On the filtering effect of iterative regularization algorithms for discrete inverse problems

Many real-world applications are addressed through a linear least-squares problem formulation, whose solution is calculated by means of an iterative approach. A huge amount of studies has been carried out in the optimization field to provide the fastest methods for the reconstruction of the solution, involving choices of adaptive parameters and scaling matrices. However, in presence of an ill-conditioned model and real data, the need of a regularized solution instead of the least-squares one changed the point of view in favour of iterative algorithms able to combine a fast execution with a stable behaviour with respect to the restoration error. In this paper we analyze some classical and recent gradient approaches for the linear least-squares problem by looking at their way of filtering the singular values, showing in particular the effects of scaling matrices and non-negative constraints in recovering the correct filters of the solution. An original analysis of the filtering effect for the image deblurring problem with Gaussian noise on the data is also provided.