Dealing with Uncertainty in the Principal Directions of Tensors

Abstract

The variability of tensor fields is usually analyzed with multivariate statistical distributions. Multivariate distributions model every component of the tensor, which are not invariant under rotation. They therefore tell very little information about the true shape of the tensor. A statistical analysis on the eigenvalues of the tensor would be more revealing. The eigenvalues determine if a tensor is uniaxial, i.e.: only one eigenvalue is different from zero, isotropic, volume preserving or purely anisotropic. However, the eigenvalues of a normally distributed tensor are not, in general, normally distributed. In this paper, we solve this problem directly for small sizes of samples by determining the probability that the maximum error is within a reasonable bound. When the error is likely to be within a reasonable bound, we consider the eigenvalues of a tensor to be normally distributed along a mean eigendirection. Monte Carlo simulation shows that the computed bound is tight and becomes tighter when the number of sample increases. An application of the method, analysis of deformations on the cortical surface, is presented in this paper. On this data, we found that 80% of the anisotropic deformations could be analyzed by modeling the eigenvalues directly. Thus, the proposed method allows formulating statistical hypothesis directly on eigenvalues in many cases of measured deformations. Although the method was used in only one application, the method could be extended to application involving diffusion MRI or other imaging technique involving tensors.