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Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian Exponential Maps
Linköpings universitet, Institutionen för medicinsk teknik, Medicinsk informatik. Linköpings universitet, Tekniska högskolan. Linköpings universitet, Centrum för medicinsk bildvetenskap och visualisering, CMIV.
Linköpings universitet, Institutionen för medicinsk teknik, Medicinsk informatik. Linköpings universitet, Tekniska högskolan. Linköpings universitet, Centrum för medicinsk bildvetenskap och visualisering, CMIV.ORCID-id: 0000-0002-9091-4724
2009 (engelsk)Inngår i: Visualization and Processing of Tensor Fields: Advances and Perspectives / [ed] Laidlaw, David H.; Weickert, Joachim, Springer Berlin/Heidelberg, 2009, s. 139-160Kapittel i bok, del av antologi (Fagfellevurdert)
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. 

sted, utgiver, år, opplag, sider
Springer Berlin/Heidelberg, 2009. s. 139-160
Serie
Mathematics and Visualization, ISSN 1612-3786
HSV kategori
Identifikatorer
URN: urn:nbn:se:liu:diva-58090DOI: 10.1007/978-3-540-88378-4_7ISBN: 978-3-540-88377-7 (tryckt)ISBN: 978-3-540-88378-4 (tryckt)OAI: oai:DiVA.org:liu-58090DiVA, id: diva2:331960
Tilgjengelig fra: 2010-08-19 Laget: 2010-07-29 Sist oppdatert: 2013-08-28bibliografisk kontrollert

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Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian Exponential Maps(1795 kB)1937 nedlastinger
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