Advertisement

Practical Considerations in Morse-Smale Complex Computation

  • Attila Gyulassy
  • Peer-Timo Bremer
  • Bernd Hamann
  • Valerio Pascucci
Chapter
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

The Morse-Smale complex is an effective topology-based representation for identifying, ordering, and selectively removing features in scalar-valued data. Several algorithms are known for its effective computation, however, common problems pose practical challenges for any feature-finding approach using the Morse-Smale complex. We identify these problems and present practical solutions: (1) we identify the cause of spurious critical points due to simulation of simplicity, and present a general technique for solving it; (2) we improve simplification performance by reordering critical point cancellation operations and introducing an efficient data structure for storing the arcs of the complex; (3) we present a practical approach for handling boundary conditions.

Keywords

Morse Function Integral Line Memory Footprint Discrete Gradient Standard Simulation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Attila Gyulassy was supported by the Lawrence Scholar Program (LSP). In addition, this research was supported in part by the National Science Foundation, under grant CCF-0702817. We would like to thank the members of the Center for Applied Scientific Computing (CASC), at LLNL, and the members of the Visualization and Computer Graphics Research Group of the Institute for Data Analysis and Visualization (IDAV), at UC Davis. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

References

  1. 1.
    P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci. A topological hierarchy for functions on triangulated surfaces. IEEE Transactions on Visualization and Computer Graphics, 10(4):385–396, 2004.CrossRefGoogle Scholar
  2. 2.
    H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Morse-Smale complexes for piecewise linear 3-manifolds. In Proc. 19th Ann. Sympos. Comput. Geom., pages 361–370, 2003.Google Scholar
  3. 3.
    H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 30(1):87–107, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    H. Edelsbrunner and E. P. Mücke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph., 9(1):66–104, 1990.zbMATHCrossRefGoogle Scholar
  5. 5.
    R. Forman. A user’s guide to discrete morse theory, 2001.Google Scholar
  6. 6.
    A. Gyulassy, P.-T. Bremer, B. Hamann, and V. Pascucci. 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
  7. 7.
    A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann. Topology-based simplification for feature extraction from 3d scalar fields. In Proc. IEEE Conf. Visualization, pages 535–542, 2005.Google Scholar
  8. 8.
    A. Gyulassy, V. Natarajan, V. Pascucci, P. T. Bremer, and B. Hamann. A topological approach to simplification of three-dimensional scalar fields. IEEE Transactions on Visualization and Computer Graphics (special issue IEEE Visualization 2005), pages 474–484, 2006.Google Scholar
  9. 9.
    A. Gyulassy, V. Natarajan, V. Pascucci, and B. Hamann. 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
  10. 10.
    H. King, K. Knudson, and N. Mramor. Generating discrete morse functions from point data. Experimental Mathematics, 14(4):435–444, 2005.zbMATHMathSciNetGoogle Scholar
  11. 11.
    T. Lewiner, H. Lopes, and G. Tavares. Applications of forman’s discrete morse theory to topology visualization and mesh compression. IEEE Transactions on Visualization and Computer Graphics, 10(5):499–508, 2004.CrossRefGoogle Scholar
  12. 12.
    S. Smale. Generalized Poincaré’s conjecture in dimensions greater than four. Ann. of Math., 74:391–406, 1961.CrossRefMathSciNetGoogle Scholar
  13. 13.
    S. Smale. On gradient dynamical systems. Ann. of Math., 74:199–206, 1961.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  • Attila Gyulassy
    • 1
    • 2
  • Peer-Timo Bremer
    • 2
  • Bernd Hamann
    • 1
  • Valerio Pascucci
    • 3
  1. 1.Institute for Data Analysis and Visualization (IDAV), Department of Computer ScienceUniversity of CaliforniaDavisUSA
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Scientific Computing and Imaging InstituteUniversity of Utah – School of ComputingSalt Lake CityUSA

Personalised recommendations