The Richardson-Lucy method is the most popular deconvolution method inastronomy because it preserves the number of counts and the non-negativity ofthe original object. Regularization is, in general, obtained by an earlystopping of Richardson-Lucy iterations. In the case of point-wise objects suchas binaries or open star clusters, iterations can be pushed to convergence.However, it is well-known that Richardson-Lucy is an inefficient method. Inmost cases, acceptable solutions are obtained at the cost of hundreds orthousands of iterations. A general optimization method, referred to as thescaled gradient projection method, has been proposed for the constrainedminimization of continuously differentiable convex functions. It is applicableto the non-negative minimization of the Kullback-Leibler divergence. If thescaling suggested by Richardson-Lucy is used in this method, then it provides aconsiderable increase in the efficiency of Richardson-Lucy. Therefore the aimof this paper is to apply the scaled gradient projection method to a number ofimaging problems in astronomy such as single image deconvolution, multipleimage deconvolution, and boundary effect correction. The correspondingalgorithms are derived and implemented in interactive data language. To attemptto achieve a further increase in efficiency, we also consider an implementationon graphic processing units. The proposed algorithms are tested on simulatedimages. The acceleration of scaled gradient projection methods achieved withrespect to the corresponding Richardson-Lucy methods strongly depends on boththe problem and the specific object to be reconstructed, and in our simulationsthe improvement achieved ranges from about a factor of 4 to more than 30.Moreover, significant accelerations of up to two orders of magnitude have beenobserved between the serial and parallel implementations of the algorithms.
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