An iterative algorithm for homology computation on simplicial shapes

Dobrina Boltcheva (1, 2) , David Canino (3) , Sara Merino Aceituno (1) , Jean-Claude Léon (4, 1) , Leila de Floriani (3) , Franck Hétroy (1, 5)

1 EVASION - Virtual environments for animation and image synthesis of natural objects
2 ALICE - Geometry and Lighting
3 DISI - Dipartimento di Informatica e Scienze dell'Informazione
4 G-SCOP_SIREP - Système d’Information, conception RobustE des Produits
5 MORPHEO - Capture and Analysis of Shapes in Motion

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.

Mots clés: Computational topology; Simplicial complexes; Shape decomposition; Z-homology; Mayer-Vietoris sequence; Generators; Shape

Domaines: Informatique et langage [cs.CL]

Dobrina Boltcheva, David Canino, Sara Merino Aceituno, Jean-Claude Léon, Leila de Floriani, et al.. An iterative algorithm for homology computation on simplicial shapes. Computer-Aided Design, 2011, Solid and Physical Modeling 2011, 43 (11), pp.1457-1467. ⟨10.1016/j.cad.2011.08.015⟩. ⟨hal-00644410v2⟩