An iterative algorithm for homology computation on simplicial shapes

TitleAn iterative algorithm for homology computation on simplicial shapes
Publication TypeJournal Articles
Year of Publication2011
AuthorsBoltcheva D, Canino D, Merino Aceituno S, Léon J-C, De Floriani L, Hétroy F
JournalComputer-Aided Design
Pagination1457 - 1467
Date Published2011/11//
ISBN Number0010-4485
KeywordsComputational topology, Generators, Mayer–Vietoris sequence, shape decomposition, simplicial complexes, Z -homology

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes. We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators.To the best of our knowledge, this is the first algorithm based on the constructive Mayer–Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.