---
_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: <i>Topological Data Analysis</i>.
    Vol 15. Springer Nature; 2020:181-218. doi:<a href="https://doi.org/10.1007/978-3-030-43408-3_8">10.1007/978-3-030-43408-3_8</a>'
  apa: Edelsbrunner, H., Nikitenko, A., Ölsböck, K., &#38; Synak, P. (2020). Radius
    functions on Poisson–Delaunay mosaics and related complexes experimentally. In
    <i>Topological Data Analysis</i> (Vol. 15, pp. 181–218). Springer Nature. <a href="https://doi.org/10.1007/978-3-030-43408-3_8">https://doi.org/10.1007/978-3-030-43408-3_8</a>
  chicago: Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak.
    “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.”
    In <i>Topological Data Analysis</i>, 15:181–218. Springer Nature, 2020. <a href="https://doi.org/10.1007/978-3-030-43408-3_8">https://doi.org/10.1007/978-3-030-43408-3_8</a>.
  ieee: H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions
    on Poisson–Delaunay mosaics and related complexes experimentally,” in <i>Topological
    Data Analysis</i>, 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.” <i>Topological Data Analysis</i>, vol.
    15, Springer Nature, 2020, pp. 181–218, doi:<a href="https://doi.org/10.1007/978-3-030-43408-3_8">10.1007/978-3-030-43408-3_8</a>.
  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
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oa_version: Submitted Version
page: 181-218
project:
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  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
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  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: '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:<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>.
  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. <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>.
  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
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  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
license: https://creativecommons.org/licenses/by-nc-sa/4.0/
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:
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    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
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type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
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.
    <i>Discrete and Computational Geometry</i>. 2020;64:759-775. doi:<a href="https://doi.org/10.1007/s00454-020-00188-x">10.1007/s00454-020-00188-x</a>
  apa: Edelsbrunner, H., &#38; Ölsböck, K. (2020). Tri-partitions and bases of an
    ordered complex. <i>Discrete and Computational Geometry</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00454-020-00188-x">https://doi.org/10.1007/s00454-020-00188-x</a>
  chicago: Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases
    of an Ordered Complex.” <i>Discrete and Computational Geometry</i>. Springer Nature,
    2020. <a href="https://doi.org/10.1007/s00454-020-00188-x">https://doi.org/10.1007/s00454-020-00188-x</a>.
  ieee: H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,”
    <i>Discrete and Computational Geometry</i>, 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.” <i>Discrete and Computational Geometry</i>, vol. 64, Springer
    Nature, 2020, pp. 759–75, doi:<a href="https://doi.org/10.1007/s00454-020-00188-x">10.1007/s00454-020-00188-x</a>.
  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'
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license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
page: 759-775
project:
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  name: IST Austria Open Access Fund
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
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  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Discrete and Computational Geometry
publication_identifier:
  eissn:
  - '14320444'
  issn:
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publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tri-partitions and bases of an ordered complex
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...
---
_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. <i>Computer
    Aided Geometric Design</i>. 2019;73:1-15. doi:<a href="https://doi.org/10.1016/j.cagd.2019.06.003">10.1016/j.cagd.2019.06.003</a>
  apa: Edelsbrunner, H., &#38; Ölsböck, K. (2019). Holes and dependences in an ordered
    complex. <i>Computer Aided Geometric Design</i>. Elsevier. <a href="https://doi.org/10.1016/j.cagd.2019.06.003">https://doi.org/10.1016/j.cagd.2019.06.003</a>
  chicago: Edelsbrunner, Herbert, and Katharina Ölsböck. “Holes and Dependences in
    an Ordered Complex.” <i>Computer Aided Geometric Design</i>. Elsevier, 2019. <a
    href="https://doi.org/10.1016/j.cagd.2019.06.003">https://doi.org/10.1016/j.cagd.2019.06.003</a>.
  ieee: H. Edelsbrunner and K. Ölsböck, “Holes and dependences in an ordered complex,”
    <i>Computer Aided Geometric Design</i>, 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.” <i>Computer Aided Geometric Design</i>, vol. 73, Elsevier, 2019,
    pp. 1–15, doi:<a href="https://doi.org/10.1016/j.cagd.2019.06.003">10.1016/j.cagd.2019.06.003</a>.
  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
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  - '000485207800001'
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project:
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  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'
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title: Holes and dependences in an ordered complex
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...