Improved image deblurring with anti-reﬂective boundary conditions and re-blurring (This is a preprint of an article published in Inverse Problems, 22 (2006) pp. 2035-2053.) M. Donatelli † , C. Estatico ‡ , A. Martinelli † , and S. Serra-Capizzano † †Dipartimento di Fisica e Matematica, Universit`a dell’Insubria - Sede di Como, Via Valleggio 11, 22100 Como, Italy ‡Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, 16146 Genova, Italy E-mail: marco.donatelli@uninsubria.it, estatico@dima.unige.it, andrea.martinelli@uninsubria.it, stefano.serrac@uninsubria.it Abstract. Anti-reﬂective boundary conditions (BCs) have been introduced recently in connection with fast deblurring algorithms. In the noise free case, it has been shown that they reduce substantially artifacts called ringing eﬀects with respect to other classical choices (zero Dirichlet, periodic, reﬂective BCs) and lead to O(n 2 log(n)) arithmetic operations, where n 2 is the size of the image. In the one-dimensional case, for noisy data, we proposed a successful approach called re-blurring: more speciﬁcally, when the PSF is symmetric, the normal equations product A T A is replaced by A 2 , where A is the blurring operator (see Donatelli et al. Inverse Problems, 21, pp. 169– 182). Our present goal is to extend the re-blurring idea to nonsymmetric PSFs in 2 dimensions. In this more general framework, suitable for real applications, the new proposal is to replace A T by A ′ in the normal equations, where A ′ is the blurring matrix related to the current BCs with PSF rotated by 180 degrees. We notice that, although with zero Dirichlet and periodic BCs the re-blurring approach is equivalent to the normal equations scheme, since there A ′ = A T , the novelty concerns both reﬂective BCs and anti-reﬂective BCs, where in general A ′ = A T . We show that the re-blurring with anti-reﬂective BCs is computationally convenient and leads to a large reduction of the ringing eﬀects arising in classical deblurring schemes. A wide set of numerical experiments concerning 2D images and nonsymmetric PSFs conﬁrms the eﬀectiveness of our proposal. AMS classiﬁcation scheme numbers: 65F10, 65F15, 65Y20 Submitted to: Inverse Problems 1. Introduction We consider the classical deblurring problem of blurred and noisy images with space invariant PSFs. More precisely, after standard discretization of a Fredholm equation of ﬁrst kind with