Previous Work

Much previous work in tensor visualization has started by simplifying the data to a scalar or vector field, to which established visualization techniques can be applied. That is, the tensor is viewed only in terms of some salient scalar or vector characteristic. For example, tensor field lines allow one to see the patterns in the vector fields composed of the eigenvectors of a tensor matrix [12]. In the medical community there is much interest in visualizing two-dimensional slices of MR diffusion tensor data by colormapping the direction of the principal eigenvector (the eigenvector associated with the largest eigenvalue) [23,15,26,8]. One tool for visualizing general (non-symmetric) second order tensor fields [5] proceeds by multiplying a fixed user-specified vector by the tensor field as sampled on some restricted domain (such as a plane) which acts as a probe to query specific regions of the field. Surface deformations or other vector visualization techniques are used to visualize the resultant vector field.

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 [39]. A recent advance along these lines was inspired by artists who vary the characteristics of discrete brush strokes to convey information [18]. 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.