  The objective of this paper is to develop methods for solving image recovery
problems subject to constraints on the solution. More precisely, we will be
interested in problems which can be formulated as the minimization over a
closed convex constraint set of the sum of two convex functions f and g, where
f may be non-smooth and g is differentiable with a Lipschitz-continuous
gradient. To reach this goal, we derive two types of algorithms that combine
forward-backward and Douglas-Rachford iterations. The weak convergence of the
proposed algorithms is proved. In the case when the Lipschitz-continuity
property of the gradient of g is not satisfied, we also show that, under some
assumptions, it remains possible to apply these methods to the considered
optimization problem by making use of a quadratic extension technique. The
effectiveness of the algorithms is demonstrated for two wavelet-based image
restoration problems involving a signal-dependent Gaussian noise and a Poisson
noise, respectively.
