Skip to main content
SpringerLink
Log in

Computational Discrete Morse Theory for Divergence-Free 2D Vector Fields

  • Chapter
  • First Online:
Book cover Topological Methods in Data Analysis and Visualization II

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

We present a simple approach to the topological analysis of divergence-free 2D vector fields using discrete Morse theory. We make use of the fact that the point-wise perpendicular vector field can be interpreted as the gradient of the stream function. The topology of the divergence-free vector field is thereby encoded in the topology of a gradient vector field. We can therefore apply a formulation of computational discrete Morse theory for gradient vector fields. The inherent consistence and robustness of the resulting algorithm is demonstrated on synthetic data and an example from computational fluid dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bauer, U., Lange, C., Wardetzky, M.: Optimal topological simplification of discrete functions on surfaces. arXiv:1001.1269v2 (2010)

    Google Scholar 

  2. Chari, M.K.: On discrete Morse functions and combinatorial decompositions. Discrete Math. 217(1-3), 101–113 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, G., Mischaikow, K., Laramee, R., Pilarczyk, P., Zhang, E.: Vector field editing and periodic orbit extraction using Morse decomposition. IEEE Trans. Visual. Comput. Graph. 13, 769–785 (2007)

    Article  Google Scholar 

  4. Chen, G., Mischaikow, K., Laramee, R.S., Zhang, E.: Efficient morse decomposition of vector fields. IEEE Trans. Visual. Comput. Graph. 14(4), 848–862 (2008)

    Article  Google Scholar 

  5. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Forman, R.: Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift 228(4), 629–681 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gyulassy, A.: Combinatorial Construction of Morse-Smale Complexes for Data Analysis and Visualization. PhD thesis, University of California, Davis (2008)

    Google Scholar 

  9. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge, U.K. (2002)

    MATH  Google Scholar 

  10. Helman, J., Hesselink, L.: Representation and display of vector field topology in fluid flow data sets. IEEE Comput. 22(8), 27–36 (1989)

    Article  Google Scholar 

  11. Helman, J., Hesselink, L.: Representation and display of vector field topology in fluid flow data sets. Computer 22(8), 27–36 (1989)

    Article  Google Scholar 

  12. Helmholtz, H.: Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858)

    Article  MATH  Google Scholar 

  13. King, H., Knudson, K., Mramor, N.: Generating discrete Morse functions from point data. Exp. Math. 14(4), 435–444 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Klein, T., Ertl, T.: Scale-space tracking of critical points in 3d vector fields. In: Helwig Hauser, H.H., Theisel, H. (eds.) Topology-based Methods in Visualization, Mathematics and Visualization, pp. 35–49. Springer, Berlin (2007)

    Chapter  Google Scholar 

  15. Laramee, R.S., Hauser, H., Zhao, L., Post, F.H.: Topology-based flow visualization, the state of the art. In: Helwig Hauser, H.H., Theisel, H. (eds.) Topology-based Methods in Visualization, Mathematics and Visualization, pp. 1–19. Springer, Berlin (2007)

    Chapter  Google Scholar 

  16. Lewiner, T.: Geometric discrete Morse complexes. PhD thesis, Department of Mathematics, PUC-Rio, 2005. Advised by Hélio Lopes and Geovan Tavares.

    Google Scholar 

  17. Milnor, J.: Morse Theory. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  18. Noack, B.R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P., Tadmor, G.: A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilibr. Thermodyn. 33(2), 103–148 (2008)

    Article  MATH  Google Scholar 

  19. Panton, R.: Incompressible Flow. Wiley, New York (1984)

    MATH  Google Scholar 

  20. Petronetto, F., Paiva, A., Lage, M., Tavares, G., Lopes, H., Lewiner, T.: Meshless Helmholtz-Hodge decomposition. Trans. Visual. Comput. Graph. 16(2), 338–342 (2010)

    Article  Google Scholar 

  21. Polthier, K., Preuß, E.: Identifying vector field singularities using a discrete Hodge decomposition. pp. 112–134. Springer, Berlin (2002)

    Google Scholar 

  22. Reininghaus, J., Günther, D., Hotz, I., Prohaska, S., Hege, H.-C.: TADD: A computational framework for data analysis using discrete Morse theory. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Mathematical Software – ICMS 2010, volume 6327 of Lecture Notes in Computer Science, pp. 198–208. Springer, Berlin (2010)

    Chapter  Google Scholar 

  23. Reininghaus, J., Hotz, I.: Combinatorial 2d vector field topology extraction and simplification. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds.) Topological Methods in Data Analysis and Visualization, Mathematics and Visualization, pp. 103–114. Springer, Berlin (2011)

    Chapter  Google Scholar 

  24. Reininghaus, J., Löwen, C., Hotz, I.: Fast combinatorial vector field topology. IEEE Trans. Visual. Comput. Graph. 17 1433–1443 (2011)

    Article  Google Scholar 

  25. 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. Learn. 33(8), 1646–1658 (2011)

    Article  Google Scholar 

  26. Shonkwiler, C.: Poincaré duality angles for Riemannian manifolds with boundary. Technical Report arXiv:0909.1967, Sep 2009. Comments: 51 pages, 6 figures.

    Google Scholar 

  27. Tong, Y., Lombeyda, S., Hirani, A.N., Desbrun, M.: Discrete multiscale vector field decomposition. In: ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH, volume 22 (2003)

    Google Scholar 

  28. Tricoche, X., Scheuermann, G., Hagen, H.: Continuous topology simplification of planar vector fields. In: VIS ’01: Proceedings of the conference on Visualization ’01, pp. 159–166. IEEE Computer Society, Washington, DC, USA (2001)

    Google Scholar 

  29. Tricoche, X., Scheuermann, G., Hagen, H., Clauss, S.: Vector and tensor field topology simplification on irregular grids. In: Ebert, D., Favre, J.M., Peikert, R. (eds.) VisSym ’01: Proceedings of the symposium on Data Visualization 2001, pp. 107–116. Springer, Wien, Austria, May 28–30 (2001)

    Google Scholar 

  30. Weinkauf, T.: Extraction of Topological Structures in 2D and 3D Vector Fields. PhD thesis, University Magdeburg (2008)

    Google Scholar 

  31. Weinkauf, T., Theisel, H., Shi, K., Hege, H.-C., Seidel, H.-P.: Extracting higher order critical points and topological simplification of 3D vector fields. In: Proceedings IEEE Visualization 2005, pp. 559–566. Minneapolis, U.S.A., October 2005

    Google Scholar 

Download references

Acknowledgements

We would like to thank David Günther, Jens Kasten, and Tino Weinkauf for many fruitful discussions on this topic. This work was funded by the DFG Emmy-Noether research programm. All visualizations in this paper have been created using AMIRA – a system for advanced visual data analysis (see http://amira.zib.de/).

Author information

Authors and Affiliations

  1. Zuse Institute Berlin, Takustr. 7, 14195, Berlin, Germany

    Jan Reininghaus & Ingrid Hotz

Authors
  1. Jan Reininghaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ingrid Hotz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Reininghaus .

Editor information

Editors and Affiliations

  1. Inst. Computational Science, CAB G 65.1, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Ronald Peikert

  2. , Dept. of Informatics, University of Bergen, Bergen, Norway

    Helwig Hauser

  3. , School of Computing, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom

    Hamish Carr

  4. , Computational Science, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Raphael Fuchs

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Reininghaus, J., Hotz, I. (2012). Computational Discrete Morse Theory for Divergence-Free 2D Vector Fields. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation