Well-posedness of two nonrigid multimodal image registration methods

Image registration methods may be designed as solutions of minimization problems on a set of geometric deformations. In the nonrigid case, solving these problems often means computing the steady state of a system of evolution equations involving the "gradient" of the error criterion, defined from the Euler-Lagrange equations of the corresponding minimization problem. The well-posedness of the registration method requires showing the existence of a solution to the minimization problem, as well as that of a stable solution of the evolution equations derived from it. We provide such proofs in the case where the error criterion is derived from two different statistical similarity measures: global mutual information and local cross-covariance. We also describe our numerical implementation for solving the corresponding evolution equations and show examples of registrations of real 2D and 3D images achieved with these algorithms. The proofs are quite general and can be applied to most of the known nonrigid image registration methods.