The extraction of a single high-quality image from a set of low-resolution images is an important problem which arises in fields such as remote sensing, surveillance, medical imaging and the extraction of still images from video. Typical approaches are based on the use of cross-correlation to register the images followed by the inversion of the transformation from the unknown high resolution image to the observed low resolution images, using regularization to resolve the ill-posed nature of the inversion process. In this paper we develop a Bayesian treatment of the super-resolution problem in which the likelihood function for the image registration parameters is based on a marginalization over the unknown high-resolution image. This approach allows us to determine the unknown point spread function, and is rendered tractable through the introduction of a Gaussian process prior over images. Results indicate a significant improvement over techniques based on MAP (maximum a-posteriori) point optimization of the high resolution image.