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2D Vector field approximation using linear neighborhoods

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Abstract

We present a vector field approximation for two-dimensional vector fields that preserves their topology and significantly reduces the memory footprint. This approximation is based on a segmentation. The flow within each segmentation region is approximated by an affine linear function. The implementation is driven by four aims: (1) the approximation preserves the original topology; (2) the maximal approximation error is below a user-defined threshold in all regions; (3) the number of regions is as small as possible; and (4) each point has the minimal approximation error. The generation of an optimal solution is computationally infeasible. We discuss this problem and provide a greedy strategy to efficiently compute a sensible segmentation that considers the four aims. Finally, we use the region-wise affine linear approximation to compute a simplified grid for the vector field.

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References

  1. Chen, C.K., Yan, S., Yu, H., Max, N., Ma, K.L.: An illustrative visualization framework for 3d vector fields. Comput. Graph. Forum 30(7), 1941–1951 (2011)

    Article  Google Scholar 

  2. Chen, J.L., Bai, Z., Hamann, B., Ligocki, T.J.: Normalized-cut algorithm for hierarchical vector field data segmentation. Proc. SPIE 5009, 79–90 (2003)

    Article  Google Scholar 

  3. Dey, T., Levine, J., Wenger, R.: A Delaunay simplification algorithm for vector fields. In: 15th Pacific Conference on Computer Graphics and Applications, pp. 281–290 (2007)

  4. Garcke, H., Preußer, T., Rumpf, M., Telea, A., Weikard, U., Van Wijk, J.: A continuous clustering method for vector fields. In: Proceedings of the Conference on Visualization 2000, pp. 351–358. IEEE Computer Society Press (2000)

  5. Griebel, M., Preusser, T., Rumpf, M., Schweitzer, M.A., Telea, A.: Flow field clustering via algebraic multigrid. In: Proceedings of the Conference on Visualization 2004, pp. 35–42. IEEE Computer Society Press (2004)

  6. Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids (1994–present) 13(11), 3365–3385 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Heckel, B., Weber, G., Hamann, B., Joy, K.I.: Construction of vector field hierarchies. In: Proceedings of the Conference on Visualization 1999, pp. 19–25. IEEE Computer Society Press, Los Alamitos (1999)

  8. 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 

  9. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, vol. 60, 2nd edn. Academic Press, San Diego (2004)

    MATH  Google Scholar 

  10. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kasten, J., Petz, C., Hotz, I., Noack, B., Hege, H.C.: Localized finite-time Lyapunov exponent for unsteady flow analysis. In: Magnor, M., Rosenhahn, B., Theisel, H. (eds.) Vision, Modeling and Visualization, pp. 265–274 (2009)

  12. Kenwright, D.N., Henze, C., Levit, C.: Feature extraction of separation and attachment lines. IEEE TVCG 5(2), 135–144 (1999)

    Google Scholar 

  13. Koch, S., Wiebel, A., Kasten, J., Hlawitschka, M.: Visualizing linear neighborhoods in non-linear vector fields. In: IEEE PacificVis 2013, pp. 249–256 (2013)

  14. Kuhn, A., Lehmann, D.J., Gaststeiger, R., Neugebauer, M., Preim, B., Theisel, H.: A clustering-based visualization technique to emphasize meaningful regions of vector fields. In: Proceedings of Vision, Modeling, and Visualization, pp. 191–198. Eurographics Assosciation (2011)

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

  16. Li, H., Chen, W., Shen, I.F.: Segmentation of discrete vector fields. IEEE TVCG 12(3), 289–300 (2006)

    MathSciNet  Google Scholar 

  17. Lodha, S.K., Renteria, J.C., Roskin, K.M.: Topology preserving compression of 2d vector fields. In: Proceedings of the Conference on Visualization 2000, pp. 343–350. IEEE Computer Society Press, Los Alamitos (2000)

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

  19. Marchesin, S., Chen, C.K., Ho, C., Ma, K.L.: View-dependent streamlines for 3d vector fields. IEEE TVCG 16(6), 1578–1586 (2010)

    Google Scholar 

  20. McKenzie, A., Lombeyda, S.V., Desbrun, M.: Vector field analysis and visualization through variational clustering. In: Proceedings of EuroVis, pp. 29–35. Eurographics Association (2005)

  21. McLoughlin, T., Laramee, R.S., Peikert, R., Post, F.H., Chen, M.: Over two decades of integration-based, geometric flow visualization. Comput. Graph. Forum 29(6), 1807–1829 (2010)

    Article  Google Scholar 

  22. Peikert, R., Roth, M.: The “Parallel Vectors” operator: a vector field visualization primitive. In: Proceedings of the Conference on Visualization 1999, pp. 263–270. IEEE Computer Society Press (1999)

  23. Post, F.H., Vrolijk, B., Hauser, H., Laramee, R.S., Doleisch, H.: The state of the art in flow visualization: feature extraction and tracking. Comput. Graph. Forum 22(4), 775–792 (2003)

    Article  Google Scholar 

  24. Rom-Kedar, V., Leonard, A., Wiggins, S.: An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salzbrunn, T., Jänicke, H., Wischgoll, T., Scheuermann, G.: The state of the art in flow visualization: partition-based techniques. In: Simulation and Visualization 2008, pp. 75–92. SCS Publishing House (2008)

  26. Schneider, D., Reich, W., Wiebel, A., Scheuermann, G.: Topology aware stream surfaces. Comput. Graph. Forum 29(3), 1153–1161 (2010)

    Article  Google Scholar 

  27. Stalling, D., Hege, H.C.: Fast and resolution-independent line integral convolution. In: Proceedings of SIGGRAPH ’95, pp. 249–256. ACM SIGGRAPH, Los Angeles (1995)

  28. Sujudi, D., Haimes, R.: Identification of Swirling Flow in 3D Vector Fields. Technical Report AIAA Paper 95–1715, American Institute of Aeronautics and Astronautics (1995)

  29. Telea, A., Van Wijk, J.: Simplified representation of vector fields. In: Proceedings of the Conference on Visualization 1999, pp. 35–507. IEEE Computer Society Press (1999)

  30. Theisel, H., Rössl, C., Seidel, H.P.: Compression of 2d vector fields under guaranteed topology preservation. Comput. Graph. Forum 22(3), 333–342 (2003)

    Article  Google Scholar 

  31. Tricoche, X., Scheuermann, G., Hagen, H.: Higher order singularities in piecewise linear vector fields. In: The Mathematics of Surfaces IX, pp. 99–113. Springer (2000)

  32. Tricoche, X., Wischgoll, T., Scheuermann, G., Hagen, H.: Topology tracking for the visualization of time-dependent two-dimensional flows. Comput. Graph. 26(2), 249–257 (2002)

    Article  Google Scholar 

  33. Wiebel, A., Koch, S., Scheuermann, G.: Glyphs for non-linear vector field singularities. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds.) Topological Methods in Visualization, pp. 177–190. Springer, Berlin (2012)

    Chapter  Google Scholar 

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Acknowledgments

We thank Markus Rütten, Guillaume Daviller, and Bernd Noack for providing the simulation datasets. Special thanks go to the FAnToM development group for providing the visualization software. We also thank Roxana Bujack and Sebastian Volke for the fruitful discussions. S. Koch and J. Kasten were supported by the European Social Fund (Appl. No. 100098251).

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Authors and Affiliations

  1. Institute of Computer Science, Leipzig University, Augustusplatz 10, 04109, Leipzig, Germany

    Stefan Koch, Jens Kasten, Gerik Scheuermann & Mario Hlawitschka

  2. Coburg University of Applied Sciences, Friedrich-Streib-Str.2, 96450, Coburg, Germany

    Alexander Wiebel

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  1. Stefan Koch
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  2. Jens Kasten
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  3. Alexander Wiebel
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  4. Gerik Scheuermann
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  5. Mario Hlawitschka
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Correspondence to Stefan Koch.

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Koch, S., Kasten, J., Wiebel, A. et al. 2D Vector field approximation using linear neighborhoods. Vis Comput 32, 1563–1578 (2016). https://doi.org/10.1007/s00371-015-1140-9

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  • DOI: https://doi.org/10.1007/s00371-015-1140-9

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