Consistent Approximation of Local Flow Behavior for 2D Vector Fields Using Edge Maps

Part of the Mathematics and Visualization book series (MATHVISUAL)


Vector fields, represented as vector values sampled on the vertices of a triangulation, are commonly used to model physical phenomena. To analyze and understand vector fields, practitioners use derived properties such as the paths of massless particles advected by the flow, called streamlines. However, currently available numerical methods for computing streamlines do not guarantee preservation of fundamental invariants such as the fact that streamlines cannot cross. The resulting inconsistencies can cause errors in the analysis, e.g., invalid topological skeletons, and thus lead to misinterpretations of the data. We propose an alternate representation for triangulated vector fields that exchanges vector values with an encoding of the transversal flow behavior of each triangle. We call this representation edge maps. This work focuses on the mathematical properties of edge maps; a companion paper discusses some of their applications[1]. Edge maps allow for a multi-resolution approximation of flow by merging adjacent streamlines into an interval based mapping. Consistency is enforced at any resolution if the merged sets maintain an order-preserving property. At the coarsest resolution, we define a notion of equivalency between edge maps, and show that there exist 23 equivalence classes describing all possible behaviors of piecewise linear flow within a triangle.


Directed Edge Consistent Approximation Destination Point Undirected Edge Mixed Graph 
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 is supported in part by the National Science Foundation awards IIS-1045032, OCI-0904631, OCI-0906379 and CCF-0702817. This work was also performed under the auspices of the U.S. Department of Energy by the University of Utah under contracts DE-SC0001922, DE-AC52-07NA27344, and DE-FC02-06ER25781, and Lawrence Livermore National Laboratory (LLNL) under contract DE-AC52-07NA27344. Attila Gyulassy and Philippe P. Pebay provided many useful comments and discussions. LLNL-CONF-468780.


  1. 1.
    Bhatia, H., Jadhav, S., Bremer, P.-T., Chen, G., Levine, J. A., Nonato, L. G., Pascucci, V.: Edge maps: Representing flow with bounded error. In: Pacific Visualization Symposium (PacificVis), 2011 IEEE, pp. 75–82, March 2011Google Scholar
  2. 2.
    Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’93, Anaheim, CA, pp. 263–270. ACM, New York (1993)Google Scholar
  3. 3.
    Chen, G., Mischaikow, K., Laramee, R.S., Pilarczyk, P., Zhang, E.: Vector field editing and periodic orbit extraction using Morse decomposition. IEEE Trans. Vis. Comput. Graph. 13(4), 769–785 (2007)CrossRefGoogle Scholar
  4. 4.
    Chen, G., Mischaikow, K., Laramee, R.S., Zhang, E.: Efficient morse decompositions of vector fields. IEEE Trans. Vis. Comput. Graph. 14(4), 848–862 (2008)CrossRefGoogle Scholar
  5. 5.
    de Leeuw, W., van Liere, R.: Collapsing flow topology using area metrics. In: Proceedings of the Conference on Visualization ’99: celebrating ten years, VIS ’99, San Francisco, CA, pp. 349–354. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  6. 6.
    Forman, R.: A user’s guide to discrete Morse theory. In: Proc. of the 2001 Internat. Conf. on Formal Power Series and Algebraic Combinatorics, A special volume of Advances in Applied Mathematics, p. 48 (2001)Google Scholar
  7. 7.
    Garth, C., Tricoche, X.: Topology- and feature-based flow visualization: Methods and applications. In: SIAM Conference on Geometric Design and Computing (2005)Google Scholar
  8. 8.
    Garth, C., Krishnan, H., Tricoche, X., Tricoche, T., Joy, K.I.: Generation of accurate integral surfaces in time-dependent vector fields. IEEE Trans. Vis. Comput. Graph. 14(6), 1404–1411 (2008)CrossRefGoogle Scholar
  9. 9.
    Globus, A., Levit, C. Lasinski, T.: A tool for visualizing the topology of three-dimensional vector fields. In: Proceedings of the 2nd Conference on Visualization ’91, VIS ’91, San Diego, CA, pp. 33–40. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  10. 10.
    Helman, J., Hesselink, L.: Representation and display of vector field topology in fluidflow data sets. IEEE Comput. 22(8), 27–36 (1989)CrossRefGoogle Scholar
  11. 11.
    Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd edn. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  12. 12.
    Janine, K.M., Bennett, J., Scheuermann, G., Hamann, B., Joy, K.I.: Topological segmentation in three-dimensional vector fields. IEEE Trans. Visual. Comput. Graph. 10, 198–205 (2004)CrossRefGoogle Scholar
  13. 13.
    Kipfer, P., Reck, F., Greiner, G.: Local exact particle tracing on unstructured grids. Comput. Graph. Forum 22, 133–142 (2003)CrossRefGoogle Scholar
  14. 14.
    Laramee, R.S., Hauser, H., Zhao, L., Post, F.H.: Topology based flow visualization: The state of the art. In: Topology-Based Methods in Visualization (Proc. Topo-in-Vis 2005), Mathematics and Visualization, pp. 1–19. Springer, Berlin (2007)Google Scholar
  15. 15.
    Lodha, S.K., Renteria, J.C., Roskin, K.M.: Topology preserving compression of 2d vector fields. In: Proceedings of the conference on Visualization ’00, VIS ’00, pp. 343–350. IEEE Computer Society Press, Los Alamitos, CA, USA (2000)Google Scholar
  16. 16.
    Nielson, G.M., Jung, I.-H.: Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains. IEEE Trans. Vis. Comput. Graph. 5(4), 360–372 (1999)CrossRefGoogle Scholar
  17. 17.
    Polthier, K., Preuß, E.: Identifying vector fields singularities using a discrete Hodge decomposition. In: Hege, H.C., Polthier, K. (eds.) Mathematical Visualization III, pp. 112–134 (2003)Google Scholar
  18. 18.
    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. Theory, Algorithms, and Applications. (TopoInVis’09), pp. 103–114. Springer, Berlin (2009)Google Scholar
  19. 19.
    Reininghaus, J., Löwen, C., Hotz, I.: Fast combinatorial vector field topology. IEEE Trans. Visual. Comput. Graph. 17(10), 1433–1443 (2010). doi:10.1109/TVCG.2010.235CrossRefGoogle Scholar
  20. 20.
    Sch¨onert, M., et al.: GAP – Groups, Algorithms, and Programming. Lehrstuhl D f¨ur Mathematik, Rheinisch Westf¨alische Technische Hochschule, 5th edn. Aachen, Germany (1995)Google Scholar
  21. 21.
    Scheuermann, G., Tricoche, X.: Topological methods in flow visualization. In: Hansen, C., Johnson, C. (eds.) Visualization Handbook, pp. 341–356. Elsevier, Amsterdam (2004)Google Scholar
  22. 22.
    Scheuermann, G., Krüger, H., Menzel, M., Rockwood, A.P.: Visualizing nonlinear vector field topology. IEEE Trans. Visual. Comput. Graph. 4(2), 109–116 (1998)CrossRefGoogle Scholar
  23. 23.
    Scheuermann, G., Bobach, T., Hagen, H., Mahrous, K., Hamann, B., Joy, K.I., Kollmann, W.: A tetrahedra-based stream surface algorithm. In: VIS ’01: Proceedings of the conference on Visualization ’01, pp. 151–158. IEEE Computer Society, Washington, DC, USA (2001)Google Scholar
  24. 24.
    Theisel, H., Weinkauf, T., Hege, H.-C., Seidel, H.-P.: On the applicability of topological methods for complex flow data. In: Hauser, H., Hagen, H., Theisel, H. (eds.) Topology-based Methods in Visualization, Mathematics and Visualization, pp. 105–120. Springer, Berlin (2007)CrossRefGoogle Scholar
  25. 25.
    Turk, G., Banks, D.: Image-guided streamline placement. In: Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, SIGGRAPH ’96, pp. 453–460. ACM, New York (1996)Google Scholar
  26. 26.
    Weinkauf, T., Theisel, H., Hege, H.-C., Seidel, H.-P.: Boundary switch connectors for topological visualization of complex 3D vector fields. In: Deussen, O., Hansen, C., Keim, D., Saupe, D. (eds.) Symposium on Visualization, pp. 183–192. Eurographics Association, Konstanz (2004)Google Scholar
  27. 27.
    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: IEEE Visualization, 2005. VIS 05. pp. 559–566 (2005)Google Scholar
  28. 28.
    Wischgoll, T., Scheuermann, G.: Detection and visualization of closed streamlines in planar fields. IEEE Trans. Visual. Comput. Graph. 7(2), 165–172 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.SCI InstituteUniversity of UtahSalt Lake CityUSA
  2. 2.Lawrence Livermore National LabLivermoreUSA
  3. 3.Universidade de São PauloSao PauloBrazil

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