---
_id: '3256'
abstract:
- lang: eng
  text: We use a distortion to define the dual complex of a cubical subdivision of
    ℝ n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated
    by the topological analysis of high-dimensional digital image data, we consider
    such subdivisions defined by generalizations of quad- and oct-trees to n dimensions.
    Assuming the subdivision is balanced, we show that mapping each vertex to the
    center of the corresponding n-cube gives a geometric realization of the dual complex
    in ℝ n.
acknowledgement: This research is partially supported by the Defense Advanced Research
  Projects Agency (DARPA) under grants HR0011-05-1-0057 and HR0011-09-0065 as well
  as the National Science Foundation (NSF) under grant DBI-0820624.
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: Michael
  full_name: Kerber, Michael
  id: 36E4574A-F248-11E8-B48F-1D18A9856A87
  last_name: Kerber
  orcid: 0000-0002-8030-9299
citation:
  ama: Edelsbrunner H, Kerber M. Dual complexes of cubical subdivisions of ℝn. <i>Discrete
    &#38; Computational Geometry</i>. 2012;47(2):393-414. doi:<a href="https://doi.org/10.1007/s00454-011-9382-4">10.1007/s00454-011-9382-4</a>
  apa: Edelsbrunner, H., &#38; Kerber, M. (2012). Dual complexes of cubical subdivisions
    of ℝn. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href="https://doi.org/10.1007/s00454-011-9382-4">https://doi.org/10.1007/s00454-011-9382-4</a>
  chicago: Edelsbrunner, Herbert, and Michael Kerber. “Dual Complexes of Cubical Subdivisions
    of ℝn.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2012. <a href="https://doi.org/10.1007/s00454-011-9382-4">https://doi.org/10.1007/s00454-011-9382-4</a>.
  ieee: H. Edelsbrunner and M. Kerber, “Dual complexes of cubical subdivisions of
    ℝn,” <i>Discrete &#38; Computational Geometry</i>, vol. 47, no. 2. Springer, pp.
    393–414, 2012.
  ista: Edelsbrunner H, Kerber M. 2012. Dual complexes of cubical subdivisions of
    ℝn. Discrete &#38; Computational Geometry. 47(2), 393–414.
  mla: Edelsbrunner, Herbert, and Michael Kerber. “Dual Complexes of Cubical Subdivisions
    of ℝn.” <i>Discrete &#38; Computational Geometry</i>, vol. 47, no. 2, Springer,
    2012, pp. 393–414, doi:<a href="https://doi.org/10.1007/s00454-011-9382-4">10.1007/s00454-011-9382-4</a>.
  short: H. Edelsbrunner, M. Kerber, Discrete &#38; Computational Geometry 47 (2012)
    393–414.
corr_author: '1'
date_created: 2018-12-11T12:02:17Z
date_published: 2012-03-01T00:00:00Z
date_updated: 2025-09-30T07:43:46Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s00454-011-9382-4
external_id:
  isi:
  - '000299057200010'
file:
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  creator: system
  date_created: 2018-12-12T10:08:15Z
  date_updated: 2020-07-14T12:46:05Z
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  file_name: IST-2016-543-v1+1_2012-J-08-HierarchyCubeComplex.pdf
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file_date_updated: 2020-07-14T12:46:05Z
fulldoi: https://doi.org/10.1007/s00454-011-9382-4
has_accepted_license: '1'
intvolume: '        47'
isi: 1
issue: '2'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Submitted Version
page: 393 - 414
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '3398'
pubrep_id: '543'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Dual complexes of cubical subdivisions of ℝn
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 47
year: '2012'
...