ABSTRACT: In many disciplines of computer vision, such as stereo vision, flow computation, medical image registration, the essential computational problem is the geometrical alignment of images. In this chapter we describe how such an alignment may be obtained as statistical optimal through solving a partial differential equation (PDE) in the matching function. We treat different choices of matching criteria such as minimal square difference, maximal correlation, maximal mutual information, and several smoothness criteria. All are treated from a Bayes point of view leading to a functional minimization problem solved through an Euler-Lagrange formulation as the solution to a PDE. We try in this chapter to collect the most used methodologies and draw conclusions on their properties and similarities.