A new filtering method to remove Rician noise from magnetic resonance images is presented. This filter relies on a robust estimation of the standard deviation of the noise and combines local linear minimum mean square error filters and partial differential equations for MRI, as the speckle reducing anisotropic diffusion did for ultrasound images. The parameters of the filter are automatically chosen from the estimated noise. This property improves the convergence rate of the diffusion while preserving contours, leading to more robust and intuitive filtering. The partial derivative equation of the filter is extended to a new matrix diffusion filter which allows a coherent diffusion based on the local structure of the image and on the corresponding oriented local standard deviations. This new filter combines volumetric, planar, and linear components of the local image structure. The numerical scheme is explained and visual and quantitative results on simulated and real data sets are presented. In the experiments, the new filter leads to the best results.