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

Xavier Tricoche, Mario Hlawitschka, Samer Barakat, Christoph Garth
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="" style="color: black;">Mathematics and Visualization</a> </i> &nbsp;
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
more &raquo; ... 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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1007/978-3-642-27343-8_5</a> <a target="_blank" rel="external noopener" href="">fatcat:pfqgqkeyfrgkfpsddeai66oazi</a> </span>
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