Diffusion tensor magnetic resonance imaging (DT-MRI) is a non-invasive imaging technique allowing to estimate the molecular self-diffusion tensors of water within surrounding tissue. Due to the low signal-to-noise ratio of magnetic resonance images, reconstructed tensor images usually require some sort of regularization in a post-processing step. Previous approaches are either suboptimal with respect to the reconstruction or regularization step. This paper presents a Bayesian approach for simultaneous reconstruction and regularization of DT-MR images that allows to resolve the disadvantages of previous approaches. To this end, estimation theoretical concepts are generalized to tensor valued images that are considered as Riemannian manifolds. Doing so allows us to derive a maximum a posteriori estimator of the tensor image that considers both the statistical characteristics of the Rician noise occurring in MR images as well as the nonlinear structure of tensor valued images. Experiments on synthetic data as well as real DT-MRI data validate the advantage of considering both statistical as well as geometrical characteristics of DT-MRI.
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