[{"citation":{"ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. AIMS, 2026. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. AIMS, pp. 30–62, 2026.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, AIMS, 2026, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2026;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). Chromatic alpha complexes. <i>Foundations of Data Science</i>. AIMS. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2026) 30–62."},"acknowledgement":"This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.","scopus_import":"1","_id":"20585","department":[{"_id":"HeEd"}],"publisher":"AIMS","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ec_funded":1,"external_id":{"arxiv":["2212.03128"]},"abstract":[{"lang":"eng","text":"Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided."}],"OA_type":"green","arxiv":1,"day":"01","mathsc":["62R40","55N31","68T09","57Q70"],"language":[{"iso":"eng"}],"author":[{"first_name":"Sebastiano","orcid":"0000-0001-6249-0832","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","orcid":"0000-0003-0464-3823","first_name":"Ondrej"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"status":"public","date_updated":"2026-08-12T06:19:48Z","publication_identifier":{"eissn":["2639-8001"]},"publication":"Foundations of Data Science","publication_status":"published","fulldoi":"https://doi.org/10.3934/fods.2025003","date_created":"2025-11-02T23:01:33Z","type":"journal_article","year":"2026","related_material":{"record":[{"status":"public","id":"15091","relation":"earlier_version"}]},"volume":8,"intvolume":"         8","article_processing_charge":"No","title":"Chromatic alpha complexes","keyword":["Topological data analysis","Delaunay mosaic","alpha complex","chromatic sets","persistent homology","kernel/image/cokernel persistent homology","radius function","discrete Morse theory","exact sequences"],"oa":1,"month":"03","corr_author":"1","page":"30-62","has_accepted_license":"1","oa_version":"Preprint","article_type":"original","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.03128"}],"ddc":["510"],"doi":"10.3934/fods.2025003","date_published":"2026-03-01T00:00:00Z","OA_place":"repository"},{"article_processing_charge":"No","title":"Structures and computations in topological data analysis","keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"supervisor":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"}],"oa":1,"month":"02","corr_author":"1","page":"140","has_accepted_license":"1","oa_version":"Published Version","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"ddc":["514","004"],"doi":"10.15479/at:ista:18979","date_published":"2025-02-03T00:00:00Z","OA_place":"publisher","date_updated":"2026-04-07T11:47:30Z","publication_status":"published","publication_identifier":{"issn":["2663-337X"]},"file":[{"access_level":"closed","file_id":"18983","file_size":11899491,"checksum":"af6567e5d35e5eb330b8925ae37f1998","date_updated":"2025-01-31T16:58:30Z","file_name":"Thesis.zip","date_created":"2025-01-31T16:58:30Z","content_type":"application/zip","creator":"odragano","relation":"source_file"},{"file_size":8857514,"file_id":"19000","checksum":"c3fef68e35b9dc2020b2ca6006da6343","access_level":"open_access","date_created":"2025-02-04T16:22:07Z","date_updated":"2025-02-04T16:22:07Z","file_name":"Thesis.pdf","content_type":"application/pdf","creator":"odragano","relation":"main_file"}],"fulldoi":"https://doi.org/10.15479/at:ista:18979","date_created":"2025-01-31T17:04:40Z","type":"dissertation","year":"2025","related_material":{"record":[{"relation":"part_of_dissertation","id":"15091","status":"public"},{"status":"public","id":"18981","relation":"part_of_dissertation"}]},"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."}],"file_date_updated":"2025-02-04T16:22:07Z","day":"03","language":[{"iso":"eng"}],"author":[{"first_name":"Ondrej","orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov"}],"project":[{"grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"status":"public","citation":{"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.","short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","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>","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>","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>.","ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025."},"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"}],"_id":"18979","degree_awarded":"PhD","publisher":"Institute of Science and Technology Austria","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","alternative_title":["ISTA Thesis"]}]