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Scale-Space Approaches to FTLE Ridges

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Topological Methods in Data Analysis and Visualization II

Part of the book series: Mathematics and Visualization ((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.

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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.

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Authors and Affiliations

  1. ETH Zurich, 8092, Zurich, Switzerland

    Raphael Fuchs, Benjamin Schindler & Ronald Peikert

Authors
  1. Raphael Fuchs
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  2. Benjamin Schindler
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  3. Ronald Peikert
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Corresponding author

Correspondence to Raphael Fuchs .

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Editors and Affiliations

  1. Inst. Computational Science, CAB G 65.1, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Ronald Peikert

  2. , Dept. of Informatics, University of Bergen, Bergen, Norway

    Helwig Hauser

  3. , School of Computing, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom

    Hamish Carr

  4. , Computational Science, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Raphael Fuchs

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Fuchs, R., Schindler, B., Peikert, R. (2012). Scale-Space Approaches to FTLE Ridges. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_19

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