Advertisement

Simplification Operators on a Dimension-Independent Graph-Based Representation of Morse Complexes

  • Lidija Čomić
  • Leila De Floriani
  • Federico Iuricich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7883)

Abstract

Ascending and descending Morse complexes are defined by the critical points and integral lines of a scalar field f defined on a manifold M. They induce a subdivision of M into regions of uniform gradient flow, thus providing a compact description of the topology of M and of the behavior of f over M. We represent the ascending and descending Morse complexes of f as a graph, that we call the Morse incidence graph (MIG). We have defined a simplification operator on the graph-based representation, which is atomic and dimension-independent, and we compare this operator with a previous approach to the simplification of 3D Morse complexes based on the cancellation operator. We have developed a simplification algorithm based on a simplification operator, which operates on the MIG, and we show results from this implementation as well as comparisons with the cancellation operator in 3D.

Keywords

geometric modeling Morse theory Morse complexes simplification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bertrand, G.: On Topological Watersheds. Journal of Mathematical Imaging and Vision 22(2-3), 217–230 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Biasotti, S., De Floriani, L., Falcidieno, B., Papaleo, L.: Morphological Representations of Scalar Fields. In: De Floriani, L., Spagnuolo, M. (eds.) Shape Analysis and Structuring, pp. 185–213. Springer (2008)Google Scholar
  3. 3.
    Biasotti, S., Floriani, L.D., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv. 40, Article 12 (2008)Google Scholar
  4. 4.
    Čomić, L., De Floriani, L., Iuricich, F.: Simplifying Morphological Representations of 2D and 3D Scalar Fields. In: 19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems, ACM-GIS 2011, pp. 437–440. ACM (2011)Google Scholar
  5. 5.
    Čomić, L., De Floriani, L., Iuricich, F.: Dimension-Independent Multi-Resolution Morse Complexes. Computers & Graphics 36(5), 541–547 (2012)CrossRefGoogle Scholar
  6. 6.
    Edelsbrunner, H., Harer, J.: The Persistent Morse Complex Segmentation of a 3-Manifold. In: Magnenat-Thalmann, N. (ed.) 3DPH 2009. LNCS, vol. 5903, pp. 36–50. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Morse-Smale Complexes for Piecewise Linear 3-Manifolds. In: Proceedings 19th ACM Symposium on Computational Geometry, pp. 361–370 (2003)Google Scholar
  8. 8.
    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
  9. 9.
    Forman, R.: Morse Theory for Cell Complexes. Advances in Mathematics 134, 90–145 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gyulassy, A., Bremer, P.-T., Hamann, B., Pascucci, V.: A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality. IEEE Transactions on Visualization and Computer Graphics 14(6), 1619–1626 (2008)CrossRefGoogle Scholar
  11. 11.
    Gyulassy, A., Bremer, P.-T., Hamann, B., Pascucci, V.: Practical Considerations in Morse-Smale Complex Computation. In: Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications. Mathematics and Visualization, pp. 67–78. Springer (2011)Google Scholar
  12. 12.
    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 (2005)Google Scholar
  13. 13.
    Gyulassy, A., Natarajan, V., Pascucci, V., Bremer, P.-T., Hamann, B.: A Topological Approach to Simplification of Three-Dimensional Scalar Functions. IEEE Transactions on Visualization and Computer Graphics 12(4), 474–484 (2006)CrossRefGoogle Scholar
  14. 14.
    Gyulassy, A., Natarajan, V., Pascucci, V., Hamann, B.: Efficient Computation of Morse-Smale Complexes for Three-dimensional Scalar Functions. IEEE Transactions on Visualization and Computer Graphics 13(6), 1440–1447 (2007)CrossRefGoogle Scholar
  15. 15.
    Matsumoto, Y.: An Introduction to Morse Theory. Translations of Mathematical Monographs, vol. 208. American Mathematical Society (2002)Google Scholar
  16. 16.
    Meyer, F.: Topographic distance and watershed lines. Signal Processing 38, 113–125 (1994)zbMATHCrossRefGoogle Scholar
  17. 17.
    Robins, V., Wood, P.J., Sheppard, A.P.: Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)CrossRefGoogle Scholar
  18. 18.
    Roerdink, J., Meijster, A.: The Watershed Transform: Definitions, Algorithms, and Parallelization Strategies. Fundamenta Informaticae 41, 187–228 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Lidija Čomić
    • 1
  • Leila De Floriani
    • 2
  • Federico Iuricich
    • 2
  1. 1.Faculty of Technical SciencesUniversity of Novi SadSerbia
  2. 2.Department of Computer ScienceUniversity of GenovaItaly

Personalised recommendations