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

dThe aim of the present paper is to give a general method allowing us to de vise maximum-likelihood multiplicative algorithms for inverse problems, and particularly for signal and image restoration with non-negativity constrai nt. We consider the case of a Gaussian additive noise and that of a Poisson process. The method is founded on the Kuhn-Tucker first-order optimality c onditions and the algorithms are developed to satisfy these conditions. The proposed method can be used for any convex function whose definition range includes the domain of constraints. It allows to obtain generalized forms of classical algorithms (ISRA and RLA) and to unify the method for obtainin g these algorithms. We give relaxed forms of the algorithms to increase the convergence speed; moreover, the effect of the constraints is clearly show n. For a better understanding of the method to take into account the constr aints, we express the non-negativity constraint using different functions a nd we reach a large class of algorithms that can be analyzed as descent alg orithms. Then. we can justify and analyze the behavior of several algorithm s suggested in the literature. The particular displacement directions appea ring in such algorithms are evidenced and the convergence speed is analyzed . The algorithms are applied for simulated data, to a two-dimensional decon volution problem, to show their performance and effectiveness. A support co nstraint is taken into account implicitly in the algorithms. Our method can be extended to more general hard constraints on the extreme values or on t he support of the solution and a regularization of the problem can be easil y introduced in the method. (C) 2001 Elsevier Science B.V. All rights reser ved.