Theoretical analysis of the effects of noise on diffusion tensor imaging

A theoretical framework is presented for understanding the effects of noise on estimates of the eigenvalues and eigenvectors of the diffusion tensor at moderate to high signal-to-noise ratios. Image noise produces a random perturbation of the diffusion tensor. Power series solutions to the eigenvalue equation are used to evaluate the effects of the perturbation to second order. It is shown that in anisotropic systems the expectation value of the largest eigenvalue is overestimated and the lowest eigenvalue is underestimated. Hence, diffusion anisotropy is overestimated in general. This result is independent of eigenvalue sorting bias. Furthermore, averaging eigenvalues over a region of interest produces greater bias than averaging tensors prior to diagonalization. Finally, eigenvector noise is shown to depend on the eigenvalue contrast and imposes a theoretical limit on the accuracy of simple fiber tracking schemes. The theoretical results are shown to agree with Monte Carlo simulations.