1 Introduction

The Large Binocular Telescope (LBT) has been designed for obtaining optical/infrared images with high sensitivity and resolution (Angel et al. 1998). It will consist of two 8.4 m mirrors on a common mount, with a spacing of 14.4 m between the centres. Adaptive optics (AO) will counteract the blurring effect of atmospheric turbulence whereas interferometry will improve the resolution of a single aperture.

For a given orientation of the telescope the diffraction-limited resolution along the centre-to-centre baseline will be equivalent to that of a 22.8 m mirror, while the resolution along the perpendicular direction will be that of a single 8 m mirror. However, a sequence of exposures, taken at sufficiently different parallactic angles, allows in principle a coverage of the u-v plane equivalent to that of a 22.8 m mirror.

As follows from this remark, an important feature of LBT is that, for a given target, it provides several different interferometric images which must be processed to obtain a high-resolution representation of the target. Thanks to the good coverage of the u-v plane, accurate imaging of complex and diffuse objects is also expected. But these achievements require the solution of a new and interesting problem which is a generalization of the classical restoration (deconvolution) problem and can be called deconvolution of multiple images of the same object.

There exists an extremely wide literature on the classical problem of image deconvolution: one object from one image. A tutorial introduction to this topic can be found in Bertero & Boccacci (1998), hereafter referred to as I. On the other hand the problem of multiple images deconvolution has been considered in a limited number of papers. We quote, for instance, Berenstein & Patrick (1990), Yaroslavsky & Caulfield (1994), Casey & Walnut (1995), Piana & Bertero (1996), as well as Hege et al. (1995), Reinheimer et al. (1997), Correia & Richichi (1999) for specific methods applied to LBT imaging.

The purpose of this paper is twofold: first, to generalize some of the most important deconvolution methods to the case of multiple images, with particular emphasis on LBT imaging; second, to test these methods on a sample of simulated LBT images obtained by assuming perfect AO correction.

In Sect. 2 we formulate the restoration problem of LBT as a least-squares problem and we recall the ill-posedness of this approach. In Sect. 3 we discuss the application of linear regularization methods and, more specifically, of the so-called Tikhonov Regularization (TR) (Engl et al. 1996). In addition we introduce the corresponding global PSF, a quantity suitable to describe the outcome of the restoration in terms of resolution. In Sect. 4 we extend to LBT iterative methods such as the Projected Landweber method (PL) (Eicke 1992) and the Image Space Reconstruction Algorithm (ISRA) (Daube-Whiterspoon & Muehllehner 1986) which regularize the solution of the least-squares problem with the constraint of positivity. In Sect. 5 we discuss methods derived from Maximum Likelihood in the case of Poisson noise such as the Lucy-Richardson method (LR) (Richardson 1972; Lucy 1974), also known in tomography as Expectation Maximization (EM) (Shepp & Vardi 1982), and an accelerated version of EM, known as Ordered Subsets - Expectation Maximization (OS-EM) (Hudson & Larkin 1995), whose extension to LBT was proposed in Bertero & Boccacci (2000). Finally in Sect. 6 we present the results obtained by applying these methods (TR, PL, ISRA, LR/EM and OS-EM) to a set of simulated LBT images. The accuracy achievable is estimated both in terms of a restoration error (defined as the Euclidean norm of the difference between the restored image and the original object) and in terms of integrated photometry, at least in the case of stellar sources. The computational cost of each method is also estimated. Our conclusions are presented in Sect. 7.