instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Computational Homology Applied to Discrete Objects
(Cotutela de tesis de la Universidad d'Aix-Marseille y la Universidad de Sevilla) 

Directores de tesis:
Alexandra Bac y Jean-Luc Mari (Univ. d'Aix-Marseille)  Pedro Real (Univ. de Sevilla)
Abstract: Homology theory formalizes the concept of hole in a space. For a given subspace of the Euclidean space, we define a sequence of homology groups, whose ranks are considered as the number of holes of each dimension. Hence, β_0, the rank of the 0-dimensional homology group, is the number of connected components, β_1 is the number of tunnels or handles and β_2 is the number of cavities. These groups are computable when the space is described in a combinatorial way, as simplicial or cubical complexes are. Given a discrete object (a set of pixels, voxels or their analog in higher dimension) we can build a cubical complex and thus compute its homology groups.
This thesis studies three approaches regarding the homology computation of discrete objects. First, we introduce the homological discrete vector field, a combinatorial structure which generalizes the discrete gradient vector field and allows to compute the homology groups. This notion allows to see the relation between different existing methods for computing homology. Next, we present a linear algorithm for computing the Betti numbers of a 3D cubical complex, which can be used for binary volumes. Finally, we introduce two measures (the thickness and the breadth) associated to the holes in a discrete object, which provide a topological and geometric signature more interesting than only the Betti numbers. This approach provides also some heuristics for localizing holes, obtaining minimal homology or cohomology generators, opening and closing holes.