Variable metric techniques for forward–backward methods in imaging

Variable metric techniques are a crucial ingredient in many first order optimization algorithms. In practice, they consist in a rule for computing, at each iteration, a suitable symmetric, positive definite scaling matrix to be multiplied to the gradient vector. Besides quasi-Newton BFGS techniques, which represented the state-of-the-art since the 70's, new approaches have been proposed in the last decade in the framework of imaging problems expressed in variational form. Such recent approaches are appealing since they can be applied to large scale problems without adding significant computational costs and they produce an impressive improvement in the practical performances of first order methods. These scaling strategies are strictly connected to the shape of the specific objective function and constraints of the optimization problem they are applied to; therefore, they are able to effectively capture the problem features. On the other side, this strict problem dependence makes difficult extending the existing techniques to more general problems. Moreover, in spite of the experimental evidence of their practical effectiveness, their theoretical properties are not well understood. The aim of this paper is to investigate these issues; in particular, we develop a unified framework for scaling techniques, multiplicative algorithms and the Majorization–Minimization approach. With this inspiration, we propose a scaling matrix rule for variable metric first order methods applied to nonnegatively constrained problems exploiting a suitable structure of the objective function. Finally, we evaluate the effectiveness of the proposed approach on some image restoration problems.