Modeling and Simplifying Morse Complexes in Arbitrary Dimensions

  • Lidija ČomićEmail author
  • Leila De Floriani
Part of the Mathematics and Visualization book series (MATHVISUAL)


Ascending and descending Morse complexes, defined by a scalar function f over a manifold domain M, decompose M into regions of influence of the critical points of f, thus representing themorphology of the scalar function f over M in a compact way. Here, we introduce two simplification operators on Morse complexes which work in arbitrary dimensions and we discuss their interpretation as n-dimensional Euler operators. We consider a dual representation of the two Morse complexes in terms of an incidence graph and we describe how our simplification operators affect the graph representation. This provides the basis for defining a multi-scale graph-based model of Morse complexes in arbitrary dimensions.


Simplicial Complex Morse Theory Morse Function Integral Line Incidence Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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This work has been partially supported by the National Science Foundation under grant CCF-0541032, by the MIUR-FIRB project SHALOM under contract number RBIN04HWR8, and by the Ministry of Science of the Republic of Serbia through Project 23036.


  1. 1.
    C. L. Bajaj and D. R. Shikore. Topology Preserving Data Simplification with Error Bounds. Computers and Graphics, 22(1):3–12, 1998.CrossRefGoogle Scholar
  2. 2.
    T. Banchoff. Critical Points and Curvature for Embedded Polyhedral Surfaces. American Mathematical Monthly, 77(5):475–485, 1970.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Bhaniramka, R. Wenger, and R. Crawfis. Isosurfacing in Higher Dimensions. In Proceedings IEEE Visualization 2000, pages 267–273. IEEE Computer Society, Oct. 2000.Google Scholar
  4. 4.
    S. Biasotti, L. De Floriani, B. Falcidieno, and L. Papaleo. Morphological Representations of Scalar Fields. In L. De Floriani and M. Spagnuolo, editors, Shape Analysis and Structuring, pages 185–213. Springer Verlag, 2008.Google Scholar
  5. 5.
    P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci. A Topological Hierarchy for Functions on Triangulated Surfaces. Transactions on Visualization and Computer Graphics, 10(4):385–396, July/August 2004.Google Scholar
  6. 6.
    F. Cazals, F. Chazal, and T. Lewiner. Molecular Shape Analysis Based upon the Morse-Smale Complex and the Connolly Function. In Proceedings of the nineteenth Annual Symposium on Computational Geometry, pages 351–360, New York, USA, 2003. ACM Press.Google Scholar
  7. 7.
    L. Čomić and L. De Floriani. Cancellation of Critical Points in 2D and 3D Morse and Morse-Smale Complexes. In Discrete Geometry for Computer Imagery (DGCI), Lecture Notes in Computer Science, volume 4992, pages 117–128, Lyon, France, Apr 16-18 2008. Springer-Verlag GmbH.Google Scholar
  8. 8.
    E. Danovaro, L. De Floriani, and M. M. Mesmoudi. Topological Analysis and Characterization of Discrete Scalar Fields. In T.Asano, R.Klette, and C.Ronse, editors, Theoretical Foundations of Computer Vision, Geometry, Morphology, and Computational Imaging, volume LNCS 2616, pages 386–402. Springer Verlag, 2003.Google Scholar
  9. 9.
    L. De Floriani and A. Hui. Shape Representations Based on Cell and Simplicial Complexes. In Eurographics 2007, State-of-the-art Report. September 2007.Google Scholar
  10. 10.
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer Verlag, Berlin, 1987.zbMATHGoogle Scholar
  11. 11.
    H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Morse-Smale Complexes for Piecewise Linear 3-Manifolds. In Proceedings 19th ACM Symposium on Computational Geometry, pages 361–370, 2003.Google Scholar
  12. 12.
    H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse Complexes for Piecewise Linear 2-Manifolds. In Proceedings 17th ACM Symposium on Computational Geometry, pages 70–79, 2001.Google Scholar
  13. 13.
    R. Forman. Morse Theory for Cell Complexes. Advances in Mathematics, 134:90–145, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Gerstner and R. Pajarola. Topology Preserving and Controlled Topology Simplifying Multi-Resolution Isosurface Extraction. In Proceedings IEEE Visualization 2000, pages 259–266, 2000.Google Scholar
  15. 15.
    A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann. Topology-Based Simplification for Feature Extraction from 3D Scalar Fields. In Proceedings IEEE Visualization’05, pages 275–280. ACM Press, 2005.Google Scholar
  16. 16.
    A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann. A Topological Approach to Simplification of Three-Dimensional Scalar Functions. IEEE Transactions on Visualization and Computer Graphics, 12(4):474–484, 2006.CrossRefGoogle Scholar
  17. 17.
    T. Illetschko, A. Ion, Y. Haxhimusa, and W. G. Kropatsch. Collapsing 3D Combinatorial Maps. In F. Lenzen, O. Scherzer, and M. Vincze, editors, Proceedings of the 30th OEAGM Workshop, pages 85–93, Obergurgl, Austria, 2006. sterreichische Computer Gesellschaft.Google Scholar
  18. 18.
    P. Magillo, E. Danovaro, L. De Floriani, L. Papaleo, and M. Vitali. Extracting Terrain Morphology: A New Algorithm and a Comparative Evaluation. In 2nd International Conference on Computer Graphics Theory and Applications, pages 13–20, March 8–11 2007.Google Scholar
  19. 19.
    M. Mantyla. An Introduction to Solid Modeling. Computer Science Press, 1988.Google Scholar
  20. 20.
    J. Milnor. Morse Theory. Princeton University Press, New Jersey, 1963.zbMATHGoogle Scholar
  21. 21.
    X. Ni, M. Garland, and J. C. Hart. Fair Morse Functions for Extracting the Topological Structure of a Surface Mesh. In International Conference on Computer Graphics and Interactive Techniques ACM SIGGRAPH, pages 613–622, 2004.Google Scholar
  22. 22.
    V. Pascucci. Topology Diagrams of Scalar Fields in Scientific Visualization. In S. Rana, editor, Topological Data Structures for Surfaces, pages 121–129. John Wiley & Sons Ltd, 2004.Google Scholar
  23. 23.
    S. Takahashi, T. Ikeda, T. L. Kunii, and M. Ueda. Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographic Elevation Data. In Computer Graphics Forum, volume 14, pages 181–192, 1995.Google Scholar
  24. 24.
    S. Takahashi, Y. Takeshima, and I. Fujishiro. Topological Volume Skeletonization and its Application to Transfer Function Design. Graphical Models, 66(1):24–49, 2004.zbMATHCrossRefGoogle Scholar
  25. 25.
    G. H. Weber, G. Schueuermann, H. Hagen, and B. Hamann. Exploring Scalar Fields Using Critical Isovalues. In Proceedings IEEE Visualization 2002, pages 171–178. IEEE Computer Society, 2002.Google Scholar
  26. 26.
    G. H. Weber, G. Schueuermann, and B. Hamann. Detecting Critical Regions in Scalar Fields. In G.-P. Bonneau, S. Hahmann, and C. D. Hansen, editors, Proceedings Data Visualization Symposium, pages 85–94. ACM Press, New York, 2003.Google Scholar
  27. 27.
    C. Weigle and D.Banks. Extracting Iso-Valued Features in 4-dimensional Scalar Fields. In Proceedings IEEE Visualization 1998, pages 103–110. IEEE Computer Society, Oct. 1998.Google Scholar
  28. 28.
    G. W. Wolf. Topographic Surfaces and Surface Networks. In S. Rana, editor, Topological Data Structures for Surfaces, pages 15–29. John Wiley & Sons Ltd, 2004.Google Scholar

Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Faculty of EngineeringUniversity of Novi SadNovi SadSerbia
  2. 2.Department of Computer ScienceUniversity of GenovaGenovaItaly

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