Conceptually, an image recovery problem can be reduced to a constrained minimization problem. In practice, however, the efficient implementation of standard optimization algorithms often encounters serious difficulties due to the complex nature of recovery problems, which not only involve a sizable amount of data and unknowns, but also a wide variety of constraints. The goal of this research is to develop, analyze, implement, and test a eeneral convex minimization algorithm that addresses the specific numerical difficulties posed by image recovery. The basic principle is to decompose the original problem of minimizing over a complex feasibility set into a sequence of simpler minimizations over the intersection of two larger half-spaces. The bundle of constraints is disintegrated into elementary components and, at every iteration, the half-spaces are constructed by activating in parallel a block of approximated (linearized) constraints. A wide range of constraints can thus be processed in a flexible manner and, moreover, fast convergence is achieved thanks to extrapolated relaxations. The three major objectives of this research are to study the proposed constrained image recovery algorithm and rigorously establish its convergence under very general conditions; to investigate the numerical issues pertaining to its optimal implementation in the context of high performance computing; to demonstrate through extensive numerical testing its flexibility and numerical superiority over existing schemes in a wide range of applied image recovery problems.
