A fast truncated Lagrange method for large-scale image restoration problems G. Landi Department of Mathematics, University of Bologna, Piazza Porta San Donato, 5, 40126 Bologna, Italy Abstract In this work, we present a new method for the restoration of images degraded by noise and spatially invariant blur. In the proposed method, the original image restoration problem is replaced by an equality constrained minimization problem. A quasi-Newton method is applied to the first-order optimality conditions of the constrained problem. In each quasi-New- ton iteration, the hessian of the Lagrangian is approximated by a circulant matrix and the Fast Fourier Transform is used to compute the quasi-Newton step. The quasi-Newton iteration is terminated according to the discrepancy principle. Results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Image restoration; Toeplitz matrix; Circulant matrix; Lagrange method; Discrepancy principle 1. Introduction Image restoration problems are modeled by the linear system y ¼ Kx þ e; ð1Þ where y 2 R n 2 represents the observed image, e 2 R n 2 models additive noise and x 2 R n 2 is the original true image. The blurring matrix K 2 R n 2 n 2 is a large ill-conditioned block Toeplitz matrix with Toeplitz blocks (BTTB). Ill-conditioned BTTB linear systems of the form (1) arise from the discretization of first-kind integral equations with spatially invariant point spread functions (PSF). Aim of the image restoration methods is to recover a good approximation of the true image x, given the degraded image y, the blurring matrix K and, eventually, some statistical information about e. In this work, we assume that the error norm d :¼kek is explicitly known. Through this paper, kÆk denotes the Euclidean norm. Typically, in image restoration appli- cations, the values of n range from 256 up to 1024 and hence, the linear system (1) has extremely large dimen- sions. Due to the BTTB structure of K, matrix–vector products involving K can be done efficiently using the 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.039 E-mail address: landig@dm.unibo.it Applied Mathematics and Computation 186 (2007) 1075–1082 www.elsevier.com/locate/amc