Abstract
Given a 3D binary voxel-based digital object V, an algorithm 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 homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). 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 description 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 homological information at no extra time processing cost.
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.
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Molina-Abril, H., Real, P. (2008). Advanced Homology Computation of Digital Volumes Via Cell Complexes . In: da Vitoria Lobo, N., et al. Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2008. Lecture Notes in Computer Science, vol 5342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89689-0_40
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