Abstract
In [4], given a binary 26-adjacency voxel-based digital volume V, the homological information (that related to n-dimensional holes: connected components, ”tunnels” and cavities) is extracted from a linear map (called homology gradient vector field) acting on a polyhedral cell complex P(V) homologically equivalent to V. We develop here an alternative way for constructing P(V) based on homological algebra arguments as well as a new more efficient algorithm for computing a homology gradient vector field based on the contractibility of the maximal cells of P(V).
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, and the Austrian Science Fund under grant P20134-N13.
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References
Delfinado, C.J.A., Edelsbrunner, H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3–Sphere. Comput. Aided Geom. Design 12, 771–784 (1995)
Eilenberg, S., MacLane, S.: Relations between homology and homotopy groups of spaces. Ann. of Math. 46, 480–509 (1945)
Forman, R.: A Discrete Morse Theory for Cell Complexes. In: Yau, S.T. (ed.) Geometry, Topology & Physics for Raoul Bott. International Press (1995)
Molina-Abril, H., Real, P.: Advanced homology computation of digital volumes via cell complexes. In: Proceedings of the Structural and Syntactic Pattern Recognition Workshop, Orlando, Florida, USA (December 2008)
Gonzalez-Diaz, R., Real, P.: On the Cohomology of 3D Digital Images. Discrete Applied Math. 147, 245–263 (2005)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)
Kenmochi, Y., Kotani, K., Imiya, A.: Marching cube method with connectivity. In: Proceedings of International Conference on Image Processing. ICIP 1999, vol. 4(4), pp. 361–365 (1999)
Kenmochi, Y., Imiya, A., Ichikawa, A.: Boundary extraction of discrete objects. Computer Vision and Image Understanding 71, 281–293 (1998)
Munkres, J.R.: Elements of Algebraic Topology. Addison–Wesley Co., London (1984)
Real, P., Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Molina-Abril, H.: Integral Operators for Computing Homology Generators at Any Dimension. In: Ruiz-Shulcloper, J., Kropatsch, W.G. (eds.) CIARP 2008. LNCS, vol. 5197, pp. 356–363. Springer, Heidelberg (2008)
Zomorodian, A., Carlsson, G.: Localized Homology. Computational Geometry: Theory and Applications archive 41(3), 126–148 (2008)
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Real, P., Molina-Abril, H. (2009). Cell AT-Models for Digital Volumes . In: Torsello, A., Escolano, F., Brun, L. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2009. Lecture Notes in Computer Science, vol 5534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02124-4_32
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DOI: https://doi.org/10.1007/978-3-642-02124-4_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02123-7
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