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Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Centrum för bildanalys. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Datoriserad bildanalys. (Centre for imaga analysis)ORCID-id: 0000-0002-4405-6888
Linköpings Universitet. (Medical Informatics, Department of Biomedical Engineering)
2009 (engelsk)Inngår i: Visualization and Processing of Tensor Fields: Advances and Perspectives / [ed] David Laidlaw, Joachim Weickert, Berlin Heidelberg: Springer , 2009, XVII, s. 139-160Kapittel i bok, del av antologi (Annet vitenskapelig)
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 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.

sted, utgiver, år, opplag, sider
Berlin Heidelberg: Springer , 2009, XVII. s. 139-160
Serie
Mathematics and Visualization, ISSN 1612-3786 ; 3
HSV kategori
Forskningsprogram
Datoriserad bildanalys
Identifikatorer
URN: urn:nbn:se:uu:diva-111488DOI: 10.1007/978-3-540-88378-4_7ISBN: 978-3-540-88377-7 (tryckt)OAI: oai:DiVA.org:uu-111488DiVA, id: diva2:281358
Tilgjengelig fra: 2009-12-15 Laget: 2009-12-15 Sist oppdatert: 2018-01-12bibliografisk kontrollert

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