Calculations for D-eigenvalues of a diffusion kurtosis tensor

Abstract

A diffusion kurtosis (abbr. DK) tensor plays a key role in the diffusion kurtosis imaging from medical engineering. Specifically, the largest, the smallest and the average D-eigenvalues of a DK tensor correspond with the largest, the smallest and the average apparent kurtosis coefficients of a water molecule in biological tissues, respectively.In order to calculate all D-eigenvalues of a DK tensor W, we first convert the D-eigenvalues of W into the Z-eigenvalues of a semi-symmetric tensor A via an invertible linear transformation. After that, we adopt this method for calculating all D-eigenvalues: we first calculate all Z-eigenvalues of A by using the existing methods, and then calculate all D-eigenvalues of W via this linear transformation. Subsequently, we show the calculation routine of D-eigenvalues in two algorithms. Finally, we report an example whose data come from MRI experiments in medical engineering to show the feasibility and effectiveness of this two algorithms in practice.