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Diffusion tensor magnetic resonance imaging (DT-MRI or DTI) is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data have to be acquired rapidly, often at the expense of the image quality. This often results in DTI datasets that are not suitable for complex postprocessing like fiber tracking. We propose a new variational framework to improve the estimation of DT-MRI in this clinical context. Most of the existing estimation methods rely on a log-Gaussian noise (Gaussian noise on the image logarithms), or a Gaussian noise, that do not reflect the Rician nature of the noise in MR images with a low signal-to-noise ratio (SNR). With these methods, the Rician noise induces a shrinking effect: the tensor volume is underestimated when other noise models are used for the estimation. In this paper, we propose a maximum likelihood strategy that fully exploits the assumption of a Rician noise. To further reduce the influence of the noise, we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself, which results in a maximum a posteriori estimation. Optimizing such a nonlinear criterion requires adapted tools for tensor computing. We show that Riemannian metrics for tensors, and more specifically the log-Euclidean metrics, are a good candidate and that this criterion can be efficiently optimized. Experiments on synthetic data show that our method correctly handles the shrinking effect even with very low SNR, and that the positive definiteness of tensors is always ensured. Results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord.