Abstract
We have proposed a complete set of basis Euler operators for updating cell complexes in arbitrary dimensions, which can be classified as homology-preserving and homology-modifying. Here, we define the effect of homology-preserving operators on the incidence graph representation of cell complexes. Based on these operators, we build a multi-resolution model for cell complexes represented in the form of the incidence graph, and we compare its 2D instance with the pyramids of 2-maps, designed for images.
Chapter PDF
Similar content being viewed by others
Keywords
References
Agoston, M.K.: Computer Graphics and Geometric Modeling: Mathematics. Springer-Verlag London Ltd. (2005) ISBN:1-85233-817-2
Brun, L., Kropatsch, W.G.: Introduction to Combinatorial Pyramids. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 108–128. Springer, Heidelberg (2001)
Čomić, L., De Floriani, L.: Topological Operators on Cell Complexes in Arbitrary Dimensions. In: Ferri, M., Frosini, P., Landi, C., Cerri, A., Di Fabio, B. (eds.) CTIC 2012. LNCS, vol. 7309, pp. 98–107. Springer, Heidelberg (2012)
Damiand, G., Gonzalez-Diaz, R., Peltier, S.: Removal Operations in nD Generalized Maps for Efficient Homology Computation. In: Ferri, M., Frosini, P., Landi, C., Cerri, A., Di Fabio, B. (eds.) CTIC 2012. LNCS, vol. 7309, pp. 20–29. Springer, Heidelberg (2012)
De Floriani, L., Magillo, P., Puppo, E.: Multiresolution Representation of Shapes Based on Cell Complexes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 3–18. Springer, Heidelberg (1999)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)
Edmonds, J.: A Combinatorial Representation for Polyhedral Surfaces. Notices Amer. Math. Soc. 7, 646 (1960)
Haxhimusa, Y., Glantz, R., Kropatsch, W.G.: Constructing Stochastic Pyramids by MIDES - Maximal Independent Directed Edge Set. In: Hancock, E.R., Vento, M. (eds.) GbRPR 2003. LNCS, vol. 2726, pp. 24–34. Springer, Heidelberg (2003)
Kovalevsky, V.: Axiomatic Digital Topology. Journal of Mathematical Imaging and Vision 26(1-2), 41–58 (2006)
Lévy, B., Mallet, J.L.: Cellular Modelling in Arbitrary Dimension Using Generalized Maps. Technical report, INRIA (1999)
Lienhardt, P.: N-Dimensional Generalized Combinatorial Maps and Cellular Quasi-Manifolds. International Journal of Computational Geometry and Applications 4(3), 275–324 (1994)
Peltier, S., Ion, A., Haxhimusa, Y., Kropatsch, W.G., Damiand, G.: Computing Homology Group Generators of Images Using Irregular Graph Pyramids. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 283–294. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Čomić, L., De Floriani, L., Iuricich, F. (2013). Multi-resolution Cell Complexes Based on Homology-Preserving Euler Operators. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-37067-0_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37066-3
Online ISBN: 978-3-642-37067-0
eBook Packages: Computer ScienceComputer Science (R0)