Some theoretical results on tensor elliptical distribution

TL;DR

This paper introduces the tensor elliptical distribution as an extension of the multilinear normal distribution, providing new theoretical properties and connections useful for tensor analysis in MRI.

Contribution

It presents the tensor elliptical distribution, explores its properties, and establishes integral representations linking it to the multilinear normal distribution.

Findings

Derived the characteristic function of TE distribution.

Established the distribution of affine transformations of TE.

Provided an integral representation connecting TE and MLN densities.

Abstract

The multilinear normal distribution is a widely used tool in tensor analysis of magnetic resonance imaging (MRI). Diffusion tensor MRI provides a statistical estimate of a symmetric 2nd-order diffusion tensor, for each voxel within an imaging volume. In this article, tensor elliptical (TE) distribution is introduced as an extension to the multilinear normal (MLN) distribution. Some properties including the characteristic function and distribution of affine transformations are given. An integral representation connecting densities of TE and MLN distributions is exhibited that is used in deriving the expectation of any measurable function of a TE variate.