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

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Topological Methods in Data Analysis and Visualization II

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

Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. SCI Institute, University of Utah, Salt Lake City, UT, 84112, USA

    Shreeraj Jadhav, Harsh Bhatia, Joshua A. Levine & Valerio Pascucci

  2. Lawrence Livermore National Lab, 7000 East Ave., Livermore, CA, 94550-9234, USA

    Peer-Timo Bremer

  3. Universidade de São Paulo, Sao Paulo, 05508-010, Brazil

    Luis Gustavo Nonato

Authors
  1. Shreeraj Jadhav
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  2. Harsh Bhatia
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  3. Peer-Timo Bremer
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  4. Joshua A. Levine
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  5. Luis Gustavo Nonato
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  6. Valerio Pascucci
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Corresponding author

Correspondence to Shreeraj Jadhav .

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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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Jadhav, S., Bhatia, H., Bremer, PT., Levine, J.A., Nonato, L.G., Pascucci, V. (2012). Consistent Approximation of Local Flow Behavior for 2D Vector Fields Using Edge Maps. 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_10

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