1 Introduction

The Large Binocular Telescope (LBT) will consist of two 8.4 m mirrors on a common mount, with a 14.4 m centre-to-centre spacing. When the beams of the two mirrors are combined interferometrically the diffraction-limited resolution of this system in the direction of the baseline will be equivalent to that of a 22.8 m mirror. Moreover, thanks to the small distance between the mirrors, by taking images with different orientations of the baseline it is possible to have a coverage of the u-v plane essentially equivalent to that of a 22.8 m mirror (Angel et al. 1998).

In order to exploit the imaging properties of LBT, the use of accurate image restoration methods will be required. This problem has been already considered by a few authors. For instance Hege et al. (1995) apply the iterative blind-deconvolution method developed by Jefferies & Christou (1993) to simulated LBT images of an extended object while Reinheimer et al. (1997) investigate the applicability to LBT of speckle masking and of the iterative building block method for the restoration of the object from its bispectrum (Hofmann & Weigelt 1993; Reinheimer et al. 1993).

More traditional restoration methods such as Tikhonov regularization, Conjugate Gradients (CG), projected Landweber method etc. (see, for instance, Bertero et al. 1995; Piana & Bertero 1996; Bertero & Boccacci 1998) have not yet been considered and we are preparing a paper on their extension to LBT images. However the extension of the Lucy-Richardson (LR) iterative method (Richardson 1972; Lucy 1974) has been used by Correia & Richichi 1999 for the restoration of simulated images and of LBT-like images taken at the TIRGO telescope. The results obtained by this method look good but the number of iterations required is exceedingly high.

The LR method is known as Expectation Maximization (EM) method (Shepp & Vardi 1982) in the literature on Computed Tomography (CT) where it is demonstrated that it provides reliable restorations both in PET and SPECT imaging. In the following we will use the denomination LR/EM for this method. Also in medical applications the number of iterations required for obtaining reliable restorations is too large. For this reason the acceleration of the method has been investigated by many authors. A very efficient technique, based on ordered subsets of projection data and called OS-EM method, has been proposed by Hudson & Larkin (1994). It provides the same accuracy as EM with a number of iterations which is smaller by a factor approximately equal to the number of subsets used by the algorithm, the computational cost of one iteration being essentially the same in EM and OS-EM.

It is interesting to point out that LBT imaging has a structure similar to that of CT imaging: a projection in CT provides information on the Fourier transform (FT) of the object in the direction of the projection while an image of LBT provides information on the FT of the object mainly in the direction of the baseline. Therefore it looks quite natural to attempt the adaptation of OS-EM to LBT. This is the purpose of this paper. For simplicity we will restrict the analysis to the case of one image per orientation of the baseline and we will also assume that the number of orientations is small. Then it is quite natural to consider subsets consisting of just one image. The extension to the case of several frames per orientation angle or to the case of a continuously changing orientation is straightforward. In these cases one could also consider subsets consisting of more than one image and apply the strategies developed in the case of CT for the choice of the subsets (Hudson & Larkin 1994).