[{"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","title":"Braiding geometry and topology to study shapes and data","OA_place":"publisher","oa":1,"has_accepted_license":"1","supervisor":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","first_name":"Uli","orcid":"0000-0002-1494-0568"}],"date_created":"2026-01-20T21:38:40Z","publication_identifier":{"issn":["2663-337X"]},"abstract":[{"text":"This thesis examines how geometry and topology intersect in the representation, transformation, and analysis of complex shapes. It considers how continuous manifolds relate to their discrete analogues, how topological structures evolve in persistence vineyards, and how tools from topological data analysis can illuminate problems in mathematical physics. Central to this exploration is the question of how structure, both geometric and topological, persists or changes under approximation, sampling, or deformation. The work develops new approaches to skeletal and grid-based representations of surfaces, reveals the full expressive capacity of persistence vineyards, and applies topological methods to the longstanding problem of equilibria in electrostatic fields. These threads braid together into a broader understanding of how topology and geometry inform one another across theory, computation, and application.","lang":"eng"}],"type":"dissertation","date_updated":"2026-04-07T11:42:49Z","file_date_updated":"2026-01-30T11:40:09Z","department":[{"_id":"GradSch"},{"_id":"HeEd"},{"_id":"UlWa"}],"publication_status":"published","day":"21","doi":"10.15479/AT-ISTA-21021","related_material":{"record":[{"status":"public","id":"20260","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"21050","status":"public"},{"relation":"part_of_dissertation","id":"21051","status":"public"}]},"month":"01","corr_author":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"acknowledged_ssus":[{"_id":"M-Shop"},{"_id":"ScienComp"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"file_name":"2025_Fillmore_Christopher_Thesis.pdf","file_id":"21046","date_created":"2026-01-26T19:44:46Z","creator":"cfillmor","file_size":55954297,"date_updated":"2026-01-30T11:40:09Z","access_level":"open_access","checksum":"4c0889130095c31d4e5088c5b8dfd607","relation":"main_file","content_type":"application/pdf"},{"date_updated":"2026-01-26T19:46:20Z","creator":"cfillmor","file_size":166080788,"file_id":"21047","date_created":"2026-01-26T19:46:20Z","file_name":"Thesis.zip","content_type":"application/x-zip-compressed","checksum":"d69afb71d82ab98f856886126ee7303a","relation":"source_file","access_level":"closed"}],"citation":{"mla":"Fillmore, Christopher D. <i>Braiding Geometry and Topology to Study Shapes and Data</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>.","chicago":"Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes and Data.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>.","short":"C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute of Science and Technology Austria, 2026.","ista":"Fillmore CD. 2026. Braiding geometry and topology to study shapes and data. Institute of Science and Technology Austria.","ieee":"C. D. Fillmore, “Braiding geometry and topology to study shapes and data,” Institute of Science and Technology Austria, 2026.","apa":"Fillmore, C. D. (2026). <i>Braiding geometry and topology to study shapes and data</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-21021\">https://doi.org/10.15479/AT-ISTA-21021</a>","ama":"Fillmore CD. Braiding geometry and topology to study shapes and data. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-21021\">10.15479/AT-ISTA-21021</a>"},"status":"public","acknowledgement":"The research presented in this thesis was funded by the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n","alternative_title":["ISTA Thesis"],"_id":"21021","author":[{"first_name":"Christopher D","last_name":"Fillmore","full_name":"Fillmore, Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"}],"date_published":"2026-01-21T00:00:00Z","page":"122","article_processing_charge":"No","ddc":["514","516"],"publisher":"Institute of Science and Technology Austria","year":"2026","degree_awarded":"PhD"},{"date_created":"2025-01-31T17:04:40Z","supervisor":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"publication_identifier":{"issn":["2663-337X"]},"abstract":[{"lang":"eng","text":"Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot."}],"type":"dissertation","date_updated":"2026-04-07T11:47:30Z","file_date_updated":"2025-02-04T16:22:07Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","title":"Structures and computations in topological data analysis","OA_place":"publisher","has_accepted_license":"1","oa":1,"keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"file_id":"18983","date_created":"2025-01-31T16:58:30Z","file_name":"Thesis.zip","date_updated":"2025-01-31T16:58:30Z","file_size":11899491,"creator":"odragano","access_level":"closed","content_type":"application/zip","relation":"source_file","checksum":"af6567e5d35e5eb330b8925ae37f1998"},{"creator":"odragano","file_size":8857514,"date_updated":"2025-02-04T16:22:07Z","file_name":"Thesis.pdf","file_id":"19000","date_created":"2025-02-04T16:22:07Z","checksum":"c3fef68e35b9dc2020b2ca6006da6343","relation":"main_file","content_type":"application/pdf","access_level":"open_access"}],"citation":{"ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>","apa":"Draganov, O. (2025). <i>Structures and computations in topological data analysis</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","mla":"Draganov, Ondrej. <i>Structures and Computations in Topological Data Analysis</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>.","chicago":"Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>.","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria."},"status":"public","acknowledgement":"The research presented in this thesis was funded with the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publication_status":"published","day":"03","doi":"10.15479/at:ista:18979","related_material":{"record":[{"id":"15091","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18981"}]},"month":"02","corr_author":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","alternative_title":["ISTA Thesis"],"_id":"18979","author":[{"last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","full_name":"Draganov, Ondrej","first_name":"Ondrej","orcid":"0000-0003-0464-3823"}],"ddc":["514","004"],"project":[{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"publisher":"Institute of Science and Technology Austria","year":"2025","degree_awarded":"PhD","date_published":"2025-02-03T00:00:00Z","page":"140","article_processing_charge":"No"},{"_id":"15094","alternative_title":["ISTA Thesis"],"author":[{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"}],"date_published":"2024-03-08T00:00:00Z","article_processing_charge":"No","page":"108","publisher":"Institute of Science and Technology Austria","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"ddc":["514","500","516"],"degree_awarded":"PhD","year":"2024","OA_place":"publisher","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","title":"Persistence and Morse theory for discrete geometric structures","oa":1,"has_accepted_license":"1","abstract":[{"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","lang":"eng"}],"publication_identifier":{"issn":["2663-337X"]},"date_created":"2024-03-08T15:28:10Z","supervisor":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"}],"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","file_date_updated":"2024-03-14T14:14:35Z","date_updated":"2026-04-07T12:58:48Z","type":"dissertation","corr_author":"1","month":"03","related_material":{"record":[{"id":"15091","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"11660","relation":"part_of_dissertation"},{"status":"public","id":"15090","relation":"part_of_dissertation"},{"status":"public","id":"15093","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"13182"},{"relation":"part_of_dissertation","id":"11658","status":"public"}]},"day":"08","ec_funded":1,"doi":"10.15479/at:ista:15094","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publication_status":"published","language":[{"iso":"eng"}],"oa_version":"Published Version","status":"public","file":[{"relation":"main_file","checksum":"1e468bfa42a7dcf04d89f4dadc621c87","content_type":"application/pdf","success":1,"access_level":"open_access","file_size":4106872,"creator":"scultrer","date_updated":"2024-03-14T08:55:07Z","file_name":"Thesis Sebastiano.pdf","file_id":"15112","date_created":"2024-03-14T08:55:07Z"},{"file_id":"15113","date_created":"2024-03-14T08:56:24Z","file_name":"Thesis (1).zip","date_updated":"2024-03-14T14:14:35Z","file_size":4746234,"creator":"scultrer","access_level":"closed","content_type":"application/zip","relation":"source_file","checksum":"bcbd213490f5a7e68855a092bbce93f1"}],"citation":{"ieee":"S. Cultrera di Montesano, “Persistence and Morse theory for discrete geometric structures,” Institute of Science and Technology Austria, 2024.","ama":"Cultrera di Montesano S. Persistence and Morse theory for discrete geometric structures. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>","apa":"Cultrera di Montesano, S. (2024). <i>Persistence and Morse theory for discrete geometric structures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>","mla":"Cultrera di Montesano, Sebastiano. <i>Persistence and Morse Theory for Discrete Geometric Structures</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:15094\">10.15479/at:ista:15094</a>.","chicago":"Cultrera di Montesano, Sebastiano. “Persistence and Morse Theory for Discrete Geometric Structures.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:15094\">https://doi.org/10.15479/at:ista:15094</a>.","short":"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."},"tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","short":"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"}},{"publication_identifier":{"isbn":["978-3-99078-052-7"],"issn":["2663-337X"]},"date_created":"2024-12-17T16:17:55Z","supervisor":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"abstract":[{"text":"Many chemical and physical properties of materials are determined by the material’s shape,\r\nfor example the size of its pores and the width of its tunnels. This makes materials science\r\na prime application area for geometrical and topological methods. Nevertheless many\r\nmethods in topological data analysis have not been satisfyingly extended to the needs of\r\nmaterials science. This thesis provides new methods and new mathematical theorems\r\ntargeted at those specific needs by answering four different research questions. While the\r\nmotivation for each of the research questions arises from materials science, the methods\r\nare versatile and can be applied in different areas as well. \r\n\r\nThe first research question is concerned with image data, for example a three-dimensional\r\ncomputed tomography (CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies for digital images and depending on the application either of them might be\r\nrequired. However, software for computing the topological data analysis method persistence\r\nhomology, usually supports only one of the two topologies. We answer the question how to\r\ncompute persistent homology of an image with respect to one of the two topologies using\r\nsoftware that is intended for the other topology. \r\n\r\nThe second research question is concerned with image data as well, and asks how much\r\nof the topological information of an image is lost when the resolution is coarsened. As\r\ncomputer tomography scanners are more expensive the higher the resolution, it is an\r\nimportant question in materials science to know which resolution is enough to get satisfying\r\npersistent homology. We give theoretical bounds on the information loss based on different\r\ngeometrical properties of the object to be scanned. In addition, we conduct experiments on\r\nsand and stone CT image data. \r\n\r\nThe third research question is motivated by comparing crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically, crystalline materials are either modeled by\r\nunmanageable infinite periodic point sets, or by one of their fundamental domains, which is\r\nunstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate properties such that comparing the crystals can be eased by comparing the\r\nfingerprints instead. We define the density fingerprint and prove the necessary properties. \r\n\r\nThe fourth research question is motivated by studying the hole-structure or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials. A common way to deal\r\nwith periodicity is to take a fundamental domain and identify opposite boundaries to form a\r\ntorus. However, computing persistent homology or merge trees on that torus loses some\r\nof the information materials scientists are interested in and is additionally not stable under\r\ncertain noise. We therefore decorate the merge tree stemming from the torus with additional\r\ninformation describing the density and growth rate of the periodic copies of a component\r\nwithin a growing spherical window. We prove all desired properties, like stability and efficient\r\ncomputability.","lang":"eng"}],"date_updated":"2026-04-07T12:54:10Z","type":"dissertation","file_date_updated":"2024-12-19T10:24:50Z","title":"New methods for applying topological data analysis to materials science","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","OA_place":"publisher","oa":1,"has_accepted_license":"1","keyword":["persistent homology","topological data analysis","periodic","crystalline materials","images","fingerprint"],"citation":{"mla":"Heiss, Teresa. <i>New Methods for Applying Topological Data Analysis to Materials Science</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:18667\">10.15479/at:ista:18667</a>.","chicago":"Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials Science.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:18667\">https://doi.org/10.15479/at:ista:18667</a>.","short":"T. Heiss, New Methods for Applying Topological Data Analysis to Materials Science, Institute of Science and Technology Austria, 2024.","ista":"Heiss T. 2024. New methods for applying topological data analysis to materials science. Institute of Science and Technology Austria.","ieee":"T. Heiss, “New methods for applying topological data analysis to materials science,” Institute of Science and Technology Austria, 2024.","apa":"Heiss, T. (2024). <i>New methods for applying topological data analysis to materials science</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18667\">https://doi.org/10.15479/at:ista:18667</a>","ama":"Heiss T. New methods for applying topological data analysis to materials science. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:18667\">10.15479/at:ista:18667</a>"},"file":[{"file_name":"Teresa_Heiss_PhD_Thesis_final.pdf","file_id":"18686","date_created":"2024-12-19T10:24:46Z","file_size":7752253,"creator":"theiss","date_updated":"2024-12-19T10:24:46Z","success":1,"access_level":"open_access","relation":"main_file","checksum":"247bb057aed2fba1cd4711917aaa2d77","content_type":"application/pdf"},{"file_name":"PhD_Thesis.zip","date_created":"2024-12-19T10:24:50Z","file_id":"18687","creator":"theiss","file_size":17197731,"date_updated":"2024-12-19T10:24:50Z","access_level":"closed","checksum":"9648b45c07a008ee11a07f99856a139d","relation":"source_file","content_type":"application/zip"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"status":"public","acknowledgement":"I was supported by the European Research Council (ERC) Horizon 2020 project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship. In addition,\r\nI am very thankful for having been able to attend the second Workshop for Women in\r\nComputational Topology in July 2019, funded by the Mathematical Sciences Institute at\r\nANU, the US National Science Foundation through the award CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association for Women in Mathematics. Two of the\r\nprojects presented in this thesis started there. One of them reached completion thanks to\r\nfunding from the MSRI Summer Research in Mathematics program awarded to me and my\r\ncollaborators in 2020.","ec_funded":1,"doi":"10.15479/at:ista:18667","day":"17","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publication_status":"published","month":"12","corr_author":"1","related_material":{"record":[{"status":"public","id":"10828","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"11440"},{"id":"18673","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"9345","relation":"part_of_dissertation"}]},"language":[{"iso":"eng"}],"oa_version":"Published Version","_id":"18667","alternative_title":["ISTA Thesis"],"author":[{"orcid":"0000-0002-1780-2689","first_name":"Teresa","last_name":"Heiss","full_name":"Heiss, Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87"}],"ddc":["514","516","004"],"publisher":"Institute of Science and Technology Austria","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"}],"degree_awarded":"PhD","year":"2024","date_published":"2024-12-17T00:00:00Z","page":"111","article_processing_charge":"No"},{"ddc":["500"],"publisher":"Institute of Science and Technology Austria","year":"2023","degree_awarded":"MS","date_published":"2023-08-24T00:00:00Z","page":"43","article_processing_charge":"No","alternative_title":["ISTA Master's Thesis"],"_id":"14226","author":[{"orcid":"0000-0002-6862-208X","first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson"}],"citation":{"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.","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>.","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>","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>","ieee":"E. R. Stephenson, “Generalizing medial axes with homology switches,” Institute of Science and Technology Austria, 2023."},"file":[{"file_name":"documents-export-2023-08-24.zip","date_created":"2023-08-24T13:02:49Z","file_id":"14227","creator":"cchlebak","file_size":15501411,"date_updated":"2024-02-26T23:30:03Z","access_level":"closed","checksum":"453caf851d75c3478c10ed09bd242a91","relation":"source_file","content_type":"application/x-zip-compressed","embargo_to":"open_access"},{"content_type":"application/pdf","checksum":"7349d29963d6695e555e171748648d9a","relation":"main_file","access_level":"open_access","embargo":"2024-02-25","date_updated":"2024-02-26T23:30:03Z","creator":"cchlebak","file_size":6854783,"date_created":"2023-08-24T13:03:42Z","file_id":"14228","file_name":"thesis_pdf_a.pdf"}],"status":"public","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"publication_status":"published","doi":"10.15479/at:ista:14226","day":"24","month":"08","corr_author":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"supervisor":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"}],"date_created":"2023-08-24T13:01:18Z","publication_identifier":{"issn":["2791-4585"]},"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."}],"type":"dissertation","date_updated":"2026-04-07T14:02:30Z","file_date_updated":"2024-02-26T23:30:03Z","title":"Generalizing medial axes with homology switches","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","OA_place":"publisher","has_accepted_license":"1","oa":1},{"author":[{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","full_name":"Osang, Georg F","last_name":"Osang","first_name":"Georg F","orcid":"0000-0002-8882-5116"}],"alternative_title":["ISTA Thesis"],"_id":"9056","year":"2021","degree_awarded":"PhD","publisher":"Institute of Science and Technology Austria","ddc":["006","514","516"],"article_processing_charge":"No","page":"134","date_published":"2021-02-01T00:00:00Z","file_date_updated":"2021-02-03T10:37:28Z","type":"dissertation","date_updated":"2026-04-08T07:01:30Z","abstract":[{"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.","lang":"eng"}],"date_created":"2021-02-02T14:11:06Z","supervisor":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"}],"publication_identifier":{"issn":["2663-337X"]},"place":"Klosterneuburg","has_accepted_license":"1","oa":1,"OA_place":"publisher","title":"Multi-cover persistence and Delaunay mosaics","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"citation":{"ieee":"G. F. Osang, “Multi-cover persistence and Delaunay mosaics,” Institute of Science and Technology Austria, Klosterneuburg, 2021.","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>","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.","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>.","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>.","ista":"Osang GF. 2021. Multi-cover persistence and Delaunay mosaics. Klosterneuburg: Institute of Science and Technology Austria."},"file":[{"date_updated":"2021-02-03T10:37:28Z","creator":"patrickd","file_size":13446994,"file_id":"9063","date_created":"2021-02-02T14:09:25Z","file_name":"thesis_source.zip","content_type":"application/zip","checksum":"bcf27986147cab0533b6abadd74e7629","relation":"source_file","access_level":"closed"},{"file_size":5210329,"creator":"patrickd","date_updated":"2021-02-02T14:09:18Z","file_name":"thesis_pdfA2b.pdf","file_id":"9064","date_created":"2021-02-02T14:09:18Z","relation":"main_file","checksum":"9cc8af266579a464385bbe2aff6af606","content_type":"application/pdf","success":1,"access_level":"open_access"}],"language":[{"iso":"eng"}],"oa_version":"Published Version","related_material":{"record":[{"relation":"part_of_dissertation","id":"187","status":"public"},{"status":"public","id":"8703","relation":"part_of_dissertation"}]},"month":"02","corr_author":"1","publication_status":"published","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"day":"01","doi":"10.15479/AT:ISTA:9056"},{"_id":"7944","alternative_title":["ISTA Thesis"],"author":[{"last_name":"Masárová","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","full_name":"Masárová, Zuzana","first_name":"Zuzana","orcid":"0000-0002-6660-1322"}],"publisher":"Institute of Science and Technology Austria","ddc":["516","514"],"degree_awarded":"PhD","year":"2020","date_published":"2020-06-09T00:00:00Z","article_processing_charge":"No","page":"160","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."}],"publication_identifier":{"isbn":["978-3-99078-005-3"],"issn":["2663-337X"]},"supervisor":[{"first_name":"Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"}],"date_created":"2020-06-08T00:49:46Z","license":"https://creativecommons.org/licenses/by-sa/4.0/","file_date_updated":"2020-07-14T12:48:05Z","date_updated":"2026-04-08T07:23:01Z","type":"dissertation","OA_place":"publisher","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","title":"Reconfiguration problems","keyword":["reconfiguration","reconfiguration graph","triangulations","flip","constrained triangulations","shellability","piecewise-linear balls","token swapping","trees","coloured weighted token swapping"],"has_accepted_license":"1","oa":1,"status":"public","citation":{"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>.","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>.","short":"Z. Masárová, Reconfiguration Problems, Institute of Science and Technology Austria, 2020.","ista":"Masárová Z. 2020. Reconfiguration problems. Institute of Science and Technology Austria.","ieee":"Z. Masárová, “Reconfiguration problems,” Institute of Science and Technology Austria, 2020.","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>"},"file":[{"relation":"main_file","checksum":"df688bc5a82b50baee0b99d25fc7b7f0","content_type":"application/pdf","access_level":"open_access","file_size":13661779,"creator":"zmasarov","date_updated":"2020-07-14T12:48:05Z","file_name":"THESIS_Zuzka_Masarova.pdf","date_created":"2020-06-08T00:34:00Z","file_id":"7945"},{"relation":"source_file","checksum":"45341a35b8f5529c74010b7af43ac188","content_type":"application/zip","access_level":"closed","file_size":32184006,"creator":"zmasarov","date_updated":"2020-07-14T12:48:05Z","file_name":"THESIS_Zuzka_Masarova_SOURCE_FILES.zip","date_created":"2020-06-08T00:35:30Z","file_id":"7946"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","image":"/images/cc_by_sa.png","short":"CC BY-SA (4.0)","name":"Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)"},"month":"06","corr_author":"1","related_material":{"record":[{"id":"7950","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"5986"}]},"doi":"10.15479/AT:ISTA:7944","day":"09","publication_status":"published","department":[{"_id":"HeEd"},{"_id":"UlWa"}],"oa_version":"Published Version","language":[{"iso":"eng"}]},{"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"6608"}]},"month":"02","corr_author":"1","department":[{"_id":"HeEd"},{"_id":"GradSch"}],"publication_status":"published","day":"10","doi":"10.15479/AT:ISTA:7460","language":[{"iso":"eng"}],"oa_version":"Published Version","status":"public","tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","short":"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"},"file":[{"file_id":"7461","date_created":"2020-02-06T14:43:54Z","file_name":"thesis_ist-final_noack.pdf","date_updated":"2020-07-14T12:47:58Z","creator":"koelsboe","file_size":76195184,"access_level":"open_access","content_type":"application/pdf","checksum":"1df9f8c530b443c0e63a3f2e4fde412e","relation":"main_file"},{"description":"latex source files, figures","access_level":"closed","checksum":"7a52383c812b0be64d3826546509e5a4","relation":"source_file","content_type":"application/x-zip-compressed","file_name":"latex-files.zip","file_id":"7462","date_created":"2020-02-06T14:52:45Z","creator":"koelsboe","file_size":122103715,"date_updated":"2020-07-14T12:47:58Z"}],"citation":{"ieee":"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>","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>","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>.","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."},"OA_place":"publisher","title":"The hole system of triangulated shapes","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","keyword":["shape reconstruction","hole manipulation","ordered complexes","Alpha complex","Wrap complex","computational topology","Bregman geometry"],"has_accepted_license":"1","oa":1,"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"}],"date_created":"2020-02-06T14:56:53Z","supervisor":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"}],"publication_identifier":{"issn":["2663-337X"]},"file_date_updated":"2020-07-14T12:47:58Z","type":"dissertation","date_updated":"2026-04-08T07:23:21Z","date_published":"2020-02-10T00:00:00Z","article_processing_charge":"No","page":"155","publisher":"Institute of Science and Technology Austria","ddc":["514"],"year":"2020","degree_awarded":"PhD","alternative_title":["ISTA Thesis"],"_id":"7460","author":[{"last_name":"Ölsböck","full_name":"Ölsböck, Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4672-8297","first_name":"Katharina"}]},{"publist_id":"7712","author":[{"first_name":"Mabel","last_name":"Iglesias Ham","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","full_name":"Iglesias Ham, Mabel"}],"_id":"201","alternative_title":["ISTA Thesis"],"pubrep_id":"1026","degree_awarded":"PhD","year":"2018","ddc":["514","516"],"publisher":"Institute of Science and Technology Austria","page":"171","article_processing_charge":"No","date_published":"2018-06-11T00:00:00Z","date_updated":"2026-04-08T14:04:03Z","type":"dissertation","file_date_updated":"2020-07-14T12:45:24Z","publication_identifier":{"issn":["2663-337X"]},"date_created":"2018-12-11T11:45:10Z","supervisor":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"abstract":[{"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.","lang":"eng"}],"has_accepted_license":"1","oa":1,"title":"Multiple covers with balls","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","OA_place":"publisher","citation":{"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.","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>.","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>.","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>","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."},"file":[{"access_level":"closed","content_type":"application/zip","relation":"source_file","checksum":"dd699303623e96d1478a6ae07210dd05","file_id":"5918","date_created":"2019-02-05T07:43:31Z","file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","date_updated":"2020-07-14T12:45:24Z","file_size":11827713,"creator":"kschuh"},{"file_size":4783846,"creator":"kschuh","date_updated":"2020-07-14T12:45:24Z","file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","date_created":"2019-02-05T07:43:45Z","file_id":"5919","relation":"main_file","checksum":"ba163849a190d2b41d66fef0e4983294","content_type":"application/pdf","access_level":"open_access"}],"status":"public","oa_version":"Published Version","language":[{"iso":"eng"}],"day":"11","doi":"10.15479/AT:ISTA:th_1026","publication_status":"published","department":[{"_id":"HeEd"}],"corr_author":"1","month":"06"},{"year":"2017","degree_awarded":"PhD","pubrep_id":"873","ddc":["514","516","519"],"publisher":"Institute of Science and Technology Austria","page":"86","article_processing_charge":"No","date_published":"2017-10-27T00:00:00Z","author":[{"first_name":"Anton","orcid":"0000-0002-0659-3201","last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton"}],"alternative_title":["ISTA Thesis"],"_id":"6287","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"content_type":"application/pdf","relation":"main_file","checksum":"ece7e598a2f060b263c2febf7f3fe7f9","access_level":"open_access","date_updated":"2020-07-14T12:47:26Z","file_size":2324870,"creator":"dernst","date_created":"2019-04-09T14:54:51Z","file_id":"6289","file_name":"2017_Thesis_Nikitenko.pdf"},{"date_created":"2019-04-09T14:54:51Z","file_id":"6290","file_name":"2017_Thesis_Nikitenko_source.zip","date_updated":"2020-07-14T12:47:26Z","file_size":2863219,"creator":"dernst","access_level":"closed","content_type":"application/zip","relation":"source_file","checksum":"99b7ad76e317efd447af60f91e29b49b"}],"citation":{"short":"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>.","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>.","ista":"Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute of Science and Technology Austria.","ieee":"A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of Science and Technology Austria, 2017.","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>","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>"},"status":"public","oa_version":"Published Version","language":[{"iso":"eng"}],"publication_status":"published","department":[{"_id":"HeEd"}],"day":"27","doi":"10.15479/AT:ISTA:th_873","related_material":{"record":[{"status":"public","id":"87","relation":"part_of_dissertation"},{"id":"5678","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"718","relation":"part_of_dissertation"}]},"month":"10","corr_author":"1","type":"dissertation","date_updated":"2026-04-08T14:19:31Z","file_date_updated":"2020-07-14T12:47:26Z","supervisor":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"date_created":"2019-04-09T15:04:32Z","publication_identifier":{"issn":["2663-337X"]},"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"}],"has_accepted_license":"1","oa":1,"title":"Discrete Morse theory for random complexes ","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","OA_place":"publisher"},{"type":"dissertation","date_updated":"2026-04-16T10:09:04Z","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."}],"date_created":"2018-12-11T11:51:48Z","supervisor":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833"}],"publication_identifier":{"issn":["2663-337X"]},"publist_id":"5808","author":[{"id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87","full_name":"Pausinger, Florian","last_name":"Pausinger","first_name":"Florian","orcid":"0000-0002-8379-3768"}],"OA_place":"publisher","title":"On the approximation of intrinsic volumes","alternative_title":["ISTA Thesis"],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","_id":"1399","year":"2015","degree_awarded":"PhD","status":"public","publisher":"Institute of Science and Technology Austria","citation":{"ieee":"F. Pausinger, “On the approximation of intrinsic volumes,” Institute of Science and Technology Austria, 2015.","ama":"Pausinger F. On the approximation of intrinsic volumes. 2015.","apa":"Pausinger, F. (2015). <i>On the approximation of intrinsic volumes</i>. Institute of Science and Technology Austria.","chicago":"Pausinger, Florian. “On the Approximation of Intrinsic Volumes.” Institute of Science and Technology Austria, 2015.","mla":"Pausinger, Florian. <i>On the Approximation of Intrinsic Volumes</i>. Institute of Science and Technology Austria, 2015.","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."},"article_processing_charge":"No","page":"144","oa_version":"None","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"1662","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"1792","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"2255"}]},"month":"06","date_published":"2015-06-01T00:00:00Z","corr_author":"1","publication_status":"published","department":[{"_id":"HeEd"}],"day":"01"}]