A Fourier-based method for the restoration of chopped and nodded images

In a series of previous papers we have proposed and validated an iterative method, known as the projected Landweber method, for the restoration of astronomical images taken in chopping and nodding mode. While the method generally provides good results, it may also generate artifacts related to the huge non-uniqueness of the solution of the restoration problem. If the image satisfies additional boundary conditions, the nonuniqueness can be reduced, or even entirely removed. In this paper we investigate the case of periodic boundary conditions, which apply, in particular, to the case of a target area surrounded by a suitable region of empty sky. Periodic boundary conditions do not entirely remove the nonuniqueness of the solution, but allow using Fourier-based techniques. We introduce a new iterative method which can be considered as a relaxed and projected version of the van Cittert method. We formally demonstrate why this method does not produce the artifacts generated by the one we previously proposed, and we present numerical simulations confirming this result. We illustrate the convergence properties of the algorithm in the case of both compact and extended sources. Finally, we briefly discuss the potential and the limitations of the proposed technique.