A Homological–Based Description of Subdivided nD Objects
We present here a topo–geometrical description of a subdivided nD object called homological spanning forest representation. This representation is a convenient tool in order to completely control not only geometrical, but also advanced topological information of a given object. By codifying the underlying algebraic topological machinery in terms of coordinate–based graphs, we progress in the task to “combinatorialize” homological information at two levels: local and global. Therefore, our method presents several advantages with respect to the existing Algebraic topological models, and techniques based in Discrete Morse Theory. A construction algorithm has been implemented, and some examples are shown.
KeywordsChain Complex Cell Complex Critical Cell Barycentric Subdivision Harmonic Complex
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- 2.Klette, R.: Cell complexes through time. Communication and Information Technology Research Technical Report 60 (2000)Google Scholar
- 3.Edmonds, J.: A combinatorial representation for polyhedral surfaces. Notices Amer. Math. Soc. 7 (1960)Google Scholar
- 4.Kropatsch, W.G., Haxhimusa, Y., Ion, A.: Multiresolution image segmentations in graph pyramids. In: Applied Graph Theory in Computer Vision and Pattern Recognition, pp. 3–41 (2007)Google Scholar
- 5.Eilenberg, S., Mac Lane, S.: On the groups h(π,n), i, ii, iii. Annals of Math. 58, 60, 60, 55–106,48–139, 513–557 (1953,1954)Google Scholar
- 8.Forman, R.: A Discrete Morse Theory for Cell Complexes. In: Yau, S.T. (ed.) Topology and Physics for Raoul Bott. International Press (1995)Google Scholar
- 12.Molina-Abril, H., Real, P.: Homological optimality in discrete morse theory through chain homotopies. Submitted to Pattern Recognition Letters (2011)Google Scholar