Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps

  • Anders Brun
  • Hans Knutsson
Part of the Mathematics and Visualization book series (MATHVISUAL)


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.


Geodesic Distance Geodesic Equation Geodesic Sphere Index Notation IEEE Visualization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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We thank Magnus Herberthson for valuable discussions on tensors and manifolds and Carl-Fredrik Westin for discussions on the application of these glyphs to Diffusion Tensor MRI data. We are also grateful for the financial support from The Manifold Valued Signal Processing project, Swedish Research Council (Vetenskapsrådet, grant 2004-4721), CMIV (, the Center for Medical Image Science and Visualization and MOVIII (, the center for Modeling, Visualization and Information Integration funded by the Swedish Foundation for Strategic Research, SSF.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Anders Brun
    • 1
    • 2
    • 3
  • Hans Knutsson
    • 1
    • 2
  1. 1.Department of Biomedical EngineeringLinköping UniversitySweden
  2. 2.Center for Medical Image Science and Visualization (CMIV)LinköpingSweden
  3. 3.Centre for Image AnalysisUppsalaSweden

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