Efficient deconvolution methods for astronomical imaging: algorithms and IDL-GPU codes

Context. The Richardson-Lucy method is the most popular deconvolution method in astronomy because it preserves the number of counts and the non-negativity of the original object. Regularization is, in general, obtained by an early stopping of Richardson-Lucy iterations. In the case of point-wise objects such as binaries or open star clusters, iterations can be pushed to convergence. However, it is well-known that Richardson-Lucy is an inefficient method. In most cases and, in particular, for low noise levels, acceptable solutions are obtained at the cost of hundreds or thousands of iterations, thus several approaches to accelerating Richardson-Lucy have been proposed. They are mainly based on Richardson-Lucy being a scaled gradient method for the minimization of the Kullback-Leibler divergence, or Csiszar I-divergence, which represents the data-fidelity function in the case of Poisson noise. In this framework, a line search along the descent direction is considered for reducing the number of iterations. Aims. A general optimization method, referred to as the scaled gradient projection method, has been proposed for the constrained minimization of continuously differentiable convex functions. It is applicable to the non-negative minimization of the Kullback-Leibler divergence. If the scaling suggested by Richardson-Lucy is used in this method, then it provides a considerable increase in the efficiency of Richardson-Lucy. Therefore the aim of this paper is to apply the scaled gradient projection method to a number of imaging problems in astronomy such as single image deconvolution, multiple image deconvolution, and boundary effect correction. Methods. Deconvolution methods are proposed by applying the scaled gradient projection method to the minimization of the Kullback-Leibler divergence for the imaging problems mentioned above and the corresponding algorithms are derived and implemented in interactive data language. For all the algorithms, several stopping rules are introduced, including one based on a recently proposed discrepancy principle for Poisson data. To attempt to achieve a further increase in efficiency, we also consider an implementation on graphic processing units. Results. The proposed algorithms are tested on simulated images. The acceleration of scaled gradient projection methods achieved with respect to the corresponding Richardson-Lucy methods strongly depends on both the problem and the specific object to be reconstructed, and in our simulations the improvement achieved ranges from about a factor of 4 to more than 30. Moreover, significant accelerations of up to two orders of magnitude have been observed between the serial and parallel implementations of the algorithms. The codes are available upon request.