Iterative image reconstruction : a point of view

Several iterative methods are available for solving the ill-posed problem of image reconstruction. They are motivated by different approaches and may derive from methods used for the solution of linear equations or the minimization of suitable functionals. In this paper we adopt the approach flowing from maximum likelihood to Bayesian formulation of image reconstruction and providing a generalization of the classical regularization theory. This approach leads to the minimization of functionals derived from properties of the noise and, possibly, from additional information on the solution. We investigate a class of scaled gradient methods, based on a suitable decomposition of the gradient, and we show that this class contains some of the methods used for the solution of maximum likelihood problems in image reconstruction. We also obtain very simple regularized versions of these methods. Constraints of non-negativity and flux conservation are taken into account by considering scaled gradient projection (SGP) methods, derived from the previous approach, and for them a convergence proof can be given. Numerical experience on a particular problem shows that SGP can provide a considerable increase in efficiency with respect to the standard algorithm used for that problem. Work is in progress in order to understand whether a similar gain can be achieved in other cases.