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DIFFUSION IMAGING FIBER BUNDLES 33

nature of the object being scanned. Other criteria (23). The third category is the distance between two
can also be used, such as the curvature or length of Euclidean space embeddings of the curves, such as the
the curve, or the signal-to-noise ratio. Gaussian kernel distance (5). Of all these proximity
Integral curves from the first eigenvector field measures, only Hausdorff distance, Fréchet distance,
can be augmented by additional information from and the distance between two Euclidean space curve
the tensor field. For example, streamtubes (23) use embeddings are metrics that satisfy triangle inequality
cross section shape and color to map the second and and symmetry. To get rid of the bias from the order
third eigenvectors and the anisotropy values along of variables, asymmetric distance measure d (x,y) can
the integral curve. be made symmetric by averaging two distances
d' (x,y) = (d (x, y) + d (y, x))/2.
FIBER CLUSTERING
Diffusion fibers can be densely generated over Many clustering algorithms can be applied to diffu-
the entire data volume. These fibers often need to sion fiber bundling—e.g., agglomerative hierarchical
be selected based on their locations and anatomical clustering method, nearest neighbor, spectral cluster-
features for further analysis. Individually selected ing method, etc. (11). As an example, agglomerative
diffusion fibers are sensitive to small changes in seed hierarchical clustering starts from putting each fiber
point location and difficult to match across subjects. into a single element cluster and progressively merges
Individual neural axons tend to group into large the two closest clusters until there is only one clus-
and coherent bundles that have a natural boundary. ter of all fibers. If the distance between two clusters
Diffusion fiber bundles can be selected to emulate is defined as the minimum distance between any
these anatomical fiber bundles by setting a region- two fibers from the two clusters, the agglomerative
of-interest to catch all fibers passing through the hierarchical clustering algorithm is also called the
targeted region (9). Multiple regions-of-interest and single-linkage algorithm. It was suggested in pre-
Boolean logic can be used to select more complicated vious studies (14) that the single linkage algorithm
fiber bundles (18). However, this approach needs con- performs well in clustering fiber bundles in the brain
siderable expert knowledge of white matter anatomy, white matter. The tree structure built from the clus-
is prone to rater error from misidentification of tracts ter merging process of the agglomerative hierarchical
or improper decisions about whether to include ana- clustering is called a dendrogram and can be used to
tomically ambiguous fibers in a specific tract, and is select the number of clusters.
susceptible to experimenter bias. Hence, automatic
algorithms for clustering diffusion fibers into fiber TRACTOGRAPHY-BASED METRICS
bundles have been proposed. An important application of diffusion imaging
Most fiber bundling algorithms work by grouping is to assess the integrity of the white matter fiber
fibers while trying to minimize in-group distances bundles. There are two categories of methods for
and maximize between-group distances. There are the integrity assessment with diffusion imaging.
two key components: a similarity measure between Voxel-based methods calculate metrics on individ-
fibers and a clustering algorithm. There are a num- ual tensor or group of tensors in a region-of-interest.
ber of similarity measures between two curves that Metrics on individual tensor include mean diffusiv-
can be roughly grouped into several categories. The ity, fractional anisotropy, linear anisotropy, planar
first category is the Euclidean distance between two anisotropy, etc. They measure the velocity, anisotropy,
selected points on each curve, such as the closest or shape of the diffusion. Tractography-based meth-
point measure, the Hausdorff distance (17), or the ods complement and extend voxel-based methods by
Fréchet distance (1). The second category is the providing detailed information about the orienta-
mean Euclidean distance along the run lengths of tion and curvature of white matter pathways as they
the curves, such as the mean distance of closest dis- course through the brain. Tractography-based metrics
tances (8) or the mean thresholded closest distances can be designed to describe the shape and diffusion