Abstract
Given a cell complex K whose geometric realization |K| is embedded in R 3 and a continuous function h: |K|→R (called the height function), we construct a graph G h (K) which is an extension of the Reeb graph R h (|K|). More concretely, the graph G h (K) without loops is a subdivision of R h (|K|). The most important difference between the graphs G h (K) and R h (|K|) is that G h (K) preserves not only the number of connected components but also the number of “tunnels” (the homology generators of dimension 1) of K. The latter is not true in general for R h (|K|). Moreover, we construct a map ψ: G h (K)→K identifying representative cycles of the tunnels in K with the ones in G h (K) in the way that if e is a loop in G h (K), then ψ(e) is a cycle in K such that all the points in |ψ(e)| belong to the same level set in |K|.
Partially supported by Junta de Andalucía (FQM-296 and TIC-02268) and Spanish Ministry for Science and Education (MTM-2006-03722).
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References
Alexandroff, P., Hopf, H.: Topologie I. Springer, Berlin (1935)
Biasotti, S., Facidieno, B., Spagnuolo, M.: Extended Reeb Graphs for Surface Understanding and Description. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 185–197. Springer, Heidelberg (2000)
Dahmen, W., Micchelli, C.A.: On the Linear Independence of Multivariate b-Splines. Triangulation of Simploids. SIAM J. Numer. Anal. 19 (1982)
Cole-McLaughlin, K., Edelsbruner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb Graphs of 2-mainifolds. Discrete Comput. Geom. 32, 231–244 (2004)
Fomenko, A.T., Kunii, T.L.: Topological Methods for Visualization. Springer, Heidelberg (1997)
Forman, R.: A discrete Morse theory for cell complexes. In: Yau, S.T.(ed.) Geometry, Topology and Physics for Raoul Bott. International Press (1995)
Forman, R.: Discrete Morse Theory and the Cohomology Ring. Transactions of the American Mathematical Society 354, 5063–5085 (2002)
Gonzalez-Diaz, R., Real, P.: Towards Digital Cohomology. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 92–101. Springer, Heidelberg (2003)
Gonzalez-Diaz, R., Real, P.: On the Cohomology of 3D Digital Images. Discrete Applied Math. 147, 245–263 (2005)
Gonzalez-Diaz, R., Jiménez, M.J., Medrano, B., Real, P.: Extending AT-Models for Integer Homology Computation. In: GbR 2007. LNCS, vol. 4538, pp. 330–339. Springer, Heidelberg (2007)
Massey, W.M.: A Basic Course in Algebraic Topology. New York (1991)
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, London, UK (1984)
Reeb, G.: Sur les Points Singuliers d’une Forme de Pfaff Complement Integrable ou d’une Function Numérique. C. Rendud Acad. Sciences 222, 847–849 (1946)
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Gonzalez-Diaz, R., Jiménez, M.J., Medrano, B., Real, P. (2007). A Graph-with-Loop Structure for a Topological Representation of 3D Objects. In: Kropatsch, W.G., Kampel, M., Hanbury, A. (eds) Computer Analysis of Images and Patterns. CAIP 2007. Lecture Notes in Computer Science, vol 4673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74272-2_63
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DOI: https://doi.org/10.1007/978-3-540-74272-2_63
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