next up previous
Next: Lit-Tensors Up: Methods Previous: Methods

Barycentric Mapping

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 $ 3 \times 3$ 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 $ \lambda_1 \geq \lambda_2
\geq \lambda_3$ [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:

$\displaystyle c_l$ $\displaystyle =$ $\displaystyle \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2 + \lambda_3}
\hspace{16pt}\raisebox{-15pt}{\epsfig{file=eps/c_l.eps, width=40pt}}$ (1)
$\displaystyle c_p$ $\displaystyle =$ $\displaystyle \frac{2(\lambda_2 - \lambda_3)}{\lambda_1 + \lambda_2 +
width=40pt}}$ (2)
$\displaystyle c_s$ $\displaystyle =$ $\displaystyle \frac{3\lambda_3}{\lambda_1 + \lambda_2 +
\lambda_3} \vspace{-30pt}
\hspace{15pt}\raisebox{-15pt}{\epsfig{file=eps/c_s.eps, width=40pt}}$ (3)

It can be shown that all the metrics fall in the range $ [0,1]$, and that they sum to unity: $ c_l + c_p + c_s = 1$. The ellipsoids drawn next to the anisotropy metrics indicate the shape of diffusion tensor for which that metric will be high; it will be near zero for the other two shapes. Where only $ c_l$ is high, the tensor field is said to be linearly anisotropic; where only $ c_p$ is high, the tensor field is planarly anisotropic. The last metric, $ c_s$ is actually for isotropy; $ c_s = 1$ only when all the eigenvalues are equal. Therefore, a single anisotropy metric called the ``anisotropy index'' is defined as:

$\displaystyle c_a = 1 - c_s = c_l + c_p = \frac{\lambda_1 + \lambda_2 - 2\lambda_3} {\lambda_1 + \lambda_2 + \lambda_3}$ (4)

Figure 2: Different anisotropy metrics on a slice.
\begin{figure}\centering {
\epsfig{file=eps/Figure02.eps, width=\figwidth}}

To see how the anisotropy can vary in measured data, Figure 2 shows the metrics $ c_l$, $ c_p$, and $ c_a$ evaluated over the same dataset slice seen in previous figures, with brighter areas indicating higher anisotropy.

In light of the normalization built into $ c_l$, $ c_p$, and $ c_s$, we propose the use of barycentric coordinates to depict the space of possible anisotropies, as shown in

Figure 3: Barycentric space of anisotropies.
\begin{figure}\centering {
\epsfig{file=eps/Figure03.eps, width=\figwidth}}

Figure 3. For every point in the triangle, there is a corresponding ellipsoid for which the anisotropy measures ($ c_l$, $ c_p$, and $ c_s$) evaluate to the point's barycentric coordinates. In the figure, the three ellipsoids accompanying the corners of the triangle are representative of the ellipsoids that correspond to those corners. At each vertex of the triangle, one of the anisotropy measures is one, while the two others are both zero. Along the sides of the triangle, one of the anisotropy measures is zero, and the other two measures sum to one.

Figure 4: Examples of barycentric opacity maps and resulting volumes.
\begin{figure}\centering {
\epsfig{file=eps/Figure04.eps, width=\figwidth}}

Barycentric opacity functions use this barycentric space of anisotropy as their domain, assigning an opacity between $ 0.0$ and $ 1.0$ 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.

Figure 5: Examples of barycentric color maps and resulting renderings.
\begin{figure}\centering {
\epsfig{file=eps/Figure05.eps, width=\figwidth}}

next up previous
Next: Lit-Tensors Up: Methods Previous: Methods
Gordon Kindlmann 2001-09-15