In the Bayesian approach to image matching (or image registration) one assumes a prior distribution on a class of perturbations of a canonical image. Given measurements based on the true image, one then defines the "best'' approximation as the solution to a statistical estimation problem. It is sometimes more convenient to define the perturbed images through random changes of variable, and then estimate the optimal change of variable. After reviewing some early approaches to this problem, we focus on an approach in which the estimation problem can be cast in terms of an optimal control problem. Among the topics to be discussed are: (i) qualitative properties of the solution to the control problem, (ii) methods of approximation, and (iii) construction of priors for which this variational problem has an approximate maximum a posteriori interpretation.