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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. To a good approximation, the diffusion rate's directional dependence can be represented with a $ 3 \times 3$ real-valued symmetric matrix. This matrix representation of the diffusion tensor can be calculated from a sequence of diffusion-weighted MRI images.

Figure 1: Matrix of images showing the individual tensor components within one dataset slice.
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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 $ 3 \times 3$ 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 representation1.

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.



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 [29].

next up previous
Next: Previous Work Up: Strategies for Direct Volume Previous: Strategies for Direct Volume
Gordon Kindlmann 2001-09-15