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

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@INCOLLECTION{bk08,
  author = {Anders Brun and Hans Knutsson},
  title = {Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian
	Exponential Maps},
  booktitle = {Visualization and Processing of Tensor Fields: Advances and Perspectives},
  publisher = {Springer},
  year = {2009},
  editor = {David H. Laidlaw and Joachim Weickert},
  series = {Mathematics and Visualization},
  chapter = {Part III},
  pages = {139-160},
  note = {ISBN:978-3-540-88377-7},
  abstract = {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 tensor shape in a local neighborhood.
	The framework is based on the expp(v i) and
	logp(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 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.}
}