Image restoration and simultaneous edge detection by optimal control methods

The present paper is concerned with the numerical solution of multidimensional control problems of Dieudonné-Rashevsky type by discretization methods and large-scale optimization techniques. We prove first a convergence theorem wherein the difference of the minimal value and the objective values along a minimizing sequence is estimated by the mesh size of the underlying triangulations. Then we apply the proposed method to the problem of the simultaneous restoration of noisy image data and edge detection therein. Instead of using an Ambrosio-Tortorelli type energy functional, we reformulate the image restoration problem as a multidimensional control problem. The edge detector can be immediately built from the control variables. The quality of our numerical results competes well with the results given by variational techniques.