Implementation of the regularized structured total least squares algorithms for blind image deblurring

The structured total least squares (STLS) problem has been introduced to handle problems involving structured matrices corrupted by noise. Often the problem is ill-posed. Recently, regularization has been proposed in the STLS framework to solve ill-posed blind ∗ Corresponding author. E-mail address: [email protected] (N. Mastronardi). 1 This work was partially supported by MIUR, grant number 2002014121. 2 Philippe Lemmerling is supported by a post-doctoral K. U. Leuven fellowship. 3 This work was supported by the National Science Foundation under Grant CCR 0204084. 4 Research supported by Research Council KUL: GOA-Mefisto 666, IDO /99/003 and /02/009 (Predictive computer models for medical classification problems using patient data and expert knowledge), several PhD/postdoc & fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects, G.0078.01 (structured matrices), G.0407.02 (support vector machines), G.0269.02 (magnetic resonance spectroscopic imaging), G.0270.02 (nonlinear Lp approximation), research communities (ICCoS, ANMMM); IWT: PhD Grants, Belgian Federal Government: DWTC (IUAP IV-02 (1996-2001) and IUAP V-22 (2002-2006): Dynamical Systems and Control: Computation, Identification & Modelling)); EU: PDT-COIL, BIOPATTERN, ETU-MOUR. 0024-3795/$ see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2004.07.006 204 N. Mastronardi et al. / Linear Algebra and its Applications 391 (2004) 203–221 deconvolution problems encountered in image deblurring when both the image and the blurring function have uncertainty. The kernel of the regularized STLS (RSTLS) problem is a least squares problem involving Block–Toeplitz–Toeplitz–Block matrices. In this paper an algorithm is described to solve this problem, based on a particular implementation of the generalized Schur Algorithm (GSA). It is shown that this new implementation improves the computational efficiency of the straightforward implementation of GSA from O(N2.5) to O(N2), where N is the number of pixels in the image. © 2004 Elsevier Inc. All rights reserved.