A hierarchical convex optimization approach for high fidelity solution selection in image recovery

The aim of this paper is to propose a hierarchical convex optimization for selecting a high fidelity image from possible solutions of a convex optimization problem associated with existing image recovery methods. Image recovery problems have been cast in certain convex optimization problems which have infinitely many solutions in general. However, existing convex optimization algorithms are designed to reach one solution randomly, and hence can not select a solution corresponding to a high fidelity image from the possible solutions. In this paper, we propose to select a high fidelity image by solving a newly-formulated hierarchical convex optimization problem. This problem is a constrained minimization of a convex criteria over the solution set of all images which are optimal in the sense of any existing image recovery method. The hierarchical convex optimization problem is efficiently solved by a proposed iterative scheme based on the hybrid steepest descent method with the help of a nonexpansive mapping related to the Douglas-Rachford splitting type algorithms. Numerical results indicate that our method appropriately selects a recovered image of high fidelity in the case of inpainting and compressed sensing recovery.