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arXiv · 1105.3620

Chain Homotopies for Object Topological Representations

Rocio Gonzalez-Diaz·Maria Jose Jimenez·Belen Medrano·Pedro Real
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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. A concept of generators which are "nicely" representative cycles 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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Rocio Gonzalez-Diaz, Maria Jose Jimenez, Belen Medrano, Pedro Real. 2011-05-18. Chain Homotopies for Object Topological Representations. https://doi.org/10.1016/j.dam.2008.05.029

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