Evaluation

Figure 18: Interpolation path between cingulum bundle and corpus callosum. Anisotropy at 100 points along this path was evaluated using the three different interpolation methods.

Figure 19: Interpolated paths between the cingulum bundle and the corpus callosum, as measured in barycentric anisotropy space.

One way to appreciate the difference between the three different interpolation methods described is to use them to interpolate along a path between two features of interest, and watch how a quantity derived from the tensor varies along the path. Because of the importance of barycentric anisotropy in our methods, we have plotted the variation in anisotropy along a path. The path we chose runs from the cingulum bundle down to the corpus callosum, as shown in Fig 18. The first point is in the cingulum bundle, an anatomic structure with linear anisotropy along the y-axis. The second point lies just seven voxels away in the corpus callosum, an anatomic structure with linear anisotropy along the x-axis. Between these two points is a region of low linear anisotropy, and somewhat higher planar anisotropy. The paths in barycentric anisotropy space shown in Figure 19 were traced by interpolating 100 uniformly spaced points in the tensor field on the line between the two points. Note that the channel and matrix interpolants follow very similar curved paths, while eigenvalue interpolation follows a straighter, simpler path.

Figure 20: Interpolation between two non-neighboring voxels.

Figure 20 demonstrates the consequences of violating the sampling density assumption of eigenvalue interpolation. It shows the paths in barycentric anisotropy space for interpolating directly between the two endpoints in Figure 18, without sampling any data in the intervening field. While there is linear anisotropy at both endpoints, the direction of the principal eigenvector varies by 90 degrees, passing through a region in the dataset where $c_l$ is near zero, so the eigenvalue correspondence induced by sorting is incorrect. The paths for channel and matrix interpolation methods do correctly veer towards low $c_l$ anisotropy (albeit at a higher value of $c_p$), while the path for eigenvalue interpolation stays strictly within the space of high linear anisotropy.

From this evaluation, we conclude that channel and matrix interpolation tend to track each other closely, and that eigenvalue interpolation can work well as long as there is sufficient sampling density. Thus, we plan to incorporate better handling of eigenvalue interpolation in future versions of our rendering system. However, for the purposes of experimenting with the different rendering strategies presented here, some of which require the underlying tensor matrix, we have found it simplest (though by no means fastest) to perform matrix interpolation. All the volume rendered figures in this paper were created this way, with any necessary eigensystem calculation being done after interpolation.