Regularized and Positive-constrained Inverse Methods in the Problem of Object Restoration

The restoration of incoherently illuminated objects, imaged by a perfect optical instrument, is a typical example of a linear ill-posed inverse problem where positive solutions are required. The purpose of this paper is two-fold: first, to discuss the limitations of regularization methods where the solution is not constrained to be positive; and second, to introduce a positive-constrained restoring method consisting in the solution of a linear programming problem. It is found that regularization methods are quite efficient in the restoration of smooth objects, while the solutions of the linear programming problem are considerably better in the restoration of objects with edges and sharp peaks. In order to justify the numerical results, the effect of positivity on numerical stability is carefully analysed. The extension of the results to other inverse problems is briefly discussed.