Optimal Control of Degenerate Parabolic Equations in Image Processing

This work is concerned with the study of an optimal control and parameter identification problem which is motivated by an interpolation task in medical image processing. The underlying model for preservation and emergence of edges involves a class of degenerate parabolic partial differential equations for which the degeneracy is controlled. Existence and uniqueness of solutions for the degenerate equations in solution spaces varying with the parameter are proven. These spaces are characterized in terms of weighted and directional Sobolev spaces, leading to existence results for associated optimization problems. First-order necessary conditions as well as a numerical realization are derived. Moreover, computations are performed, showing the appropriateness of the proposed model for image interpolation.