Abstract
The aim of this work is to present a new and efficient optimization method for the solution of blind deconvolution problems with data corrupted by Gaussian noise, which can be reformulated as a constrained minimization problem whose unknowns are the point spread function (PSF) of the acquisition system and the true image. The objective function we consider is the weighted sum of the least-squares fit-to-data discrepancy and possible regularization terms accounting for specific features to be preserved in both the image and the PSF. The solution of the corresponding minimization problem is addressed by means of a proximal alternating linearized minimization (PALM) algorithm, in which the updating procedure is made up of one step of a gradient projection method along the arc and the choice of the parameter identifying the steplength in the descent direction is performed automatically by exploiting the optimality conditions of the problem. The resulting approach is a particular case of a general scheme whose convergence to stationary points of the constrained minimization problem has been recently proved. The effectiveness of the iterative method is validated in several numerical simulations in image reconstruction problems.
Highlights
Image deconvolution is an extremely prolific field which on one hand finds applications in a large variety of areas and on the other hand rounds up the efforts of a large community of mathematicians working on inverse problems and optimization methods
The regularization terms we chose in this case are RHS for the image and RT0 for the point spread function (PSF); 2. the satellite image frequently used in several papers on image deconvolution, artificially blurred with an out-of-focus PSF with radius equal to 4
The regularization term we chose in this case is RHS for both the image and the PSF; 3. the Hubble Space Telescope (HST) image of the crab nebula NGC 19521, artificially blurred with an Airy function [30] mimicking the ideal acquisition of one mirror of the Large Binocular Telescope (LBT - http://www.lbto.org)
Summary
Image deconvolution is an extremely prolific field which on one hand finds applications in a large variety of areas (physics, medicine, engineering,...) and on the other hand rounds up the efforts of a large community of mathematicians working on inverse problems and optimization methods. Most of the resulting works deal with the ill-conditioned discrete problem in which the blurring matrix is assumed to be known and the goal is to find a good approximation of the unknown image by means of some regularization approaches [1]. In many real applications the blurring matrix is not completely known due to a lack of information on the acquisition model and/or to external agents which corrupt the measured image (atmospheric turbulence, thermal blooming,...). This situation is known as blind deconvolution and most strategies to approach this problem are based on a simultaneous recovery of both the image approximation and the point spread function (PSF) of the acquisition system. An alternative formulation involves the unconstrained minimization of the same objective function with the addition of the indicator functions of the feasible sets, which can be numerically solved with forward–backward
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