--- _id: '15094' abstract: - lang: eng text: "Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in\r\ndiscrete geometry that have captivated mathematicians for centuries, if not millennia. This\r\nthesis seeks to cast new light on these structures by illustrating specific instances where a\r\ntopological perspective, specifically through discrete Morse theory and persistent homology,\r\nprovides valuable insights.\r\n\r\nAt first glance, the topology of these geometric objects might seem uneventful: point sets\r\nessentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which\r\nis a contractible space, and the topology of a network primarily involves the enumeration\r\nof connected components and cycles within the network. However, beneath this apparent\r\nsimplicity, there lies an array of intriguing structures, a small subset of which will be uncovered\r\nin this thesis.\r\n\r\nFocused on three case studies, each addressing one of the mentioned objects, this work\r\nwill showcase connections that intertwine topology with diverse fields such as combinatorial\r\ngeometry, algorithms and data structures, and emerging applications like spatial biology.\r\n\r\n" alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 citation: ama: Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:10.15479/at:ista:15094 apa: Cultrera di Montesano, S. (2024). Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:15094 chicago: Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:15094. ieee: S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024. ista: Cultrera di Montesano S. 2024. Persistence and Morse theory for discrete geometric structures. Institute of Science and Technology Austria. mla: Cultrera di Montesano, Sebastiano. Persistence and Morse Theory for Discrete Geometric Structures. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:15094. short: S. Cultrera di Montesano, Persistence and Morse Theory for Discrete Geometric Structures, Institute of Science and Technology Austria, 2024. date_created: 2024-03-08T15:28:10Z date_published: 2024-03-08T00:00:00Z date_updated: 2024-03-20T09:36:57Z day: '08' ddc: - '514' - '500' - '516' degree_awarded: PhD department: - _id: GradSch - _id: HeEd doi: 10.15479/at:ista:15094 ec_funded: 1 file: - access_level: open_access checksum: 1e468bfa42a7dcf04d89f4dadc621c87 content_type: application/pdf creator: scultrer date_created: 2024-03-14T08:55:07Z date_updated: 2024-03-14T08:55:07Z file_id: '15112' file_name: Thesis Sebastiano.pdf file_size: 4106872 relation: main_file success: 1 - access_level: closed checksum: bcbd213490f5a7e68855a092bbce93f1 content_type: application/zip creator: scultrer date_created: 2024-03-14T08:56:24Z date_updated: 2024-03-14T14:14:35Z file_id: '15113' file_name: Thesis (1).zip file_size: 4746234 relation: source_file file_date_updated: 2024-03-14T14:14:35Z has_accepted_license: '1' language: - iso: eng license: https://creativecommons.org/licenses/by-nc-sa/4.0/ month: '03' oa: 1 oa_version: Published Version page: '108' project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication_identifier: issn: - 2663 - 337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '11660' relation: part_of_dissertation status: public - id: '11658' relation: part_of_dissertation status: public - id: '13182' relation: part_of_dissertation status: public - id: '15090' relation: part_of_dissertation status: public - id: '15091' relation: part_of_dissertation status: public - id: '15093' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Persistence and Morse theory for discrete geometric structures tmp: image: /images/cc_by_nc_sa.png legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) short: CC BY-NC-SA (4.0) type: dissertation user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2024' ... --- _id: '14226' abstract: - lang: eng 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." alternative_title: - ISTA Master's Thesis article_processing_charge: No author: - first_name: Elizabeth R full_name: Stephenson, Elizabeth R id: 2D04F932-F248-11E8-B48F-1D18A9856A87 last_name: Stephenson orcid: 0000-0002-6862-208X citation: ama: Stephenson ER. Generalizing medial axes with homology switches. 2023. doi:10.15479/at:ista:14226 apa: Stephenson, E. R. (2023). Generalizing medial axes with homology switches. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:14226 chicago: Stephenson, Elizabeth R. “Generalizing Medial Axes with Homology Switches.” Institute of Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14226. ieee: E. R. Stephenson, “Generalizing medial axes with homology switches,” Institute of Science and Technology Austria, 2023. ista: Stephenson ER. 2023. Generalizing medial axes with homology switches. Institute of Science and Technology Austria. mla: Stephenson, Elizabeth R. Generalizing Medial Axes with Homology Switches. Institute of Science and Technology Austria, 2023, doi:10.15479/at:ista:14226. short: E.R. Stephenson, Generalizing Medial Axes with Homology Switches, Institute of Science and Technology Austria, 2023. date_created: 2023-08-24T13:01:18Z date_published: 2023-08-24T00:00:00Z date_updated: 2024-02-26T23:30:04Z day: '24' ddc: - '500' degree_awarded: MS department: - _id: GradSch - _id: HeEd doi: 10.15479/at:ista:14226 file: - access_level: closed checksum: 453caf851d75c3478c10ed09bd242a91 content_type: application/x-zip-compressed creator: cchlebak date_created: 2023-08-24T13:02:49Z date_updated: 2024-02-26T23:30:03Z embargo_to: open_access file_id: '14227' file_name: documents-export-2023-08-24.zip file_size: 15501411 relation: source_file - access_level: open_access checksum: 7349d29963d6695e555e171748648d9a content_type: application/pdf creator: cchlebak date_created: 2023-08-24T13:03:42Z date_updated: 2024-02-26T23:30:03Z embargo: 2024-02-25 file_id: '14228' file_name: thesis_pdf_a.pdf file_size: 6854783 relation: main_file file_date_updated: 2024-02-26T23:30:03Z has_accepted_license: '1' language: - iso: eng month: '08' oa: 1 oa_version: Published Version page: '43' publication_identifier: issn: - 2791-4585 publication_status: published publisher: Institute of Science and Technology Austria status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Generalizing medial axes with homology switches type: dissertation user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2023' ... --- _id: '9056' 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." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Georg F full_name: Osang, Georg F id: 464B40D6-F248-11E8-B48F-1D18A9856A87 last_name: Osang orcid: 0000-0002-8882-5116 citation: ama: Osang GF. Multi-cover persistence and Delaunay mosaics. 2021. doi:10.15479/AT:ISTA:9056 apa: Osang, G. F. (2021). Multi-cover persistence and Delaunay mosaics. Institute of Science and Technology Austria, Klosterneuburg. https://doi.org/10.15479/AT:ISTA:9056 chicago: Osang, Georg F. “Multi-Cover Persistence and Delaunay Mosaics.” Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/AT:ISTA:9056. ieee: G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021. ista: 'Osang GF. 2021. Multi-cover persistence and Delaunay mosaics. Klosterneuburg: Institute of Science and Technology Austria.' mla: Osang, Georg F. Multi-Cover Persistence and Delaunay Mosaics. Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9056. short: G.F. Osang, Multi-Cover Persistence and Delaunay Mosaics, Institute of Science and Technology Austria, 2021. date_created: 2021-02-02T14:11:06Z date_published: 2021-02-01T00:00:00Z date_updated: 2023-09-07T13:29:01Z day: '01' ddc: - '006' - '514' - '516' degree_awarded: PhD department: - _id: HeEd - _id: GradSch doi: 10.15479/AT:ISTA:9056 file: - access_level: closed checksum: bcf27986147cab0533b6abadd74e7629 content_type: application/zip creator: patrickd date_created: 2021-02-02T14:09:25Z date_updated: 2021-02-03T10:37:28Z file_id: '9063' file_name: thesis_source.zip file_size: 13446994 relation: source_file - access_level: open_access checksum: 9cc8af266579a464385bbe2aff6af606 content_type: application/pdf creator: patrickd date_created: 2021-02-02T14:09:18Z date_updated: 2021-02-02T14:09:18Z file_id: '9064' file_name: thesis_pdfA2b.pdf file_size: 5210329 relation: main_file success: 1 file_date_updated: 2021-02-03T10:37:28Z has_accepted_license: '1' language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: '134' place: Klosterneuburg publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '187' relation: part_of_dissertation status: public - id: '8703' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Multi-cover persistence and Delaunay mosaics 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) type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2021' ... --- _id: '7460' abstract: - lang: eng 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." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Katharina full_name: Ölsböck, Katharina id: 4D4AA390-F248-11E8-B48F-1D18A9856A87 last_name: Ölsböck orcid: 0000-0002-4672-8297 citation: ama: Ölsböck K. The hole system of triangulated shapes. 2020. doi:10.15479/AT:ISTA:7460 apa: Ölsböck, K. (2020). The hole system of triangulated shapes. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7460 chicago: Ölsböck, Katharina. “The Hole System of Triangulated Shapes.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7460. ieee: K. Ölsböck, “The hole system of triangulated shapes,” Institute of Science and Technology Austria, 2020. ista: Ölsböck K. 2020. The hole system of triangulated shapes. Institute of Science and Technology Austria. mla: Ölsböck, Katharina. The Hole System of Triangulated Shapes. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7460. short: K. Ölsböck, The Hole System of Triangulated Shapes, Institute of Science and Technology Austria, 2020. date_created: 2020-02-06T14:56:53Z date_published: 2020-02-10T00:00:00Z date_updated: 2023-09-07T13:15:30Z day: '10' ddc: - '514' degree_awarded: PhD department: - _id: HeEd - _id: GradSch doi: 10.15479/AT:ISTA:7460 file: - access_level: open_access checksum: 1df9f8c530b443c0e63a3f2e4fde412e content_type: application/pdf creator: koelsboe date_created: 2020-02-06T14:43:54Z date_updated: 2020-07-14T12:47:58Z file_id: '7461' file_name: thesis_ist-final_noack.pdf file_size: 76195184 relation: main_file - access_level: closed checksum: 7a52383c812b0be64d3826546509e5a4 content_type: application/x-zip-compressed creator: koelsboe date_created: 2020-02-06T14:52:45Z date_updated: 2020-07-14T12:47:58Z description: latex source files, figures file_id: '7462' file_name: latex-files.zip file_size: 122103715 relation: source_file file_date_updated: 2020-07-14T12:47:58Z has_accepted_license: '1' keyword: - shape reconstruction - hole manipulation - ordered complexes - Alpha complex - Wrap complex - computational topology - Bregman geometry language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: '155' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '6608' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: The hole system of triangulated shapes tmp: image: /images/cc_by_nc_sa.png legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) short: CC BY-NC-SA (4.0) type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2020' ... --- _id: '7944' abstract: - lang: eng 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." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Zuzana full_name: Masárová, Zuzana id: 45CFE238-F248-11E8-B48F-1D18A9856A87 last_name: Masárová orcid: 0000-0002-6660-1322 citation: ama: Masárová Z. Reconfiguration problems. 2020. doi:10.15479/AT:ISTA:7944 apa: Masárová, Z. (2020). Reconfiguration problems. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7944 chicago: Masárová, Zuzana. “Reconfiguration Problems.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7944. ieee: 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. Reconfiguration Problems. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7944. short: Z. Masárová, Reconfiguration Problems, Institute of Science and Technology Austria, 2020. date_created: 2020-06-08T00:49:46Z date_published: 2020-06-09T00:00:00Z date_updated: 2023-09-07T13:17:37Z day: '09' ddc: - '516' - '514' degree_awarded: PhD department: - _id: HeEd - _id: UlWa doi: 10.15479/AT:ISTA:7944 file: - access_level: open_access checksum: df688bc5a82b50baee0b99d25fc7b7f0 content_type: application/pdf creator: zmasarov date_created: 2020-06-08T00:34:00Z date_updated: 2020-07-14T12:48:05Z file_id: '7945' file_name: THESIS_Zuzka_Masarova.pdf file_size: 13661779 relation: main_file - access_level: closed checksum: 45341a35b8f5529c74010b7af43ac188 content_type: application/zip creator: zmasarov date_created: 2020-06-08T00:35:30Z date_updated: 2020-07-14T12:48:05Z file_id: '7946' file_name: THESIS_Zuzka_Masarova_SOURCE_FILES.zip file_size: 32184006 relation: source_file file_date_updated: 2020-07-14T12:48:05Z has_accepted_license: '1' keyword: - reconfiguration - reconfiguration graph - triangulations - flip - constrained triangulations - shellability - piecewise-linear balls - token swapping - trees - coloured weighted token swapping language: - iso: eng license: https://creativecommons.org/licenses/by-sa/4.0/ month: '06' oa: 1 oa_version: Published Version page: '160' publication_identifier: isbn: - 978-3-99078-005-3 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '7950' relation: part_of_dissertation status: public - id: '5986' relation: part_of_dissertation status: public status: public supervisor: - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Reconfiguration problems tmp: image: /images/cc_by_sa.png legal_code_url: https://creativecommons.org/licenses/by-sa/4.0/legalcode name: Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) short: CC BY-SA (4.0) type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2020' ... --- _id: '201' 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.' alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: Iglesias Ham M. Multiple covers with balls. 2018. doi:10.15479/AT:ISTA:th_1026 apa: Iglesias Ham, M. (2018). Multiple covers with balls. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_1026 chicago: Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:th_1026. ieee: M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018. ista: Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria. mla: Iglesias Ham, Mabel. Multiple Covers with Balls. Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:th_1026. short: M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018. date_created: 2018-12-11T11:45:10Z date_published: 2018-06-11T00:00:00Z date_updated: 2023-09-07T12:25:32Z day: '11' ddc: - '514' - '516' degree_awarded: PhD department: - _id: HeEd doi: 10.15479/AT:ISTA:th_1026 file: - access_level: closed checksum: dd699303623e96d1478a6ae07210dd05 content_type: application/zip creator: kschuh date_created: 2019-02-05T07:43:31Z date_updated: 2020-07-14T12:45:24Z file_id: '5918' file_name: IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip file_size: 11827713 relation: source_file - access_level: open_access checksum: ba163849a190d2b41d66fef0e4983294 content_type: application/pdf creator: kschuh date_created: 2019-02-05T07:43:45Z date_updated: 2020-07-14T12:45:24Z file_id: '5919' file_name: IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf file_size: 4783846 relation: main_file file_date_updated: 2020-07-14T12:45:24Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: '171' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria publist_id: '7712' pubrep_id: '1026' status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: Multiple covers with balls type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2018' ... --- _id: '6287' abstract: - lang: eng 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. alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Anton full_name: Nikitenko, Anton id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87 last_name: Nikitenko orcid: 0000-0002-0659-3201 citation: ama: Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:10.15479/AT:ISTA:th_873 apa: Nikitenko, A. (2017). Discrete Morse theory for random complexes . Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_873 chicago: Nikitenko, Anton. “Discrete Morse Theory for Random Complexes .” Institute of Science and Technology Austria, 2017. https://doi.org/10.15479/AT:ISTA:th_873. ieee: 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. mla: Nikitenko, Anton. Discrete Morse Theory for Random Complexes . Institute of Science and Technology Austria, 2017, doi:10.15479/AT:ISTA:th_873. short: A. Nikitenko, Discrete Morse Theory for Random Complexes , Institute of Science and Technology Austria, 2017. date_created: 2019-04-09T15:04:32Z date_published: 2017-10-27T00:00:00Z date_updated: 2023-09-15T12:10:34Z day: '27' ddc: - '514' - '516' - '519' degree_awarded: PhD department: - _id: HeEd doi: 10.15479/AT:ISTA:th_873 file: - access_level: open_access checksum: ece7e598a2f060b263c2febf7f3fe7f9 content_type: application/pdf creator: dernst date_created: 2019-04-09T14:54:51Z date_updated: 2020-07-14T12:47:26Z file_id: '6289' file_name: 2017_Thesis_Nikitenko.pdf file_size: 2324870 relation: main_file - access_level: closed checksum: 99b7ad76e317efd447af60f91e29b49b content_type: application/zip creator: dernst date_created: 2019-04-09T14:54:51Z date_updated: 2020-07-14T12:47:26Z file_id: '6290' file_name: 2017_Thesis_Nikitenko_source.zip file_size: 2863219 relation: source_file file_date_updated: 2020-07-14T12:47:26Z has_accepted_license: '1' language: - iso: eng month: '10' oa: 1 oa_version: Published Version page: '86' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria pubrep_id: '873' related_material: record: - id: '718' relation: part_of_dissertation status: public - id: '5678' relation: part_of_dissertation status: public - id: '87' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: 'Discrete Morse theory for random complexes ' 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) type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2017' ... --- _id: '1399' abstract: - lang: eng 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. alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Florian full_name: Pausinger, Florian id: 2A77D7A2-F248-11E8-B48F-1D18A9856A87 last_name: Pausinger orcid: 0000-0002-8379-3768 citation: ama: Pausinger F. On the approximation of intrinsic volumes. 2015. apa: Pausinger, F. (2015). On the approximation of intrinsic volumes. Institute of Science and Technology Austria. 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. ista: Pausinger F. 2015. On the approximation of intrinsic volumes. Institute of Science and Technology Austria. mla: Pausinger, Florian. On the Approximation of Intrinsic Volumes. Institute of Science and Technology Austria, 2015. short: F. Pausinger, On the Approximation of Intrinsic Volumes, Institute of Science and Technology Austria, 2015. date_created: 2018-12-11T11:51:48Z date_published: 2015-06-01T00:00:00Z date_updated: 2023-09-07T11:41:25Z day: '01' degree_awarded: PhD department: - _id: HeEd language: - iso: eng month: '06' oa_version: None page: '144' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria publist_id: '5808' related_material: record: - id: '1662' relation: part_of_dissertation status: public - id: '1792' relation: part_of_dissertation status: public - id: '2255' relation: part_of_dissertation status: public status: public supervisor: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 title: On the approximation of intrinsic volumes type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2015' ...