The measurement of the distance between diffusion tensors is the foundation on which any subsequent analysis or processing of these quantities, such as registration, regularization, interpolation, or statistical inference is based. In recent years a family of Riemannian tensor metrics based on geometric considerations has been introduced for this purpose. In this work we examine the properties one would use to select metrics for diffusion tensors, diffusion coefficients, and diffusion weighted MR image data. We show that empirical evidence supports the use of a Euclidean metric for diffusion tensors, based upon Monte Carlo simulations. Our findings suggest that affine invariance is not a desirable property for a diffusion tensor metric because it leads to substantial biases in tensor data. Rather, the relationship between distribution and distance is suggested as a novel criterion for metric selection.
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