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.

Template-like Tensor Domain Operations to Enhancing Diffusion Datasets Quality

“…2. Each of them is restricted to a cylinder of ρ = 5µ and length L = 5mm, where the particles diffuse at 1/1500 mm 2 /s as in [19]. This cylindrical model for diffusion mimics the preferential direction of water motion in the free space between cells, voxel by voxel.…”

Symmetric Positive-Definite Cartesian Tensor Orientation Distribution Functions (CT-ODF)

“…1 needs to be re-parametrized such that this positivity property is adhered to. In this work, we use the higher-order positive-definite tensor parametrization that has been recently proposed in [9]. According to this parametrization, any non-negative spherical function can be written as a positive-definite L th order homogeneous polynomial in 3 variables, which is expressed as a sum of squares of (L/2) th order homogeneous polynomials p(g 1 , g 2 , g 3 ; c), where c is a vector that contains the polynomial coefficients.…”

“…2 are non-negative weights. This parametrization approximates the space of L th order SPD tensors and the approximation accuracy depends on how well the set of vectors c j sample the space of unit vectors c. It has been shown that by constructing a large enough set of well sampled vectors c j , we can achieve any desired level of accuracy [9].…”

“…4 th and 6 th ) that generalize the 2 nd order tensors and have the ability to approximate multilobed functions [6]. Several methods have been proposed for estimating 4 th order tensors with positive definite constraints [7,8,9] as well as for processing higher order tensor fields [10]. This approach is attractive not only because the rich set of processing and analysis algorithms developed for second order tensor fields can be extended for higher order tensors but also unlike spherical harmonics basis, the local maxima of higher order tensors can be easily computed [11,12] due to their simple polynomial form.…”

Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions

“…However, existing ODF estimation methods based on a spherical harmonic (SH) representation of the ODF [1][2][3][4][5][6][7][8][9] do not enforce the non-negativity constraint. As a consequence, due to noise and low order SH representation, the estimated ODFs may contain negative values.…”