When the tensor visualization is not accomplished by showing only some of the information at all locations, it is often done by showing all the tensor information in a restricted subset of locations. A natural choice has been the ellipsoid representation of the tensor [25,28,16,34], though rectangular prisms (with geometry determined by the eigensystem) also work very well . A recent advance along these lines was inspired by artists who vary the characteristics of discrete brush strokes to convey information . Through a carefully designed mapping from tensor attributes to brush stroke qualities, a two-dimensional MR diffusion tensor dataset can be rendered as an image with rich information content. Furthermore, the image can be understood at a range of scales, showing both the overall shape of the anisotropic regions, as well as the degree and direction of anisotropy at one particular location.
Another method of tensor visualization by explicit representation is hyperstreamlines [10,11]. Streamlines are advected through a vector field of one of the eigenvectors, but instead of simply drawing a line to indicate the path, a surface is formed whose cross-section indicates the orientation of the other two eigenvectors and their associated eigenvalues. As with the ellipsoids, this type of representation must be unobstructed to be interpreted, so the density of hyperstreamlines in the volume must be low in order to avoid visual cluttering. Also, as with any scheme in which high-dimensional information is carefully packed into a single image, it can take some time to learn how to ``read'' these visualizations.
One could argue that density of visual information is what limits the number of hyperstreamlines that can go into a single visualization, or prevents a stack of ellipsoid-based two-dimensional visualizations from being readily composited to form a volume rendering. However, volume rendering is precisely what is needed for our application. Three-dimensional rendering of tensor fields will almost certainly require the elision of some of the tensor information; the challenge is to choose which tensor characteristics to display and how to do so.