---
OA_place: repository
OA_type: green
_id: '20585'
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.
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Ondrej
  full_name: Draganov, Ondrej
  id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
  last_name: Draganov
  orcid: 0000-0003-0464-3823
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  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>
  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.
  ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic
    alpha complexes. Foundations of Data Science. 8, 30–62.
  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>.
  short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations
    of Data Science 8 (2026) 30–62.
corr_author: '1'
date_created: 2025-11-02T23:01:33Z
date_published: 2026-03-01T00:00:00Z
date_updated: 2026-08-12T06:19:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.3934/fods.2025003
ec_funded: 1
external_id:
  arxiv:
  - '2212.03128'
fulldoi: https://doi.org/10.3934/fods.2025003
has_accepted_license: '1'
intvolume: '         8'
keyword:
- Topological data analysis
- Delaunay mosaic
- alpha complex
- chromatic sets
- persistent homology
- kernel/image/cokernel persistent homology
- radius function
- discrete Morse theory
- exact sequences
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2212.03128
mathsc:
- 62R40
- 55N31
- 68T09
- 57Q70
month: '03'
oa: 1
oa_version: Preprint
page: 30-62
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Foundations of Data Science
publication_identifier:
  eissn:
  - 2639-8001
publication_status: published
publisher: AIMS
quality_controlled: '1'
related_material:
  record:
  - id: '15091'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Chromatic alpha complexes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2026'
...