Modeling diffusion‐weighted MRI as a spatially variant Gaussian mixture: Application to image denoising

Purpose: This work describes a spatially variant mixture model constrained by a Markov random field to model high angular resolution diffusion imaging (HARDI) data. Mixture models suit HARDI well because the attenuation by diffusion is inherently a mixture. The goal is to create a general model that can be used in different applications. This study focuses on image denoising and segmentation (primarily the former). Methods: HARDI signal attenuation data are used to train a Gaussian mixture model in which the mean vectors and covariance matrices are assumed to be independent of spatial locations, whereas the mixture weights are allowed to vary at different lattice positions. Spatial smoothness of the data is ensured by imposing a Markov random field prior on the mixture weights. The model is trained in an unsupervised fashion using the expectation maximization algorithm. The number of mixture components is determined using the minimum message length criterion from information theory. Once the model has been trained, it can be fitted to a noisy diffusion MRI volume by maximizing the posterior probability of the underlying noiseless data in a Bayesian framework, recovering a denoised version of the image. Moreover, the fitted probability maps of the mixture components can be used as features for posterior image segmentation. Results: The model‐based denoising algorithm proposed here was compared on real data with three other approaches that are commonly used in the literature: Gaussian filtering, anisotropic diffusion, and Rician‐adapted nonlocal means. The comparison shows that, at low signal‐to‐noise ratio, when these methods falter, our algorithm considerably outperforms them. When tractography is performed on the model‐fitted data rather than on the noisy measurements, the quality of the output improves substantially. Finally, ventricle and caudate nucleus segmentation experiments also show the potential usefulness of the mixture probability maps for classification tasks. Conclusions: The presented spatially variant mixture model for diffusion MRI provides excellent denoising results at low signal‐to‐noise ratios. This makes it possible to restore data acquired with a fast (i.e., noisy) pulse sequence to acceptable noise levels. This is the case in diffusion MRI, where a large number of diffusion‐weighted volumes have to be acquired under clinical time constraints.