Diffusion kurtosis imaging: Robust estimation from DW-MRI using homogeneous polynomials

Several tensor-based models have been presented in literature for parameterizing the water diffusion in Diffusion-Weighted MRI datasets, namely Diffusion Tensor Imaging (DTI), Generalized Tensor Imaging (GTI), and Diffusion Kurtosis Imaging (DKI). In this paper we use homogeneous trivariate polynomials to show that GTI is a special case of DKI for single angular shell acquisitions, and then we employ the theory for imposing positive semi-definite (PSD) constraints to GTIs in order to perform robust estimation of the DKI parameters. We propose a novel framework for DKI estimation that simultaneously imposes constraints to the diffusivity function, diffusion tensor and diffusion kurtosis. These three constraints are parameterized explicitly as a set of linear systems that can be efficiently solved using the non-negative least squares technique. The robustness of our framework is demonstrated using synthetic and real data from a human brain.