:54:51Z","file_id":"6290","checksum":"99b7ad76e317efd447af60f91e29b49b","creator":"dernst","content_type":"application/zip","relation":"source_file","access_level":"closed"}],"title":"Discrete Morse theory for random complexes ","status":"public","date_updated":"2023-09-15T12:10:34Z","abstract":[{"text":"The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.","lang":"eng"}],"publication_status":"published","page":"86","oa_version":"Published Version","day":"27","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"degree_awarded":"PhD"},{"month":"06","publication_identifier":{"issn":["2663-337X"]},"publist_id":"5808","supervisor":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"title":"On the approximation of intrinsic volumes","related_material":{"record":[{"relation":"part_of_dissertation","id":"1662","status":"public"},{"id":"1792","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"2255","relation":"part_of_dissertation"}]},"date_created":"2018-12-11T11:51:48Z","department":[{"_id":"HeEd"}],"citation":{"chicago":"Pausinger, Florian. “On the Approximation of Intrinsic Volumes.” Institute of Science and Technology Austria, 2015.","ieee":"F. Pausinger, “On the approximation of intrinsic volumes,” Institute of Science and Technology Austria, 2015.","apa":"Pausinger, F. (2015). <i>On the approximation of intrinsic volumes</i>. Institute of Science and Technology Austria.","short":"F. Pausinger, On the Approximation of Intrinsic Volumes, Institute of Science and Technology Austria, 2015.","ista":"Pausinger F. 2015. On the approximation of intrinsic volumes. Institute of Science and Technology Austria.","mla":"Pausinger, Florian. <i>On the Approximation of Intrinsic Volumes</i>. Institute of Science and Technology Austria, 2015.","ama":"Pausinger F. On the approximation of intrinsic volumes. 2015."},"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","alternative_title":["ISTA Thesis"],"degree_awarded":"PhD","author":[{"full_name":"Pausinger, Florian","first_name":"Florian","last_name":"Pausinger","id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8379-3768"}],"_id":"1399","day":"01","date_published":"2015-06-01T00:00:00Z","page":"144","oa_version":"None","publication_status":"published","abstract":[{"text":"This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold.","lang":"eng"}],"language":[{"iso":"eng"}],"year":"2015","publisher":"Institute of Science and Technology Austria","date_updated":"2023-09-07T11:41:25Z","status":"public","type":"dissertation"}]