Diffusion Tensor Representations and Their Applications to DTI Error Propagation

INTRODUCTION The diffusion tensor is a 3x3 positive definite matrix and, therefore, possesses several distinct matrix decompositions, e.g. the Cholesky, and the Eigenvalue decompositions. To date, the Eigenvalue decomposition has been used only in computing tensor-derived quantities [1-4] but not as a parametrization (or equivalently, a representation) in DTI error propagation [5]. Treating a matrix decomposition of the diffusion tensor as a representation is a very useful strategy not only in tensor estimation, constrained or otherwise, [1-4,6-8] but also, as we will show, in DTI error propagation. Specifically, we propose to use the Eigenvalue decomposition as a representation in DTI error propagation and show how the proposed technique can address important questions related to the uncertainty of the eigenvalues and, more importantly, of the eigenvectors. For completeness, we introduce three representations to DTI error propagation — the ordinary, the Cholesky, and the Euler representations. This work provides a unified framework in which the uncertainty of any tensor-derived quantity such as eigenvalues, eigenvectors, trace, fractional anisotropy (FA), and relative anisotropy (RA) can be analytically derived and estimated. Although the diffusion representations are essentially equivalent when dealing with the tensor estimate itself, the variances of interests derived from them may be different. In other words, one representation may be more accurate than the other in variance estimation. Furthermore, one representation may be more convenient and analytically tractable than the other. Theoretical analysis and simulations are carried out to investigate this issue.