A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints

Abstract The aim of the present paper is to give a general method allowing us to devise maximum-likelihood multiplicative algorithms for inverse problems, and particularly for signal and image restoration with non-negativity constraint. We consider the case of a Gaussian additive noise and that of a Poisson process. The method is founded on the Kuhn}Tucker "rst-order optimality conditions and the algorithms are developed to satisfy these conditions. The proposed method can be used for any convex function whose de"nition range includes the domain of constraints. It allows to obtain generalized forms of classical algorithms (ISRA and RLA) and to unify the method for obtaining these algorithms. We give relaxed forms of the algorithms to increase the convergence speed; moreover, the e!ect of the constraints is clearly shown. For a better understanding of the method to take into account the constraints, we express the non-negativity constraint using di!erent functions and we reach a large class of algorithms that can be analyzed as descent algorithms. Then, we can justify and analyze the behavior of several algorithms suggested in the literature. The particular displacement directions appearing in such algorithms are evidenced and the convergence speed is analyzed. The algorithms are applied for simulated data, to a two-dimensional deconvolution problem, to show their performance and e!ectiveness. A support constraint is taken into account implicitly in the algorithms. Our method can be extended to more general hard constraints on the extreme values or on the support of the solution and a regularization of the problem can be easily introduced in the method