Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps
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)
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.
Place, publisher, year, edition, pages
Berlin Heidelberg: Springer , 2009, XVII. 139-160 p.
, Mathematics and Visualization, ISSN 1612-3786 ; 3
Computer Vision and Robotics (Autonomous Systems)
Research subject Computerized Image Analysis
IdentifiersURN: 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