Diffusion tensor field restoration and segmentation

Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a relative new MRI modality that can be used to study tissue microstructure, e.g., white matter connectivity in brain in vivo. It has attracted vast attention during the past few years. In this dissertation, novel solutions to two important problems in DT-MRI analysis: diffusion tensor field (aka diffusion tensor image, DTI) restoration and segmentation are presented. For DTI restoration, we first develop a model for estimating the accuracy of a nonlinear estimator used in estimating the apparent diffusivity coefficient (ADC). We also use this model to design optimal diffusion weighting factors by accounting for the fact that the ground truth of the ADC is a distribution instead of a single value. The proposed method may be extended to study the statistical properties of DTI estimators and to design corresponding optimal acquisition parameters. We then present a novel constrained variational principle for simultaneous smoothing and estimation of the DTI from complex valued diffusion weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of Lp smoothness 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 DTI. 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. For DTI segmentation, we present a novel definition of tensor “distance” grounded in concepts from information theory and incorporate it in the segmentation of DTI. Diffusion tensor is a symmetric positive definite (SPD) tensor at each point of a DTI and can be interpreted as the covariance matrix of a local Gaussian distribution. Thus, a natural measure of dissimilarity between diffusion tensors would be the Kullback-Leibler (KL) divergence or its relative. In particular, we define a tensor “distance” as the square root of the J-divergence (symmetrized KL) between two Gaussian distributions corresponding to the tensors being compared. Our definition leads to novel closed form expressions for the “distance” itself and the mean value of a DTI. We then incorporate this new tensor “distance” in a region based active contour model for DTI segmentation and the closed expressions we derived greatly help the computation.