This paper presents an efficient algorithm for large deformation diffeomorphic metric mapping (LDDMM) with geodesic shooting for image registration. We introduce a novel finite dimensional Fourier representation of diffeomorphic deformations based on the key fact that the high frequency components of a diffeomorphism remain stationary throughout the integration process when computing the deformation associated with smooth velocity fields. We show that manipulating high dimensional diffeomorphisms can be carried out entirely in the bandlimited space by integrating the nonstationary low frequency components of the displacement field. This insight substantially reduces the computational cost of the registration problem. Experimental results show that our method is significantly faster than the state-of-the-art diffeomorphic image registration methods while producing equally accurate alignment. We demonstrate our algorithm in two different applications of image registration: neuroimaging and in-utero imaging.