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

  • Shreeraj Jadhav
  • Harsh Bhatia
  • Peer-Timo Bremer
  • Joshua A. Levine
  • Luis Gustavo Nonato
  • Valerio Pascucci
Chapter
Part of the Mathematics and Visualization book series (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.

Keywords

Directed Edge Consistent Approximation Destination Point Undirected Edge Mixed Graph 
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

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Shreeraj Jadhav
    • 1
  • Harsh Bhatia
    • 1
  • Peer-Timo Bremer
    • 2
  • Joshua A. Levine
    • 1
  • Luis Gustavo Nonato
    • 3
  • Valerio Pascucci
    • 1
  1. 1.SCI InstituteUniversity of UtahSalt Lake CityUSA
  2. 2.Lawrence Livermore National LabLivermoreUSA
  3. 3.Universidade de São PauloSao PauloBrazil

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