A UNIFIED FRAMEWORK FOR ESTIMATING DIFFUSION TENSORS OF ANY ORDER WITH SYMMETRIC POSITIVE-DEFINITE CONSTRAINTS

Abstract

Cartesian tensors of various orders have been employed for either modeling the diffusivity or the orientation distribution function in Diffusion-Weighted MRI datasets. In both cases, the estimated tensors have to be positive-definite since they model positive-valued functions. In this paper we present a novel unified framework for estimating positive-definite tensors of any order, in contrast to the existing methods in literature, which are either order-specific or fail to handle the positive-definite property. The proposed framework employs a homogeneous polynomial parametrization that covers the full space of any order positive-definite tensors and explicitly imposes the positive-definite constraint on the estimated tensors. We show that this parametrization leads to a linear system that is solved using the non-negative least squares technique. The framework is demonstrated using synthetic and real data from an excised rat hippocampus.

Fig. 1

Comparison of the proposed method with the technique in [2] for various levels of Riccian noise in the data.

Fig. 2

DW-MRI dataset from an isolated rat hippocampus. The S₀ image is shown on the top left. The 6^th-order diffusion tensors estimated by the proposed method are shown as a field of spherical functions. One of the depicted regions of interest contains tensors that model 3-fiber structures, which can not be reconstructed by 2^nd or 4^th-order tensors.