Characterising a Population Distribution of Diffusion Tensors

Group analyses of DT-MRI data have so far focused on comparison of scalar variables derived from the tensor, such as the trace or anisotropy. An increasingly popular approach is to spatially normalise such scalar data sets and perform comparisons on a voxel-by-voxel basis. Here, we show how this approach can be extended to comparisons of the whole tensor and suggest statistical measures for characterizing a population distribution of tensors. In particular, we show how to determine the mean, median and mode of a distribution of tensors together with measures of tensor dispersion. Theory: Pennec and Ayache' have shown how the statistical concepts of mean, median and mode are relevant not only to scalar data but also to more complex geometrical data. The key idea is to redefine such measures in a more abstract form. Frkchet' defined a continuum of central locations pr, ( r E W ) , where the location p, is the element of the domain that minimizes the sum of the distances, raised to the power r, to the samples, e.g. p2 is the closest to the samples in a distancesquared sense. Griffin3 has shown that, for scalar variables, the Frtchet-defined pz is the mean, p1 is the median and (with appropriate taking of limits), represents the mode(s). To apply this approach to tensor data, a distance metric, d, is required for which we propose the measure: d (A,B) :=JM. Using this metric, it can be shown that pz coincides with the obvious definition of the mean of a set of tensors i.e. The median tensor is computed by starting with the mean tensor and gradient descending on the absolute distance function. The mode is determined by starting with median tensor and repeatedly gradient descending on the u-distance function. For the first gradient descent, Y is set to 0.9, for the second 0.8 and so on. At the completion of gradient descent with r = 0.1, the sample closest to is selected as the mode tensor. This approach also provides a means for generating dispersion