A normal distribution for tensor-valued random variables to analyze diffusion tensor MRI data

Diffusion Tensor MRI (DT-MRI) provides a statistical estimate of a symmetric 2nd-order diffusion tensor, D, for each voxel within an imaging volume. We propose a new normal distribution, p(D) ∼ exp(- 1/2 D:A:D), for a tensor random variable, D. The scalar invariant, D:A:D, is the contraction of a positive definite symmetric 4th-order precision tensor, A, and D. A formal correspondence is established between D:A:D and the elastic strain energy density function in continuum mechanics. We show that A can then be classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic tensor, an explicit expression for p(D), and for the distribution of its three eigenvalues, p(γ₁,γ₂,γ₃), are derived, which are confirmed by Monte Carlo simulations. Sample estimates of A are also obtained using synthetic DT-MRI data. Estimates of p(D) should be useful in feature extraction and in classification of noisy, discrete tensor data.