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Compute and Visualize Discontinuity Among Neighboring Integral Curves of 2D Vector Fields

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Topological Methods in Data Analysis and Visualization IV (TopoInVis 2015)

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

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Abstract

This paper studies the discontinuity in the behavior of neighboring integral curves. The discontinuity is measured by a number of selected attributes of integral curves. A variety of attribute fields are defined. The attribute value at any given spatio-temporal point in these fields is assigned by the attribute of the integral curve that passes through this point. This encodes the global behavior of integral curves into a number of scalar fields in an Eulerian fashion, which differs from the previous pathline attribute approach that focuses on the discrete representation of individual pathlines. With this representation, the discontinuity of the integral curve behavior now corresponds to locations in the derived fields where the attribute values have sharp gradients. We show that based on the selected attributes, the extracted discontinuity from the corresponding attribute fields may relate to a number of flow features, such as LCS, vortices, and cusp-like seeding curves. In addition, we study the correlations among different attributes via their pairwise scatter plots. We also study the behavior of the combined attribute fields to understand the spatial correlation that cannot be revealed by the scatter plots. Finally, we integrate our attribute field computation and their discontinuity detection into an interactive system to guide the exploration of various 2D flows.

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References

  1. Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8(6), 679–698 (1986)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. Edmunds, M., Laramee, R.S., Chen, G., Max, N., Zhang, E., Ware, C.: Surface-based flow visualization. Comput. Graph. 36(8), 974–990 (2012)

    Article  Google Scholar 

  5. Haller, G.: Lagrangian coherent structures and the rate of strain in two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Haller, G., Sapsis, T.: Lagrangian coherent structures and the smallest finite-time Lyapunov exponent. Chaos 21(2), 023115 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  8. Kasten, J., Reininghaus, J., Hotz, I., Hege, H.C.: Two-dimensional time-dependent vortex regions based on the acceleration magnitude. IEEE Trans. Vis. Comput. Graph. 17(12), 2080–2087 (2011)

    Article  Google Scholar 

  9. Lu, K., Chaudhuri, A., Lee, T.Y., Shen, H.W., Wong, P.C.: Exploring vector fields with distribution-based streamline analysis. In: Proceedings IEEE Pacific Visualization Symposium, Sydney, pp. 257–264 (2013)

    Google Scholar 

  10. McLoughlin, T., Jones, M.W., Laramee, R.S., Malki, R., Masters, I., Hansen, C.D.: Similarity measures for enhancing interactive streamline seeding. IEEE Trans. Vis. Comput. Graph. 19(8), 1342–1353 (2013)

    Article  Google Scholar 

  11. Pobitzer, A., Peikert, R., Fuchs, R., Schindler, B., Kuhn, A., Theisel, H., Matkovic, K., Hauser, H.: The state of the art in topology-based visualization of unsteady flow. Comput. Graph. Forum 30(6), 1789–1811 (2011)

    Article  Google Scholar 

  12. Pobitzer, A., Lez, A., Matkovic, K., Hauser, H.: A statistics-based dimension reduction of the space of path line attributes for interactive visual flow analysis. In: Proceedings IEEE Pacific Visualization Symposium, pp. 113–120. IEEE, Piscataway (2012)

    Google Scholar 

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

    Article  Google Scholar 

  14. Salzbrunn, T., Scheuermann, G.: Streamline predicates. IEEE Trans. Vis. Comput. Graph. 12(6), 1601–1612 (2006)

    Article  MATH  Google Scholar 

  15. Salzbrunn, T., Garth, C., Scheuermann, G., Meyer, J.: Pathline predicates and unsteady flow structures. Vis. Comput. 24(12), 1039–1051 (2008)

    Article  Google Scholar 

  16. Shadden, S., Lekien, F., Marsden, J.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212(3), 271–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shi, K., Theisel, H., Weinkauf, T., Hege, H.C., Seidel, H.P.: Visualizing transport structures of time-dependent flow fields. IEEE Comput. Graph. Appl. 28(5), 24–36 (2008)

    Article  Google Scholar 

  18. Shi, K., Theisel, H., Hauser, H., Weinkauf, T., Matkovic, K., Hege, H.C., Seidel, H.P.: Path line attributes - an information visualization approach to analyzing the dynamic behavior of 3D time-dependent flow fields. In: Hege, H.C., Polthier, K., Scheuermann, G. (eds.) Topology-Based Methods in Visualization II. Mathematics and Visualization, pp. 75–88. Springer, Berlin (2009)

    Chapter  Google Scholar 

  19. Theisel, H., Weinkauf, T., Seidel, H.P.: Grid-independent detection of closed stream lines in 2D vector fields. In: Proceedings Conference on Vision, Modeling and Visualization, pp. 421–428 (2004)

    Google Scholar 

  20. Weinkauf, T., Theisel, H.: Streak lines as tangent curves of a derived vector field. IEEE Trans. Vis. Comput. Graph. 16(6), 1225–1234 (2010)

    Article  Google Scholar 

  21. Weinkauf, T., Theisel, H., Van Gelder, A., Pang, A.: Stable feature flow fields. IEEE Trans. Vis. Comput. Graph. 17(6), 770–780 (2011)

    Article  Google Scholar 

  22. Weinkauf, T., Theisel, H., Sorkine, O.: Cusps of characteristic curves and intersection-aware visualization of path and streak lines. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds.) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization, pp. 161–176. Springer, Berlin (2012)

    Chapter  Google Scholar 

  23. Wischgoll, T., Scheuermann, G.: Detection and visualization of closed streamlines in planar fields. IEEE Trans. Vis. Comput. Graph. 7(2), 165–172 (2001)

    Article  Google Scholar 

  24. Yu, H., Wang, C., Shene, C.K., Chen, J.H.: Hierarchical streamline bundles. IEEE Trans. Vis. Comput. Graph. 18(8), 1353–1367 (2012)

    Article  Google Scholar 

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Acknowledgements

We thank Jackie Chen, Mathew Maltude, Tino Weinkauf for the data. This research was in part supported by NSF IIS-1352722 and IIS-1065107.

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Authors and Affiliations

  1. University of Houston, Houston, TX, USA

    Lei Zhang & Guoning Chen

  2. Swansea University, Wales, UK

    Robert S. Laramee

  3. Mississippi State University, Starkville, MS, USA

    David Thompson & Adrian Sescu

Authors
  1. Lei Zhang
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  2. Robert S. Laramee
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  3. David Thompson
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  4. Adrian Sescu
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  5. Guoning Chen
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Corresponding author

Correspondence to Guoning Chen .

Editor information

Editors and Affiliations

  1. University of Leeds, Leeds, United Kingdom

    Hamish Carr

  2. Department of Computer Science, Technical University of Kaiserslautern, Kaiserslautern, Germany

    Christoph Garth

  3. School of Computer Science and Communication, KTH Royal Institute of Technology, Stockholm, Sweden

    Tino Weinkauf

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Zhang, L., Laramee, R.S., Thompson, D., Sescu, A., Chen, G. (2017). Compute and Visualize Discontinuity Among Neighboring Integral Curves of 2D Vector Fields. In: Carr, H., Garth, C., Weinkauf, T. (eds) Topological Methods in Data Analysis and Visualization IV. TopoInVis 2015. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-44684-4_11

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