The Scaled Gradient Projection Method: An Application to Nonconvex Optimization

Abstract — The scaled gradient projection (SGP) method is a variable metric forward-backward algorithm designed for constrained differentiable optimization problems, as those obtained by reformulating several signal and image processing problems according to standard statistical ap-proaches. The main SGP features are a variable scaling matrix multiplying the gradient direction at each iteration and an adaptive steplength parameter chosen by generalizing the well-known Barzilai-Borwein rules. An interesting result is that SGP can be exploited within an alternating minimization approach in order to address optimization problems in which the unknown can be splitted in several blocks, each with a given convex and closed feasible set. Classical examples of applications belonging to this class are the non-negative matrix factorization and the blind deconvolution problems. In this work we applied this method to the blind deconvolution of multiple images of the same target obtained with different PSFs. In particular, for our experiments we considered the NASA funded Fizeau interferometer LBTI of the Large Binocular Telescope, which is already operating on Mount Graham and has provided the first Fizeau images, demonstrating the possibility of reaching the resolution of a 22.8m telescope. Due to the Poisson nature of the noise affecting the measured images, the resulting optimization problem consists in the minimization of the sum of several Kullback-Leibler divergences, constrained in suitable feasible sets accounting for the different features to be preserved in the object and the PSFs. 1