Combinatorial 2D Vector Field Topology Extraction and Simplification
This paper investigates a combinatorial approach to vector field topology. The theoretical basis is given by Robin Forman’s work on a combinatorial Morse theory for dynamical systems defined on general simplicial complexes. We formulateForman’s theory in a graph theoretic setting and provide a simple algorithm for the construction and topological simplification of combinatorial vector fields on 2D manifolds. Given a combinatorial vector field we are able to extract its topological skeleton including allperiodic orbits. Due to the solid theoretical foundation we know that the resulting structure is always topologically consistent. We explore the applicability and limitations of this combinatorial approach with several examples and determine its robustness with respect to noise.
KeywordsPeriodic Orbit Simplicial Graph Morse Theory Singular Homology Morse Decomposition
Unable to display preview. Download preview PDF.
We would like to thank David Günther and Tino Weinkauf for many fruitful discussions on this topic and Christian Löwen for his great implementational efforts. 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/).
- 2.Herbert Edelsbrunner and J. Harer. Persistent homology — a survey. In J. E. Goodman, J. Pach, and R. Pollack, editors, Surveys on Discrete and Computational Geometry: Twenty Years Later, volume 458, pages 257–282. AMS Bookstore, 2008.Google Scholar
- 3.Herbert Edelsbrunner, John Harer, Vijay Natarajan, and Valerio Pascucci. Morse-smale complexes forpiecewise linear 3-manifolds. In SCG ’03: Proceedings of the nineteenth annual symposium on Computational geometry, pages 361–370, New York, NY, USA, 2003. ACM.Google Scholar
- 6.Robin Forman. A user’s guide to discrete morse theory. In Proceedings of the 2001 Internat. Conf. on Formal Power Series and Algebraic Combinatorics, Advances in Applied Mathematics, 2001.Google Scholar
- 9.Thomas Klein and Thomas Ertl. Scale-space tracking of critical points in 3d vector fields. In Hans Hagen Helwig Hauser and Holger Theisel, editors, Topology-based Methods in Visualization, Mathematics and Visualization, pages 35–49. Springer Berlin Heidelberg, May 2007.Google Scholar
- 10.Robert S. Laramee, Helwig Hauser, Lingxiao Zhao, and Frits H. Post. Topology-based flow visualization, the state of the art. In Hans Hagen Helwig Hauser and Holger Theisel, editors, Topology-based Methods in Visualization, Mathematics and Visualization, pages 1–19. Springer Berlin Heidelberg, May 2007.Google Scholar
- 11.Thomas Lewiner. Geometric discrete Morse complexes. PhD thesis, Department of Mathematics, PUC-Rio, 2005. Advised by Hlio Lopes and Geovan Tavares.Google Scholar
- 14.Alexander Schrijver. Combinatorial Optimization. Springer, 2003.Google Scholar
- 15.Holger Theisel, Tino Weinkauf, Hans-Christian Hege, and Hans-Peter Seidel. Grid-independent detection of closed stream lines in 2d vector fields. In Proceedings of the VMV Conference 2004, page 665, Stanford, USA, November 2004.Google Scholar
- 16.Xavier Tricoche, Gerik Scheuermann, and Hans Hagen. Continuous topology simplification of planar vector fields. In VIS ’01: Proceedings of the conference on Visualization ’01, pages 159–166, Washington, DC, USA, 2001. IEEE Computer Society.Google Scholar
- 17.Xavier Tricoche, Gerik Scheuermann, Hans Hagen, and Stefan Clauss. Vector and tensor field topology simplification on irregular grids. In D. Ebert, J. M. Favre, and R. Peikert, editors, VisSym ’01: Proceedings of the symposium on Data Visualization 2001, pages 107–116, Wien, Austria, May 28–30 2001. Springer-Verlag.Google Scholar
- 18.Tino Weinkauf, Holger Theisel, K. Shi, Hans-Christian Hege, and Hans-Peter Seidel. Extracting higher order critical points and topological simplification of 3D vector fields. In Proc. IEEE Visualization 2005, pages 559–566, Minneapolis, U.S.A., October 2005.Google Scholar