This paper proposes a non-rigid, symmetric and diffeomorphic algorithm for the registration of MR images. A log-Euclidean framework was used to model diffeomorphisms. Multiple features were used to enhance the speed of the proposed algorithm, including compactly supported Wendland’s basis functions as models of velocity fields, a greedy optimization strategy, and a coarse-to-fine hierarchical framework. Symmetry was ensured through the invertibility of diffeomorphisms and the symmetry of the objective function and optimization algorithm. The proposed algorithm was evaluated using the results of 1560 pairwise registrations of 40 T1-weighted MRIs in the LPBA40 dataset. Our results demonstrated that the proposed algorithm is diffeomorphic with sub-voxel accuracy in terms of symmetry. The proposed algorithm outperforms 14 other registration algorithms in terms of speed (compared with diffeomorphic algorithms) and accuracy.