Minimizing Compositions of Functions Using Proximity Algorithms with Application in Image Deblurring

We consider minimization of functions that are compositions of functions having closed-form proximity operators with linear transforms. A wide range of image processing problems including image deblurring can be formulated in this way. We develop proximity algorithms based on the fixed point characterization of the solution to the minimization problems. We further refine the proposed algorithms when the outer functions of the composed objective functions are separable. The convergence analysis of the developed algorithms is established. Numerical experiments in comparison with the well-known Chambolle-Pock algorithm and Zhang-Burger-Osher scheme for image deblurring are given to demonstrate that the proposed algorithms are efficient and robust.