Constrained TV $$_p$$ p - $$\ell _2$$ ℓ 2 Model for Image Restoration

Abstract

The popular total variation (TV) model for image restoration (Rudin et al. in Phys D 60(1---4):259-268, 1992) can be formulated as a Maximum A Posteriori estimator which uses a half-Laplacian image-independent prior favoring sparse image gradients. We propose a generalization of the TV prior, referred to as TV$$_p$$p, based on a half-generalized Gaussian distribution with shape parameter p. An automatic estimation of p is introduced so that the prior better fits the real images' gradient distribution; we will show that, in general, the estimated p value does not necessarily require to be close to zero. The restored image is computed by using an alternating directions methods of multipliers procedure. In this context, a novel result in multivariate proximal calculus is presented which allows for the efficient solution of the proposed model. Numerical examples show that the proposed approach is particularly efficient and well suited for images characterized by a wide range of gradient distributions.