A Majorization-Minimization Golub-Kahan bidiagonalization method for <inline-formula><tex-math id="M1">$ \ell_{2}-\ell_{q} $</tex-math></inline-formula> mimimization with applications in image restorization

Abstract

An image restorization problem is often modelled as a discrete ill-posed problem. In general, its solution, even if it exists, is very sensitive to the perturbation in the data. Regularization methods reduces the sensitivity by replacing this problem with a minimization problem with a fidelity term and $ \ell_{q} $ regularization term. In order to improve the sparsity of the solution, we only consider the case of $ 0<q\leq1 $ in this paper. This paper presents a majorization-minimization Golub-Kahan bidiagonalization algorithm to solve this kind of minimization problems. The solution subspace is extended by the Golub-Kahan bidiagonalization process. The restarted case is also considered. The regularization parameter is determined by using the discrepancy principle. Several examples in image restorization are shown for the proposed methods.