Towards digital cohomology

We propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particula...

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Detalles Bibliográficos
Autores: González Díaz, Rocío, Real Jurado, Pedro
Tipo de recurso: capítulo de libro
Fecha de publicación:2003
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/30780
Acceso en línea:http://hdl.handle.net/11441/30780
https://doi.org/10.1007/978-3-540-39966-7_8
Access Level:acceso abierto
Palabra clave:Digital topology
chain complexes
cohomology ring
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spelling Towards digital cohomologyGonzález Díaz, RocíoReal Jurado, PedroDigital topologychain complexescohomology ringWe propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2–cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed.Matemática Aplicada I2003info:eu-repo/semantics/bookPartapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/30780https://doi.org/10.1007/978-3-540-39966-7_8reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésDiscrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 2886 p. 92-101info:eu-repo/semantics/openAccessoai:idus.us.es:11441/307802026-06-17T12:51:07Z
dc.title.none.fl_str_mv Towards digital cohomology
title Towards digital cohomology
spellingShingle Towards digital cohomology
González Díaz, Rocío
Digital topology
chain complexes
cohomology ring
title_short Towards digital cohomology
title_full Towards digital cohomology
title_fullStr Towards digital cohomology
title_full_unstemmed Towards digital cohomology
title_sort Towards digital cohomology
dc.creator.none.fl_str_mv González Díaz, Rocío
Real Jurado, Pedro
author González Díaz, Rocío
author_facet González Díaz, Rocío
Real Jurado, Pedro
author_role author
author2 Real Jurado, Pedro
author2_role author
dc.contributor.none.fl_str_mv Matemática Aplicada I
dc.subject.none.fl_str_mv Digital topology
chain complexes
cohomology ring
topic Digital topology
chain complexes
cohomology ring
description We propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2–cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed.
publishDate 2003
dc.date.none.fl_str_mv 2003
dc.type.none.fl_str_mv info:eu-repo/semantics/bookPart
format bookPart
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/30780
https://doi.org/10.1007/978-3-540-39966-7_8
url http://hdl.handle.net/11441/30780
https://doi.org/10.1007/978-3-540-39966-7_8
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Vol. 2886 p. 92-101
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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