--- _id: '8135' abstract: - lang: eng text: Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics. acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF). alternative_title: - Abel Symposia article_processing_charge: No author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Anton full_name: Nikitenko, Anton id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87 last_name: Nikitenko - first_name: Katharina full_name: Ölsböck, Katharina id: 4D4AA390-F248-11E8-B48F-1D18A9856A87 last_name: Ölsböck - first_name: Peter full_name: Synak, Peter id: 331776E2-F248-11E8-B48F-1D18A9856A87 last_name: Synak citation: ama: 'Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: Topological Data Analysis. Vol 15. Springer Nature; 2020:181-218. doi:10.1007/978-3-030-43408-3_8' apa: Edelsbrunner, H., Nikitenko, A., Ölsböck, K., & Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In Topological Data Analysis (Vol. 15, pp. 181–218). Springer Nature. https://doi.org/10.1007/978-3-030-43408-3_8 chicago: Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In Topological Data Analysis, 15:181–218. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-43408-3_8. ieee: H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in Topological Data Analysis, 2020, vol. 15, pp. 181–218. ista: Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218. mla: Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” Topological Data Analysis, vol. 15, Springer Nature, 2020, pp. 181–218, doi:10.1007/978-3-030-43408-3_8. short: H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218. date_created: 2020-07-19T22:00:59Z date_published: 2020-06-22T00:00:00Z date_updated: 2021-01-12T08:17:06Z day: '22' ddc: - '510' department: - _id: HeEd doi: 10.1007/978-3-030-43408-3_8 ec_funded: 1 file: - access_level: open_access checksum: 7b5e0de10675d787a2ddb2091370b8d8 content_type: application/pdf creator: dernst date_created: 2020-10-08T08:56:14Z date_updated: 2020-10-08T08:56:14Z file_id: '8628' file_name: 2020-B-01-PoissonExperimentalSurvey.pdf file_size: 2207071 relation: main_file success: 1 file_date_updated: 2020-10-08T08:56:14Z has_accepted_license: '1' intvolume: ' 15' language: - iso: eng month: '06' oa: 1 oa_version: Submitted Version page: 181-218 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2533E772-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '638176' name: Efficient Simulation of Natural Phenomena at Extremely Large Scales - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Topological Data Analysis publication_identifier: eissn: - '21978549' isbn: - '9783030434076' issn: - '21932808' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Radius functions on Poisson–Delaunay mosaics and related complexes experimentally type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2020' ... --- _id: '7666' abstract: - lang: eng text: Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups. acknowledgement: This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF). article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Katharina full_name: Ölsböck, Katharina id: 4D4AA390-F248-11E8-B48F-1D18A9856A87 last_name: Ölsböck orcid: 0000-0002-4672-8297 citation: ama: Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 2020;64:759-775. doi:10.1007/s00454-020-00188-x apa: Edelsbrunner, H., & Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00188-x chicago: Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00188-x. ieee: H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 759–775, 2020. ista: Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775. mla: Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 759–75, doi:10.1007/s00454-020-00188-x. short: H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775. date_created: 2020-04-19T22:00:56Z date_published: 2020-03-20T00:00:00Z date_updated: 2023-08-21T06:13:48Z day: '20' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-020-00188-x ec_funded: 1 external_id: isi: - '000520918800001' file: - access_level: open_access checksum: f8cc96e497f00c38340b5dafe0cb91d7 content_type: application/pdf creator: dernst date_created: 2020-11-20T13:22:21Z date_updated: 2020-11-20T13:22:21Z file_id: '8786' file_name: 2020_DiscreteCompGeo_Edelsbrunner.pdf file_size: 701673 relation: main_file success: 1 file_date_updated: 2020-11-20T13:22:21Z has_accepted_license: '1' intvolume: ' 64' isi: 1 language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '03' oa: 1 oa_version: Published Version page: 759-775 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete and Computational Geometry publication_identifier: eissn: - '14320444' issn: - '01795376' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Tri-partitions and bases of an ordered complex 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: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 64 year: '2020' ... --- _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: '6608' abstract: - lang: eng text: We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram. article_processing_charge: No author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Katharina full_name: Ölsböck, Katharina id: 4D4AA390-F248-11E8-B48F-1D18A9856A87 last_name: Ölsböck orcid: 0000-0002-4672-8297 citation: ama: Edelsbrunner H, Ölsböck K. Holes and dependences in an ordered complex. Computer Aided Geometric Design. 2019;73:1-15. doi:10.1016/j.cagd.2019.06.003 apa: Edelsbrunner, H., & Ölsböck, K. (2019). Holes and dependences in an ordered complex. Computer Aided Geometric Design. Elsevier. https://doi.org/10.1016/j.cagd.2019.06.003 chicago: Edelsbrunner, Herbert, and Katharina Ölsböck. “Holes and Dependences in an Ordered Complex.” Computer Aided Geometric Design. Elsevier, 2019. https://doi.org/10.1016/j.cagd.2019.06.003. ieee: H. Edelsbrunner and K. Ölsböck, “Holes and dependences in an ordered complex,” Computer Aided Geometric Design, vol. 73. Elsevier, pp. 1–15, 2019. ista: Edelsbrunner H, Ölsböck K. 2019. Holes and dependences in an ordered complex. Computer Aided Geometric Design. 73, 1–15. mla: Edelsbrunner, Herbert, and Katharina Ölsböck. “Holes and Dependences in an Ordered Complex.” Computer Aided Geometric Design, vol. 73, Elsevier, 2019, pp. 1–15, doi:10.1016/j.cagd.2019.06.003. short: H. Edelsbrunner, K. Ölsböck, Computer Aided Geometric Design 73 (2019) 1–15. date_created: 2019-07-07T21:59:20Z date_published: 2019-08-01T00:00:00Z date_updated: 2023-09-07T13:15:29Z day: '01' ddc: - '000' department: - _id: HeEd doi: 10.1016/j.cagd.2019.06.003 ec_funded: 1 external_id: isi: - '000485207800001' file: - access_level: open_access checksum: 7c99be505dc7533257d42eb1830cef04 content_type: application/pdf creator: kschuh date_created: 2019-07-08T15:24:26Z date_updated: 2020-07-14T12:47:34Z file_id: '6624' file_name: Elsevier_2019_Edelsbrunner.pdf file_size: 2665013 relation: main_file file_date_updated: 2020-07-14T12:47:34Z has_accepted_license: '1' intvolume: ' 73' isi: 1 language: - iso: eng license: https://creativecommons.org/licenses/by-nc-nd/4.0/ month: '08' oa: 1 oa_version: Published Version page: 1-15 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Computer Aided Geometric Design publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '7460' relation: dissertation_contains status: public scopus_import: '1' status: public title: Holes and dependences in an ordered complex tmp: image: /images/cc_by_nc_nd.png legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) short: CC BY-NC-ND (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 73 year: '2019' ...