To provide a feel for measured diffusion tensor data, a slice of a human brain dataset is portrayed in Figure 1. Each sub-image in the matrix of images is a gray-scale representation of the corresponding component of the tensor matrix, with medium gray representing zero. In the brain interior, the on-diagonal components of the tensor matrix are positive, while the off-diagonal components can be either positive or negative. This method of portraying the raw tensor data is not novel, nor is it a very intuitive way to display the orientation and shape of the diffusion tensors (in the same way that looking at the individual components of a vector field gives a poor sense of the field's structure).

All
real-valued symmetric matrices have three real
eigenvalues and three real-valued orthogonal
eigenvectors [29]. The diffusion tensor matrix enjoys
the additional constraint of having non-negative eigenvalues, implying
it can be unambiguously represented as an ellipsoid. The ellipsoid's
major, medium, and minor axes are along the tensor's eigenvectors,
with the scalings along the axes being the eigenvalues. Such an
ellipsoid is the image of the unit sphere under the linear transform
induced by the tensor's matrix representation^{1}.

The ellipsoid provides a concise and elegant way to visualize the
tensor because it has a simple shape, and it has just as many
degrees of freedom as the diffusion tensor. As such, we will use the
ellipsoid representation for demonstration purposes in figures. Also,
previous work in diffusion tensor visualization has used arrays of
ellipsoids to depict the tensor field within a two dimensional region.
Another tensor visualization method, hyperstreamlines, succeeds in
faithfully depicting the tensor along one-dimensional paths in a
volume dataset. These methods are useful because they produce a means
of visually decoding *all* the tensor's degrees of freedom at some
set of locations in the field.

We believe that in this context, however, the most informative
visualizations do not necessarily come from striving to pack many
dimensions of information into one image. Rather, it may be desirable
to create renderings of tensor datasets by displaying only *some*
of the information, but *everywhere* within a volume. The goal of
this research is creating an understanding of the fibrous structure of
white matter throughout the brain. Because the white matter fiber
tracts connect major regions of the brain, a detailed understanding of
their structure could foster advances in neuroanatomy, in surgical
planning, and cognitive
science [20,7,19]. Fortunately,
developments in magnetic resonance imaging have made it possible to
accurately measure the water diffusion tensor within living brain
tissue [2]. The white matter fiber tracts can be
distinguished from their surroundings based on properties of the
measured diffusion tensor, such as its anisotropy. Visualizing the
fiber tracts is inherently a three-dimensional problem because of
their curving, intricate structure. A technique that allows us to
visualize the large scale patterns across the entire dataset is
ideal.

Since this has historically been the goal of direct volume rendering
for *scalar* data, we have explored the use of direct volume
rendering for diffusion tensor visualization. To make this possible,
the various ingredients of the direct volume rendering algorithm need
to be supplied from the tensor data. The barycentric opacity map,
lit-tensor, and hue-ball techniques introduced in [17] are
specific approaches to performing three tasks fundamental to volume
rendering: determining opacity, calculating shading and assigning
material color.

In the current paper, we generalize our previous techniques to offer
more choices in how to accomplish the basic tasks of volume rendering.
While barycentric opacity maps can control which types of anisotropy
appear in the final image, *barycentric color maps* can add
information about how anisotropy varies across structures. Hue-balls
can take the role of assigning color, but the underlying principle of
deflection can also be used to assign opacity with a *deflection
opacity map*. Lit-tensors are one method of shading, but a more
simplistic method based on the *gradient of opacity* is also
described, as well as mixtures of the two shading approaches.
For simplicity, we have taken a ray casting approach
to volume rendering tensor fields. After interpolating the tensor
information at each sample point along a ray cast through the volume,
we apply any of the various methods described in this paper to
determinine opacity, color, and shading. We then composite
the results to determine the pixel color.

In addition, we describe a new method for visualizing diffusion tensor fields. By simulating a reaction-diffusion process between two interacting chemicals, on a domain effectively warped by the underlying tensor data, we are able to produce a volumetric solid texture that follows the structure of the diffusion tensor field. The texture is composed of a large number of pseudo-ellipsoidal ``spots'', each of which reflect the magnitude and orientation of the eigenvectors in a local neighborhood. This texture can then be mapped onto the surfaces generated by a diffusion tensor volume rendering, or it can be inspected as a stand-alone visualization.

We finish with a brief discussion of an issue highly relevant to the task of volume rendering: interpolation. With scalar data, the matter of interpolation usually becomes a choice among the wide variety of available reconstruction kernels- trilinear, tricubic, windowed sinc, etc. In all cases it is obvious that the original sampled data values are the quantity to be interpolated. With diffusion tensor data, there is still the same choice of reconstruction kernel, but there is the independent question of which tensor-related quantity should be interpolated. One can interpolate the raw diffusion-weighted images acquired by the MRI scanner, the individual components of the tensor matrix calculated from them, or a quantity derived from the diffusion tensor, such as an eigenvector. We discuss and analyze three distinct interpolation schemes based on these different options.

- ... representation
^{1} - This is not the only unambiguous ellipsoidal representation. One could also represent the tensor with the unit sphere's pre-image, or, for a tensor one could also use the set of points such that , as is done by Strang [29].