A Constrained Variational Principle for Direct Estimation and Smoothing of the Diffusion Tensor Field From Complex DWI

Abstract

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L(P) norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm.

Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections

“…This technique relies on the fact, that anatomical units occupy at least several neighboring voxels in a brain, and that it is possible to detect those regions ROI of " homogeneity". Such methods are applied to reduce the sorting bias of tensor eigenvalues [21,22], or to filter the spatial DWI fields [23,24], more global assumptions are involved in the denoising methods [25,26,27,28].…”

“…Denoising of tensor fields is described in [25,26], see also Chapter 18 by Pajevic et al, Chapter 19 and 25 by Weickert et al, and Chapter 24 by Westin et al In [25] B-splines are applied to a discrete set of noisy DT-MRI measurements to obtain a continuous representation of the tensor field, see Chaper 18 for edge preserving representations by NURBS. In such representations differential geometric quantities, like curvature or torsion of fiber tracts, but also the tangent field could be derived directly.…”

“…in regions where fiber tracts cross. In [26] the Stejskal Tanner Equations ST for complex signals are used; to achieve them formally, replace the DWIs in Eq(2) by the corresponding complex signals or complex DWIs. Assuming that sufficiently many complex DWIs are given by the measurements, a smoothed tensor field d(x) and S 0 (x) are derived by a variational principle ensuring positive definiteness of the tensor.…”

Efficient and robust computation of PDF features from diffusion MR signal

“…The iterative gradient descent is then applied until convergence (typically when ε ∈ R + , ε → 0, ∂A ∂t < ε), Similar methods have been proposed for the regularization of the DTI [62,63,64,65], of the ADC [66,67] and the ODF [43]. Yet none of these methods is able to take full advantage of the information provided by the HYDI sampling.…”

A Convex Semi-definite Positive Framework for DTI Estimation and Regularization

“…Tensors are assumed to be positive-definite matrices which was taken into account in [6] where an anisotropic filtering of the L 2 norm of the gradient of the diffusion tensor was considered and their proposed PDE scheme constrains the estimation to lie on this space. Such a concept was further developed in [7] where a variational method was proposed that aimed to minimize the L p norm of the spatial gradient of the diffusion tensor under a constraint involving the non-linear form of Stejskal-Tanner equation. A non linear diffusion scheme is described in [8] where smoothing is made directiondependent using a diffusion matrix in the PDE system.…”