Inversion of second-difference operators with application to infrared astronomy

Ground-based astronomical imaging at thermal-infrared wavelengths requires a differential technique, known as chopping and nodding, to extract the weak astronomical signal from the huge background due to the atmosphere and telescope emission. The resulting image is the second difference of the intensity distribution of the astronomical target, and leads to an image restoration problem that can be formulated as the inversion of a second-difference operator. In general, the problem is affected by a huge non-uniqueness, but the degeneracy is reduced when convenient boundary conditions can be used. In particular, if the target field is surrounded by empty sky, it is natural to require that the solution is zero at the boundary of the image. In this paper we investigate the problem of inverting a second-difference operator with the addition of Dirichlet boundary conditions. We show that the related discrete problem can be reduced to the inversion of a non-singular positive definite matrix whose eigenvalues and eigenvectors can be explicitly given. We also give an inversion formula and we investigate the numerical stability of the solution. Since in most practical situations the inversion problem is ill-conditioned, we give a reformulation as a least-squares problem. The advantage is that it is possible to introduce additional constraints such as the non-negativity of the solution. Moreover, we introduce an iterative algorithm converging to the unique non-negative leastsquares solution. Since the latter can be still affected by numerical instability, we show that early stopping of the iterations has a regularization effect. We conclude with a discussion of the observational implications of our analysis.