Tree-Based Encoding for Cancellations on Morse Complexes

  • Lidija Čomić
  • Leila De Floriani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5852)


A scalar function f, defined on a manifold M, can be simplified by applying a sequence of removal and contraction operators, which eliminate its critical points in pairs, and simplify the topological representation of M, provided by Morse complexes of f. The inverse refinement operators, together with a dependency relation between them, enable a construction of a multi-resolution representation of such complexes. Here, we encode a sequence of simplification operators in a data structure called an augmented cancellation forest, which will enable procedural encoding of the inverse refinement operators, and reduce the dependency relation between modifications of the Morse complexes. In this way, this representation will induce a high flexibility of the hierarchical representation of the Morse complexes, producing a large number of Morse complexes at different resolutions that can be obtained from the hierarchy.


Morse theory Morse complexes simplification operators graph-based representation augmented cancellation forest 


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  1. 1.
    Bremer, P.-T., Pascucci, V., Hamann, B.: Maximizing Adaptivity in Hierarchical Topological Models. In: Belyaev, A.G., Pasko, A.A., Spagnuolo, M. (eds.) Proc. of the Int. Conf. on Shape Modeling and Applications 2005 (SMI 2005), pp. 298–307. IEEE Computer Society Press, Los Alamitos (2005)CrossRefGoogle Scholar
  2. 2.
    Bremer, P.-T., Edelsbrunner, H., Hamann, B., Pascucci, V.: A Topological Hierarchy for Functions on Triangulated Surfaces. IEEE Trans. on Visualization and Computer Graphics 10(4), 385–396 (2004)CrossRefGoogle Scholar
  3. 3.
    Čomić, L., De Floriani, L.: Modeling and Simplifying Morse Complexes in Arbitrary Dimensions. In: Int. Workshop on Topological Methods in Data Analysis and Visualization (TopoInVis), Snowbird, Utah, February 23-24 (2009)Google 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.
    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)CrossRefGoogle Scholar
  6. 6.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)MATHGoogle Scholar
  7. 7.
    Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse Complexes for Piecewise Linear 2-Manifolds. In: Proceedings 17th ACM Symposium on Computational Geometry, pp. 70–79 (2001)Google Scholar
  8. 8.
    Gyulassy, A., Bremer, P.-T., Hamann, B., Pascucci, V.: A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality. IEEE Trans. on Visualization and Computer Graphics 14(6), 1619–1626 (2008)CrossRefGoogle Scholar
  9. 9.
    Gyulassy, A., Natarajan, V., Pascucci, V., Bremer, P.-T., Hamann, B.: Topology-Based Simplification for Feature Extraction from 3D Scalar Fields. In: Proceedings IEEE Visualization 2005, pp. 275–280. ACM Press, New York (2005)Google Scholar
  10. 10.
    Gyulassy, A., Natarajan, V., Pascucci, V., Hamann, B.: Efficient Computation of Morse-Smale Complexes for Three-dimensional Scalar Functions. IEEE Trans. on Visualization and Computer Graphics 13(6), 1440–1447 (2007)CrossRefGoogle Scholar
  11. 11.
    Matsumoto, Y.: An Introduction to Morse Theory, American Mathematical Society, Translations of Mathematical Monographs 208 (2002)Google Scholar
  12. 12.
    Milnor, J.: Morse Theory. Princeton University Press, New Jersey (1963)MATHGoogle Scholar
  13. 13.
    Takahashi, S.: Algorithms for Extracting Surface Topology from Digital Elevation Models. In: Rana, S. (ed.) Topological Data Structures for Surfaces, pp. 31–51. John Wiley & Sons Ltd., Chichester (2004)CrossRefGoogle Scholar
  14. 14.
    Wolf, G.W.: Topographic Surfaces and Surface Networks. In: Rana, S. (ed.) Topological Data Structures for Surfaces, pp. 15–29. John Wiley & Sons Ltd., Chichester (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

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