This paper describes the theoretic roadmap of least relative entropy matching of two point sets. The novel feature is to align two point sets without needing to establish explicit point correspondences. The recovery of transformational geometry is achieved using a mixture of principal axes registrations, whose parameters are estimated by minimizing the relative entropy between the two point distributions and using the expectation-maximization algorithm. We give evidence of the optimality of the method and we then evaluate the algorithm's performance in both rigid and nonrigid image registration cases.