Data Augmentation under Rician Noise Model in Diffusion MRI with Applications to Human Brain Studies

Abstract

Diffusion magnetic resonance imaging (diffusion MRI) is capable of measuring the displacement diffusion of water molecules and of providing a unique insight by means of image contrasts from measurements to probe non-invasively the microscopic anatomical architectures of organic tissues in vivo. Many diffusion imaging approaches have been developed to measure the underlying diffusion function, among which diffusion tensor imaging (DTI) is the most popular. A conventional modeling approach postulates a Gaussian displacement distribution at each voxel characterized by means of a second order symmetric and positive definite diffusion tensor. Typically, the inference is based on a linearized log-normal regression model. However, such an approximation fails to fit the high frequency and/or the low signal to ratio (SNR) measurements containing important information on water diffusion. The diffusion weighted MR measurements are sparse and noisy, and after the Fourier inversion they yield a non-linear regression problem. However, working with the non-linear model for the data directly leads to heavy computation. In this thesis, I present a series of novel statistical methodologies to solve the computational problem. By using data augmentation, the non-linear regression problem under the Rician noise model is reduced to the generalized linear modeling (GLM) framework. For different purposes, we use both Bayesian and frequentist statistical inferences: A Bayesian hierarchical model is established to estimate the marginal posterior distribution of every parameter of interest, where we apply the Markov chain Monte Carlo (McMC) method, exploring the state space to compute averages under the joint posterior distribution of the unknown parameters and latent variables. Moreover, we also implement Variational Bayes (VB) algorithms as a faster scheme for converging to the optimum of each posterior distribution. Under the Bayesian framework, a regularization technique is developed for modeling the contextual dependence (interaction) between the tensors. This is done by constructing an isotropic prior for the tensor fields through the Gaussian Markov random fields (GMRF). This model is intended to smooth and denoise the image. In terms of computational issues in practice, we further employ the expectation-maximization (EM) algorithm under the joint likelihood in GLM by the data augmentation to reduce computational burden in both Bayesian and frequentist frameworks. This deterministic algorithm is implemented under the assumption of voxel independence in both maximum a posterior (MAP) estimation and maximum likelihood estimation (MLE). Furthermore, we apply the stabilized Fisher scoring method for achieving fast convergence in the calculation of the tensor parameter. In addition, we address the essential difference between these two inferences working in dMRI. All these methodologies are described in four papers under several popular signal decay models in dMRI, implemented and experimented both with synthetic and real data of the human brain, and compared with different popular methods in dMRI and in the recent literature.