Constrained Minimization Algorithms

Abstract

In this paper, we consider the inverse problem of restoring an unknown signal or image, knowing the transformation suffered by the unknowns. More specifically we deal with transformations described by a linear model linking the unknown signal to an unnoisy version of the data. The measured data are generally corrupted by noise. This aspect of the problem is presented in the introduction for general models. In Section 2, we introduce the linear models, and some examples of linear inverse problems are presented. The specificities of the inverse problems are briefly mentionned and shown on a simple example. In Section 3, we give some information on classical distances or divergences. Indeed, an inverse problem is generally solved by minimizing a discrepancy function (divergence or distance) between the measured data and the model (here linear) of such data. Section 4 deals with the likelihood maximization and with their links with divergences minimization. The physical constraints on the solution are indicated and the Split Gradient Method (SGM) is detailed in Section 5. A constraint on the inferior bound of the solution is introduced at first; the positivity constraint is a particular case of such a constraint. We show how to obtain strictly, the multiplicative form of the algorithms. In a second step, the so-called flux constraint is introduced, and a complete algorithmic form is given. In Section 6 we give some brief information on acceleration method of such algorithms. A conclusion is given in Section 7.