Multi-Modal Statistical Image-Matching techniques look for a deformation field that minimizes some error criterion between two images. This is achieved by computing a solution of the parabolic system obtained from the Euler-Lagrange equations of the error criterion. We prove the existence and uniqueness of a \em classical solution of this parabolic system in eight cases corresponding to the following alternatives. We consider that the images are realizations of spatial random processes that are either stationary or nonstationary. In each case we measure the similarity between the two images either by their mutual information or by their correlation ratio. In each case we regularize the deformation field either by borrowing from the field of Linear elasticity or by using the Nagel-Enkelmann tensor. Our proof uses the Hille-Yosida theorem and the theory of analytical semi-grou- ps. We then briefly describe our numerical scheme and show some experimental results.