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Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian Exponential Maps
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).ORCID iD: 0000-0002-9091-4724
2009 (English)In: Visualization and Processing of Tensor Fields: Advances and Perspectives / [ed] Laidlaw, David H.; Weickert, Joachim, Springer Berlin/Heidelberg, 2009, 139-160 p.Chapter in book (Refereed)
Abstract [en]

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 equations, 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 additions make it possible not only to visualize tensor anisotropy, but also the curvature and change in tensorshape in a local neighborhood. The framework is based on the exp 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 non-linear 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 smooth tensor fields. 

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2009. 139-160 p.
Series
Mathematics and Visualization, ISSN 1612-3786
National Category
Biomedical Laboratory Science/Technology
Identifiers
URN: urn:nbn:se:liu:diva-58090DOI: 10.1007/978-3-540-88378-4_7ISBN: 978-3-540-88377-7 (print)ISBN: 978-3-540-88378-4 (print)OAI: oai:DiVA.org:liu-58090DiVA: diva2:331960
Available from: 2010-08-19 Created: 2010-07-29 Last updated: 2013-08-28Bibliographically approved

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Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian Exponential Maps(1795 kB)1819 downloads
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Brun, AndersKnutsson, Hans

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