In volume rendering scalar data, the domain of the transfer function is nearly always the range of scalar data values. In volume rendering three-dimensional diffusion tensor data, however, it makes little sense to use the data values as the domain of the transfer function, since they live in a six-dimensional space: a symmetric matrix has six degrees of freedom. From the tensor data, we can derive a simpler quantity living in a lower dimensional space, then specify transfer functions that map from this space to color and opacity. Therefore, the derived quantity has to vary significantly between the regions of interest and the regions that would only serve to obscure or cloud a visualization.
In the context of visualizing the shape of the white matter tracts in the human brain, such a quantity is anisotropy, since the fibers have anisotropy distinct from the isotropic gray matter that surrounds them. Assigning opacity to regions with high anisotropy while assigning low or no opacity to isotropic regions helps visualize the fiber tracts and ignore the gray matter on the exterior of the brain.
The literature provides various metrics for anisotropy based on the
tensor matrix's three sorted eigenvalues
[33,37,25]. We have chosen to
use the ones by Westin et al. due to the
simple geometric motivation behind them. Metrics for three different
kinds of anisotropy are given:
In light of the normalization built into , , and , we propose the use of barycentric coordinates to depict the space of possible anisotropies, as shown in
Barycentric opacity functions use this barycentric space of anisotropy as their domain, assigning an opacity between and to each location inside the triangle (or to each entry in a two-dimensional lookup table that represents the triangle). During rendering, a given sample point's opacity is found by looking up the opacity at the location determined by the anisotropy of the diffusion tensor at that point.
Figure 4 demonstrates some barycentric opacity maps. Each opacity map is depicted by gray-scale representation: brighter regions in the triangle correspond to higher opacity assignment. For the purposes of this figure, the effect of the opacity map is demonstrated by applying the map to the the tensor dataset, resulting in a scalar volume of opacity values. This new scalar volume is visualized with a linear opacity function, and shaded according to the gradient of opacity values. One can see that, analogous to Figure 2, appropriately chosen opacity functions allow one to see the form of structures in the dataset that have one predominant type of anisotropy.
Because of its expressive power, the barycentric space also makes sense as the domain of the color function, which assigns color to each sample point in the volume rendering according to its anisotropy. Most importantly, different kinds of anisotropy receiving equal opacity can be disambiguated by assigning different colors. Also, to the extent that various classes of white-matter tissue are found to have a characteristic anisotropy throughout the volume, they can be color-coded with an appropriate barycentric color map. Volume renderings made with both barycentric opacity and color maps allow an extra dimension of information about the diffusion tensor to be represented in the volume rendering. Figure 5 shows two examples of these.