Abstract

Computation of the mean of a collection of symmetric pos- itive definite (SPD) matrices is a fundamental ingredient of many algo- rithms in diffusion tensor image (DTI) processing. For instance, in DTI segmentation, clustering, etc. In this paper, we present novel recursive algorithms for computing the mean of a set of diffusion tensors using several distance/divergence measures commonly used in DTI segmenta- tion and clustering such as the Riemannian distance and symmetrized Kullback-Leibler divergence. To the best of our knowledge, to date, there are no recursive algorithms for computing the mean using these measures in literature. Recursive algorithms lead to a gain in computation time of several orders in magnitude over existing non-recursive algorithms. The key contributions of this paper are: (i) we present novel theoretical results on a recursive estimator for Karcher expectation in the space of SPD matrices, which in effect is a proof of the law of large numbers (with some restrictions) for the manifold of SPD matrices. (ii) We also present a recursive version of the symmetrized KL-divergence for computing the mean of a collection of SPD matrices. (iii) We present comparative tim- ing results for computing the mean of a group of SPD matrices (diffusion tensors) depicting the gains in compute time using the proposed recur- sive algorithms over existing non-recursive counter parts. Finally, we also show results on gains in compute times obtained by applying these re- cursive algorithms to the task of DTI segmentation.