Chain homotopies for object topological representations |
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Authors: | R González-Díaz MJ Jiménez P Real |
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Institution: | Depto. de Matemática Aplicada I, Escuela Superior de Ingeniería Informática, Universidad de Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain |
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Abstract: | This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in R. González-Díaz, P. Real, On the cohomology of 3D digital images, Discrete Appl. Math. 147 (2005) 245-263] in which the ground ring was a field. The concept of generators which are “nicely” representative is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse). |
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Keywords: | Simplicial complexes Chain homotopy Cohomology ring Digital topology |
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