  Purpose: To facilitate the implementation/validation of signal
representations and models using parametric matrix-variate distributions to
approximate the diffusion tensor distribution (DTD) $\mathcal{P}(\mathbf{D})$.
Theory: We establish practical mathematical tools, the matrix moments of the
DTD, enabling to compute the mean diffusion tensor and covariance tensor
associated with any parametric matrix-variate DTD whose moment-generating
function is known. As a proof of concept, we apply these tools to the
non-central matrix-variate Gamma (nc-mv-Gamma) distribution, whose covariance
tensor was so far unknown, and design a new signal representation capturing
intra-voxel heterogeneity via a single nc-mv-Gamma distribution: the
matrix-variate Gamma approximation. Methods: Furthering this proof of concept,
we evaluate the matrix-variate Gamma approximation in silico and in vivo, in a
human-brain 'tensor-valued' diffusion MRI dataset. Results: The matrix-variate
Gamma approximation fails to capture the heterogeneity arising from orientation
dispersion and from simultaneous variances in the trace (size) and anisotropy
(shape) of the underlying diffusion tensors, which is explained by the
structure of the covariance tensor associated with the nc-mv-Gamma
distribution. Conclusion: The matrix moments promote a more widespread use of
matrix-variate distributions as plausible approximations of the DTD by
alleviating their intractability, thereby facilitating the design/validation of
matrix-variate microstructural techniques.
