Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm (LDDMM) can be numerically challenging when the number of particles is large. In this paper, we introduce and discuss numerical schemes useful for surface and image matching, which are based on representing the Eulerian velocity over a finite-dimensional basis that deforms in time. The method is described within the optimal control formalism, and optimality conditions are derived, together with the equations that are needed to implement gradient-based methods. Experimental results are shown, both with surfaces and images.