We examine the problem of image registration when images have a sparse representation in a dictionary of geometric features. We propose a novel algorithm for aligning images by pairing their sparse components. We show numerically that this algorithm works well in practice and analyze key properties on the dictionary that drive the registration performance. We compare these properties to existing characterizations of redundant dictionaries (i.e., coherence, restricted isometry property) and show that the newly introduced properties finely capture the behaviour of our registration algorithm.