The acquisition of 3D MRIs is adversely affected by many degrading factors including low spatial resolution and noise. Image enhancement techniques are commonplace, but there are few proposals that address the increase of the spatial resolution and noise removal at the same time. An algorithm to address this vital need is proposed in this presented work. The proposal tiles the 3D image space into parallelepipeds, so that a median filter is applied in each parallelepiped. The results obtained from several such tilings are then combined by a subsequent median computation. The convergence properties of the proposed method are formally proved. Experimental results with both synthetic and real images demonstrate our approach outperforms its competitors for images with high noise levels. Moreover, it is demonstrated that our algorithm does not generate any hallucinations.