Abstract
Forman theory, which is a discrete alternative for cell complexes to the well-known Morse theory, is currently finding several applications in areas where the data to be handled are discrete, such as image processing and computer graphics. Here, we show that a discrete scalar field f, defined on the vertices of a triangulated multidimensional domain Σ, and its gradient vector field Grad f through the Smale-like decomposition of f [6], are both the restriction of a Forman function F and its gradient field Grad F that extends f over all the simplexes of Σ. We present an algorithm that gives an explicit construction of such an extension. Hence, the scalar field f inherits the properties of Forman gradient vector fields and functions from field Grad F and function F.
Chapter PDF
Similar content being viewed by others
References
Cazals, F., Chazal, F., Lewiner, T.: Molecular Shape Analysis Based upon the Morse-Smale Complex and the Connolly Function. In: Proceedings of the nineteenth Annual Symposium on Computational Geometry, pp. 351–360 (2003)
Čomić, L., De Floriani, L.: Multi-Scale 3D Morse Complexes. In: International Conference on Computational Science and its Applications (ICCSA), Workshop on Computational Geometry and Applications, pp. 441–451 (2008)
Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Collapses and Watersheds in Pseudomanifolds. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 397–410. Springer, Heidelberg (2009)
Danovaro, E., De Floriani, L., Mesmoudi, M.M.: Topological Analysis and Characterization of Discrete Scalar Fields. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 386–402. Springer, Heidelberg (2003)
Danovaro, E., De Floriani, L., Vitali, M., Magillo, P.: Multi-Scale Dual Morse Complexes for Representing Terrain Morphology. In: GIS 2007: Proceedings of the 15th Annual ACM International Symposium on Advances in Geographic Information Systems, pp. 1–8. ACM, New York (2007)
De Floriani, L., Mesmoudi, M.M., Danovaro, E.: Smale-Like Decomposition for Discrete Scalar Fields. In: Proceedings International Conference on Pattern Recognition, ICPR (2002)
Forman, R.: Combinatorial Vector Fields and Dynamical Systems. Mathematische Zeitschrift 228, 629–681 (1998)
Forman, R.: Morse Theory for Cell Complexes. Advances in Mathematics 134, 90–145 (1998)
Gyulassy, A., Bremer, P.T., Hamann, B., Pascucci, V.: A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality. Transactions on Visualization and Computer Graphics 14(6), 1619–1626 (2008)
Jerše, G., Mramor Kosta, N.: Ascending and descending regions of a discrete morse function. Comput. Geom. Theory Appl. 42(6-7), 639–651 (2009)
King, H., Knudson, K., Mramor, N.: Generating Discrete Morse Functions from Point Data. Experimental Mathematics 14(4), 435–444 (2005)
Lewiner, T., Lopes, H., Tavares, G.: Applications of Forman’s Discrete Morse Theory to Topology Visualization and Mesh Compression. Transactions on Visualization and Computer Graphich 10(5), 499–508 (2004)
Matsumoto, Y.: An Introduction to Morse Theory, Translations of Mathematical Monographs, vol. 208. American Mathematical Society, Providence (2002)
Milnor, J.: Morse Theory. Princeton University Press, New Jersey (1963)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Čomić, L., Mesmoudi, M.M., De Floriani, L. (2011). Smale-Like Decomposition and Forman Theory for Discrete Scalar Fields. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2011. Lecture Notes in Computer Science, vol 6607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19867-0_40
Download citation
DOI: https://doi.org/10.1007/978-3-642-19867-0_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19866-3
Online ISBN: 978-3-642-19867-0
eBook Packages: Computer ScienceComputer Science (R0)