Chain Homotopies for Object Topological Representations

Details Chain Homotopies for Object Topological Representations Chain Homotopies for Object Topological Representations

2011-05-18

arxiv.org/abs/1105.3620v2

Discrete Applied Mathematics, Volume 157, Issue 3, 2009, Pages 490-499

Computer Science

Discrete Mathematics

Scientific paper

10.1016/j.dam.2008.05.029

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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