Efficient Recursive Algorithms for Computing the Mean Diffusion Tensor and Applications to DTI Segmentation★

Computation of the mean of a collection of symmetric positive definite (SPD) matrices is a fundamental ingredient of many algorithms 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 segmentation 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 timing results for computing the mean of a group of SPD matrices (diffusion tensors) depicting the gains in compute time using the proposed recursive algorithms over existing non-recursive counter parts. Finally, we also show results on gains in compute times obtained by applying these recursive algorithms to the task of DTI segmentation.