We consider the framework, which corresponds to the so-called "large deformation diffeomorphic metric matching" family of algorithm, in which the problem of finding an optimal registration between two shapes, or images, is formulated as an optimal control problem, where the control specifies an Eulerian velocity associated to a time-dependent diffeomorphism, with a cost represented by the norm of the velocity in a suitably chosen Hilbert space of vector fields. Because this Hilbert norm can also interpreted as the expression of a right-invariant Riemannian metric in the Lie algebra of the diffeomorphism group, this directly relates to a well-known geodesic equation, often called EPDiff,that expresses momentum conservation. In this talk, we describe an approach in which additional contraints are placed on the Eulerian velocity that ensures that it belongs to a specific finite dimensional subspace of the originally considered Hilbert space. This space, which evolves with the motion, is generated by a finite number of well chosen time-dependent vector fields that we call diffeons. We will describe the resulting maximum principle and optimization algorithms for the resulting registration problems and other related algorithms, and provide some preliminary numerical experiments in two dimensions.