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Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential MapsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)In: Visualization and Processing of Tensor Fields: Advances and Perspectives / [ed] David Laidlaw, Joachim Weickert, Berlin Heidelberg: Springer , 2009, XVII, 139-160 p.Chapter in book (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Berlin Heidelberg: Springer , 2009, XVII. 139-160 p.
##### Series

, Mathematics and Visualization, ISSN 1612-3786 ; 3
##### National Category

Computer Vision and Robotics (Autonomous Systems)
##### Research subject

Computerized Image Analysis
##### Identifiers

URN: urn:nbn:se:uu:diva-111488DOI: 10.1007/978-3-540-88378-4_7ISBN: 978-3-540-88377-7OAI: oai:DiVA.org:uu-111488DiVA: diva2:281358
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Available from: 2009-12-15 Created: 2009-12-15 Last updated: 2016-04-22Bibliographically approved

The Riemannian exponential map, and its inverse the Riemannian logarithm map, can be used to visualize metric tensor fields. In this chapter we first derive the well-known metric sphere glyph from the geodesic equation, where the tensor field to be visualized is regarded as the metric of a manifold. These glyphs capture the appearance of the tensors relative to the coordinate system of the human observer. We then introduce two new concepts for metric tensor field visualization: geodesic spheres and geodesically warped glyphs. These extensions make it possible not only to visualize tensor anisotropy, but also the curvature and change in tensor-shape in a local neighborhood. The framework is based on the exp p (v i ) and log p (q) maps, which can be computed by solving a second-order ordinary differential equation (ODE) or by manipulating the geodesic distance function. The latter can be found by solving the eikonal equation, a nonlinear partial differential equation (PDE), or it can be derived analytically for some manifolds. To avoid heavy calculations, we also include first- and second-order Taylor approximations to exp and log. In our experiments, these are shown to be sufficiently accurate to produce glyphs that visually characterize anisotropy, curvature, and shape-derivatives in sufficiently smooth tensor fields where most glyphs are relatively similar in size.

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