Smale-Like Decomposition and Forman Theory for Discrete Scalar Fields

  • Lidija Čomić
  • Mohammed Mostefa Mesmoudi
  • Leila De Floriani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)

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.

Keywords

Morse Theory Forman Theory Morse Decomposition 

References

  1. 1.
    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)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    De Floriani, L., Mesmoudi, M.M., Danovaro, E.: Smale-Like Decomposition for Discrete Scalar Fields. In: Proceedings International Conference on Pattern Recognition, ICPR (2002)Google Scholar
  7. 7.
    Forman, R.: Combinatorial Vector Fields and Dynamical Systems. Mathematische Zeitschrift 228, 629–681 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Forman, R.: Morse Theory for Cell Complexes. Advances in Mathematics 134, 90–145 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Jerše, G., Mramor Kosta, N.: Ascending and descending regions of a discrete morse function. Comput. Geom. Theory Appl. 42(6-7), 639–651 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    King, H., Knudson, K., Mramor, N.: Generating Discrete Morse Functions from Point Data. Experimental Mathematics 14(4), 435–444 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Matsumoto, Y.: An Introduction to Morse Theory, Translations of Mathematical Monographs, vol. 208. American Mathematical Society, Providence (2002)Google Scholar
  14. 14.
    Milnor, J.: Morse Theory. Princeton University Press, New Jersey (1963)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Lidija Čomić
    • 1
  • Mohammed Mostefa Mesmoudi
    • 2
  • Leila De Floriani
    • 2
  1. 1.Faculty of EngineeringUniversity of Novi SadSerbia
  2. 2.Department of Computer ScienceUniversity of GenovaItaly

Personalised recommendations