Talk by Sgallari Fiorella, Depth. Mathematics, University of Bologna, Italy

A Unified Framework for the Restoration of Images Corrupted by Additive White Noise

Abstract

A real captured image may be distorted by many expected or unexpected factors among which blur and random noise are typical and often unavoidable examples. Hence, image deblurring and denoising are fundamental tasks in the field of image processing. Over the years, one of the most studied class of noises is that of additive, independent identically distributed (i.i.d.) noises, which affect all the pixels by independent random corruptions coming from the same distribution. This class includes important noises such as those characterized by Gaussian, uniform, Laplacian and Cauchy distributions, which can be found in many applications, such as e.g. medical and astronomic imaging. For any of these noise distributions, ad hoc variational models have been devised in the past for image restoration.

However, in many practical applications it is difficult to know a priori the noise distribution and, in any case, the noise might be the outcome of several sources thus giving raise to mixed noise models with very specific/complex distributions.

To overcome these inherent difficulties, in this talk we discuss a robust variational model aimed at restoring images corrupted by blur and by the generic wide class of additive white - or uncorrelated - noises, which include i.i.d noises. The solution of the non-trivial optimization problem, due to the non-smooth nonconvex proposed model, is efficiently obtained by means of a numerical algorithm based on the Alternating Directions Method of Multipliers, which in particular reduces the solution to a sequence of convex optimization sub-problems. Numerical results show the potentiality of the proposed model for restoring blurred images corrupted by several kinds of additive white noises.

Joint work with Alessandro Lanza, Serena Morigi and Federica Schiacchitano.