A Novel Gradient Projection Approach for Fourier-Based Image Restoration

This work deals with the ill-posed inverse problem of reconstructing a two-dimensional image of an unknown object starting from sparse and nonuniform measurements of its Fourier Transform. In particular, if we consider a priori information about the target image (e.g., the nonnegativity of the pixels), this inverse problem can be reformulated as a constrained optimization problem, in which the stationary points of the objective function can be viewed as the solutions of a deconvolution problem with a suitable kernel. We propose a fast and effective gradient-projection iterative algorithm to provide regularized solutions of such a deconvolution problem by early stopping the iterations. Preliminary results on a real-world application in astronomy are presented.