This paper proposes a novel approach to optimally solve volumetric registration problems. The proposed framework exploits parametric dictionaries for sparse volumetric representations, ℓ1 dissimilarities and DC (Difference of Convex functions) decomposition. The SAD (sum of absolute differences) criterion is applied to the sparse representation of the reference volume and a DC decomposition of this criterion with respect to the transformation parameters is derived. This permits to employ a cutting plane algorithm for determining the optimal relative transformation parameters of the query volume. It further provides a guarantee for the global optimality of the obtained solution, which-to the best of our knowledge-is not offered by any other existing approach. A numerical validation demonstrates the effectiveness and the large potential of the proposed method.