@article {13274, title = {An iterative algorithm for homology computation on simplicial shapes}, journal = {Computer-Aided Design}, volume = {43}, year = {2011}, month = {2011/11//}, pages = {1457 - 1467}, abstract = {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{\textendash}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. }, keywords = {Computational topology, Generators, Mayer{\textendash}Vietoris sequence, shape decomposition, simplicial complexes, Z -homology}, isbn = {0010-4485}, doi = {10.1016/j.cad.2011.08.015}, url = {http://www.sciencedirect.com/science/article/pii/S0010448511002144}, author = {Boltcheva,Dobrina and Canino,David and Merino Aceituno,Sara and L{\'e}on,Jean-Claude and De Floriani, Leila and H{\'e}troy,Franck} } @conference {13273, title = {A two-level topological decomposition for non-manifold simplicial shapes}, booktitle = {Proceedings of the 2007 ACM symposium on Solid and physical modeling}, series = {SPM {\textquoteright}07}, year = {2007}, month = {2007///}, pages = {355 - 360}, publisher = {ACM}, organization = {ACM}, address = {New York, NY, USA}, abstract = {Modeling and understanding complex non-manifold shapes is a key issue in shape analysis. Geometric shapes are commonly discretized as two- or three-dimensional simplicial complexes embedded in the 3D Euclidean space. The topological structure of a nonmanifold simplicial shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we present a topological decomposition of a shape at two different levels, with different degrees of granularity. We discuss the topological properties of the components at each level, and we present algorithms for computing such decompositions. We investigate the relations among the components, and propose a graph-based representation for such relations.}, keywords = {Non-manifold modeling, shape decomposition, shape modeling, simplicial complexes}, isbn = {978-1-59593-666-0}, doi = {10.1145/1236246.1236297}, url = {http://doi.acm.org/10.1145/1236246.1236297}, author = {Hui,Annie and De Floriani, Leila} }