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Mapping white matter tracts is an essential step towards understanding brainfunction. Diffusion Magnetic Resonance Imaging (dMRI) is the only noninvasivetechnique which can detect in vivo anisotropies in the 3-dimensional diffusionof water molecules, which correspond to nervous fibers in the living brain. Inthis process, spectral data from the displacement distribution of watermolecules is collected by a magnetic resonance scanner. From the statisticalpoint of view, inverting the Fourier transform from such sparse and noisyspectral measurements leads to a non-linear regression problem. Diffusiontensor imaging (DTI) is the simplest modeling approach postulating a Gaussiandisplacement distribution at each volume element (voxel). Typically theinference is based on a linearized log-normal regression model that can fit thespectral data at low frequencies. However such approximation fails to fit thehigh frequency measurements which contain information about the details of thedisplacement distribution but have a low signal to noise ratio. In this paper,we directly work with the Rice noise model and cover the full range of$b$-values. Using data augmentation to represent the likelihood, we reduce thenon-linear regression problem to the framework of generalized linear models.Then we construct a Bayesian hierarchical model in order to performsimultaneously estimation and regularization of the tensor field. Finally theBayesian paradigm is implemented by using Markov chain Monte Carlo.

Data augmentation in Rician noise model and Bayesian Diffusion Tensor Imaging