Optimization Methods for Image Regularization from Poisson Data

This work regards optimization techniques for image restoration problems in presence of Poisson noise. In several imaging applications (e.g. Astronomy, Microscopy, Medical Imaging) such noise is predominant; hence regularization techniques are needed in order to obtain satisfying restored images. In a variational framework, the image restoration problem consists in finding a minimum of a functional, which is the sum of two terms,: the fit–to–data and the regularization one. The trade–off between these two terms is measured by a regularization parameter. The estimation of such a parameter is very difficult due to the presence of Poisson noise. In this thesis we investigate three models regarding this parameter: a Discrepancy Model, Constrained Model and the Bregman procedure. The former two provide an estimation for the regularization parameter, but in some cases, such as low counts images, they do not allow to obtain satisfactory results. On the other hand, in presence of such images the Bregman procedure provides reliable results and, moreover, it allows to use an overestimation of the regularization parameter, giving satisfying restored images; furthermore, this procedure permits to gain a contrast enhancement on the final result. In the first part of the work, the basics on image restoration problems are recalled, and a survey on the state–of–the–art methods is given, with an original contribution regarding scaling techniques in ε–subgradient methods. Then, the Discrepancy and the Constrained Models are analyzed from both theoretical and practical point of view, developing suitable numerical techniques for their solution; furthermore, an inexact version of the Bregman procedure is introduced: such a version allows to have a minor computational cost and maintains the same theoretical features of the exact version. Finally, in the last part, a wide experimentation shows the computational efficiency of the inexact Bregman procedure; furthermore, the three models are compared, showing that in high counts images they provide similar results, while in case of low counts images the Bregman procedure provides reliable restored images. This last consideration is evident not only on test problems, but also in problems coming from Astronomy imaging, particularly in case of High Dynamic Range images, as shown in the final part of the experimental section.