Beyond Topology: A Lagrangian Metaphor to Visualize the Structure of 3D Tensor Fields

  • Xavier Tricoche
  • Mario Hlawitschka
  • Samer Barakat
  • Christoph Garth
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Topology was introduced in the visualization literature some 15 years ago as a mathematical language to describe and capture the salient structures of symmetric second-order tensor fields. Yet, despite significant theoretical and algorithmic advances, this approach has failed to gain wide acceptance in visualization practice over the last decade. In fact, the very idea of a versatile visualization methodology for tensor fields that could transcend application domains has been virtually abandoned in favor of problem-specific feature definitions and visual representations. We propose to revisit the basic idea underlying topology from a different perspective. To do so, we introduce a Lagrangian metaphor that transposes to the structural analysis of eigenvector fields a perspective that is commonly used in the study of fluid flows. Indeed, one can view eigenvector fields as the local superimposition of two vector fields, from which a bidirectional flow field can be defined. This allows us to analyze the structure of a tensor field through the behavior of fictitious particles advected by this flow. Specifically, we show that the separatrices of 3D tensor field topology can in fact be captured in a fuzzy and numerically more robust setting as ridges of a trajectory coherence measure. As a result, we propose an alternative structure characterization strategy for the visual analysis of practical 3D tensor fields, which we demonstrate on several synthetic and computational datasets.


Vector Field Fractional Anisotropy Diffusion Tensor Image Tensor Field Lagrangian Coherent Structure 
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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Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  • Xavier Tricoche
    • 1
  • Mario Hlawitschka
    • 2
  • Samer Barakat
    • 1
  • Christoph Garth
    • 2
  1. 1.Purdue UniversityLafayetteUSA
  2. 2.University of California at DavisDavisUSA

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