A constrained forward-backward algorithm for image recovery problems

In the solution of inverse problems, the objective is often to minimize the sum of two convex functions f and g subject to convex constraints. Recently, many works have been devoted to this problem in the unconstrained case, when f is possibly non-smooth and g is differentiable with a Lipschitz-continuous gradient. The use of a non-smooth penalizing function arises in particular in wavelet regularization techniques in connection with sparsity issues. In this paper, we propose a modification of the standard forward-backward algorithm, which allows us to minimize f + g over a convex constraint set C. The effectiveness of the proposed approach is illustrated in an image restoration problem involving signal-dependent noise.