This paper addresses large-displacement-diffeomorphic mapping registration from an optimal control perspective. This viewpoint leads to two complementary formulations. One approach requires the explicit computation of coordinate maps, whereas the other is formulated strictly in the image domain (thus making it also applicable to manifolds which require multiple coordinate charts). We discuss their intrinsic relation as well as the advantages and disadvantages of the two approaches. Further, we propose a novel formulation for unbiased image registration, which naturally extends to the case of time-series of images. We discuss numerical implementation details and carefully evaluate the properties of the alternative algorithms.