Introduction

A fundamental property of biological tissue is the ability of water molecules to move within it by the action of Brownian motion. Rather than being one fixed velocity, this movement, called diffusion, is often anisotropic - happening faster in some directions than others. We use the term anisotropy to describe how different the rates of diffusion can be. Anisotropy is high when the rate can vary greatly as a function of direction. It is low when the rate is the same, regardless of direction. A complete description of the diffusion rate's directional dependence is afforded by a second-order tensor, representable as a three-by-three real-valued symmetric matrix.

Figure 1: Matrix of images showing the individual tensor components within one dataset slice

To provide a feel for measured tensor data, a slice of a human brain diffusion tensor 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.

All three-by-three real-valued symmetric matrices have three real eigenvalues and three real-valued orthogonal eigenvectors [17]. 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 representation¹. More intuitively, if one were to put a drop of ink into a diffusive material, the ink might be drawn in some directions faster that others, and the resulting shape would approximate the ellipsoid described.

The ellipsoid provides an elegant and powerful way to visualize the tensor because it has a simple shape, and yet it has just as many degrees of freedom as the diffusion tensor. Indeed, 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, and other previous approaches to this problem, 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.

In some cases, however, it may be desirable to create renderings of tensor datasets by displaying only some of the information, but everywhere within a volume. The application motivating this research is to create 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 surgical planning, neurophysiology, and cognitive science [12,15]. 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 intricate structure of the fiber tracts is inherently a three-dimensional problem. A technique that makes the large scale patterns within the diffusion tensor field visually apparent would be 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. We propose methods for assigning color and opacity to each location within the dataset, as well as a way to illuminate the tensors in a way that can be integrated into a direct volume rendering algorithm such as raycasting. Hue-balls permit coloring of the diffusion tensor based on the linear transform of the tensor's matrix form. A user-specified unit vector is mapped by the tensor to an output vector, whose direction is then visualized by a two-dimensional colormap on the sphere. For the opacity assignment, we use a two-dimensional barycentric space of anisotropy (based on three existing anisotropy measures) as the domain of an opacity function. Finally, lit-tensors provide a way of illuminating a tensor according to the type and orientation of anisotropy which it exhibits. Lit-tensors provide a shading model for the one-parameter family of anisotropy between linear anisotropy (where the shading model coincides with illuminated streamlines), to planar anisotropy (where it coincides with traditional surface illumination).

Footnotes

... representation¹

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 $\mathrm{M}$ one could also use the set of points $\mathbf{x}$ such that $\mathbf{x}^\mathsf{T} \mathrm{M} \mathbf{x} = 1$, as is done by Strang [17].