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Scale-Space Approaches to FTLE Ridges

  • Raphael Fuchs
  • Benjamin Schindler
  • Ronald Peikert
Chapter
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

The finite-time Lyapunov Exponent (FTLE) is useful for the visualization of time-dependent velocity fields. The ridges of this derived scalar field have been shown to correspond well to attracting or repelling material structures, so-called Lagrangian coherent structures (LCS). There are two issues involved in the computation of FTLE for this purpose. Firstly, it is often not practically possible to refine the grid for sampling the flow map until convergence of FTLE is reached. Slow conversion is mostly caused by gradient underestimation. Secondly, there is a parameter, the integration time, which has to be chosen sensibly. Both of these problems call for an examination in scale-space. We show that a scale-space approach solves the problem of gradient underestimation. We test optimal-scale ridges for their usefulness with FTLE fields, obtaining a negative result. However, we propose an optimization of the time parameter for a given scale of observation. Finally, an incremental method for computing smoothed flow maps is presented.

Keywords

Integration Time Lyapunov Exponent Lagrangian Coherent Structure Slide Time Window Neighbor Trajectory 
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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Notes

Acknowledgements

We thank Filip Sadlo for the simulation of the air convection. The project SemSeg acknowledges the financial support of the Future and Emerging Technologies (FET) program within the Seventh Framework Program for Research of the European Commission, under FET-Open grant number 226042.

References

  1. 1.
    Bauer, D., Peikert, R.: Vortex tracking in scale space. In: Ebert, D., Brunet, P., Navazo, I. (eds.) Data Visualization 2002: Proceedings of the 4th Joint EurographicsIEEE TCVG Symposium on Visualization (VisSym 2002), Eurographics, pp. 233–240 (2002)Google Scholar
  2. 2.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory. Meccanica 15, 9–20 (1980)Google Scholar
  3. 3.
    Brunton, S.L., Rowley, C.W.: Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20(1), 017503.1–017503.9 (2010)Google Scholar
  4. 4.
    Eberly, D.: Ridges in Image and Data Analysis. Computational Imaging and Vision. Kluwer Academic Publishers, Dordrecht (1996)zbMATHGoogle Scholar
  5. 5.
    Florack, L., Kuijper, A.: The topological structure of scale-space images. J. Math. Imag. Vis. 11(1), 365–79 (2000)MathSciNetGoogle Scholar
  6. 6.
    Garth, C., Gerhardt, F., Tricoche, X., Hagen, H.: Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans. Visual. Comput. Graph. 13(6), 1464–1471 (2007)CrossRefGoogle Scholar
  7. 7.
    Garth, C., Wiebel, A., Tricoche, X., Joy, K., Scheuermann, G.: Lagrangian visualization of flow-embedded surface structures. Comput. Graph. Forum 27(3), 1007–1014 (2008)CrossRefGoogle Scholar
  8. 8.
    Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10(1), 99–108 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248–277 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Haller, G.: Lagrangian coherent structures from approximate velocity data. Physics of Fluids 14, 1851–1861 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Phys. D: Nonlinear Phenom. 240(7), 574–598 (2010)CrossRefGoogle Scholar
  12. 12.
    Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147(3), 352–370 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jimenez, R., Vankerschaver, J.: Optimization of FTLE Calculations Using nVidia’s CUDA. Technical Report, California Institute of Technology (2009)Google Scholar
  14. 14.
    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, vol. 1, pp. 265–274, Universität Magdeburg, Inst. f. Simulation u. Graph (2009)Google Scholar
  15. 15.
    Klein, T., Ertl, T.: Scale-space tracking of critical points in 3D vector fields. In: Hauser, H., Hagen, H., Theisel, H. (eds.) Topology-Based Methods in Visualization, pp. 35–49. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Kindlmann, G.L., San Jose Estepar, R., Smith, S.M., Westin, C.-F.: Sampling and visualizing creases with scale-space particles. IEEE Trans. Visual. Comput. Graph. 15(6), 1415–1424 (2009)Google Scholar
  17. 17.
    Kinsner, M., Capson, D., Spence, A.: Scale-space ridge detection with GPU acceleration. In: Canadian Conference on Electrical and Computer Engineering, 2008. CCECE 2008, Niagara Falls, pp. 1527–1530 (2008)Google Scholar
  18. 18.
    Laramee, R., Hauser, H., Zhao, L., Post, F.: Topology-based flow visualization, the state of the art. In: Hauser, H., Hagen, H., Theisel, H. (eds.) Topology-Based Methods in Visualization, pp. 1–20. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Lekien, F., Ross, S.D.: The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos 20(1), 017505.1–017505.20 (2010)Google Scholar
  20. 20.
    Lindeberg, T.: Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Dordrecht (1994)Google Scholar
  21. 21.
    Lindeberg, T.: Scale-space: a framework for handling image structures at multiple scales. In: Proceedings of CERN School of Computing (1996).Google Scholar
  22. 22.
    Lindeberg T.: Edge detection and ridge detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 117–154 (1998)CrossRefGoogle Scholar
  23. 23.
    Lipinski, D., Mohseni, K.: A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. Chaos 20(1), 017504.1–017504.9 (2010)Google Scholar
  24. 24.
    Peacock, T., Dabiri, J.: Introduction to focus issue: Lagrangian coherent structures. Chaos 20(1), 017501.1–017501.3 (2010)Google Scholar
  25. 25.
    Pobitzer, A., Peikert, R., Fuchs, R., Schindler, B., Kuhn, A., Theisel, H., Matkovic, K., Hauser, H.: On the way towards topology-based visualization of unsteady flow—the state of the art. In: Hauser, H., Reinhard, E. (eds.) State of the Art Reports, Eurographics Association, pp. 137–154 (2010)Google Scholar
  26. 26.
    Sadlo, F.: Computational Visualization of Physics and Topology in Unsteady Flow. Ph.D. Dissertation No. 19284, ETH Zurich (2010)Google Scholar
  27. 27.
    Sadlo, F., Peikert, R.: Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans. Visual. Comput. Graph. 13(6), 1456–1463 (2007)CrossRefGoogle Scholar
  28. 28.
    Sadlo, F., Rigazzi, A., Peikert, R.: Time-dependent visualization of Lagrangian coherent structures by grid advection. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds.) Topological Data Analysis and Visualization: Theory, Algorithms and Applications, pp. 151–166. Springer, Heidelberg (2010)Google Scholar
  29. 29.
    Schindler, B., Peikert, R., Fuchs, R., Theisel, H.: Ridge concepts for the visualization of Lagrangian coherent structures. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds.) Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization, pp. 221-236, Springer, Heidelberg (2012)Google Scholar
  30. 30.
    Shadden, S., Lekien, F., Marsden, J.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D: Nonlinear Phenom. 212, 271–304 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Tang, W., Chan, P. W., Haller, G.: Accurate extraction of LCS over finite domains, with application to flight safety analysis over Hong Kong International Airport. Chaos 20(1), 017502.1–017502.8 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Raphael Fuchs
    • 1
  • Benjamin Schindler
    • 1
  • Ronald Peikert
    • 1
  1. 1.ETH ZurichZurichSwitzerland

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