A Fokker–Planck Model for Optical Flow Estimation and Image Registration

The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form 12|y(T)−y1|2+α∫0T|u(t)|44dt, where y1 is the observed final state and y=yu is the solution to the state control system. Here, we prove the existence of a solution and obtain also the Euler–Lagrange optimality conditions which generate a gradient type algorithm for the above optimal control problem. A conceptual algorithm to compute the approximating optimal control and numerical implementation of this algorithm is discussed.