Advanced Homology Computation of Digital Volumes Via Cell Complexes ⋆ Helena Molina-Abril and Pedro Real Dpto. Matematica Aplicada I, E.T.S.I. Informatica, Universidad de Sevilla, Avda. Reina Mercedes, s/n 41012 Sevilla (Spain) {habril,real}@us.es Abstract. Given a 3D binary voxel-based digital object V , an algo- rithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homol- ogy numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology opera- tions,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2 ×2 ×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a de- scription of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5],AT-model method [7,5]) . Saving this vector field, we go further obtaining homo- logical information at no extra time processing cost. 1 Introduction One possible way for developing techniques for volume recognition can be through relevant structural descriptors, mainly based on topological-geometrical proper- ties. Topology deals with connectivity and separability features and although the global nature of topological properties makes their computation difficult, topol- ogy seems to be an essential part of the vocabulary by which human visual system represent and characterize objects [2]. Homology gives us the simplest topological version of a volume in terms of information about connected components, tun- nels or holes and cavities on it. Homological numbers commonly used in Pattern Recognition as robust descriptors are Betti numbers (number of connected com- ponents, holes and cavities), local topological characterization of voxels and Euler characteristic. Nowadays, the current interactions between Pattern Recognition and Computational Topology are mainly at graph-based level and it includes is- sues such that topological skeletonization and Reeb graphs [1]. Examples of image analysis and pattern recognition problems that benefit from the use of topologi- cal considerations include, for example, the use of homology invariants to compare ⋆ This work has been partially supported by ”Computational Topology and Applied Mathematics” PAICYT research project FQM-296, ”Andalusian research project PO6-TIC-02268) and Spanish MEC project MTM2006-03722. N. da Vitora Lobo et al. (Eds.): SSPR&SPR 2008, LNCS 5342, pp. 361–371, 2008. c  Springer-Verlag Berlin Heidelberg 2008