On the choice of regularization matrix for an ℓ2-ℓq minimization method for image restoration

Abstract

Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. To reduce this sensitivity, the original problem may be replaced by a minimization problem with a fidelity term and a regularization term. We consider minimization problems of this kind, in which the fidelity term is the square of the ℓ2-norm of a discrepancy and the regularization term is the qth power of the ℓq-norm of the size of the computed solution measured in some manner. We are interested in the situation when 0<q≤1, because such a choice of q promotes sparsity of the computed solution. The regularization term is determined by a regularization matrix. Novati and Russo let q=2 and proposed in (2014) [13] a regularization matrix that is a finite difference approximation of a differential operator applied to the computed approximate solution after reordering. This gives a Tikhonov regularization problem in general form. We show that this choice of regularization matrix also is well suited for minimization problems with 0<q≤1. Applications to image restoration are presented.