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We present a new, direct way to register three-dimensional (3D) surfaces given the respective 3D points and surface triangulations. Our method is non-iterative and does not require any initial solution. The idea is to compute 3D invariants based on local surface moments. The resulting local surface descriptors are invariant with respect to Euclidean or to similarity transformations, by choice. In the final step we use the Hungarian method to find a minimum cost assignment of the computed descriptors. The method is robust against different point densities, noise and partial overlap. Our experiments with real data also show that the method can serve as automatic initialization of the iterative-closest-point (ICP) algorithm and, hence, extends the field of applications for this standard registration method.