An image reconstruction method from Fourier data with uncertainties on the spatial frequencies

In this paper the reconstruction of a two-dimensional image from a nonuniform sampling of its Fourier transform is considered, in the presence of uncertainties on the frequencies corresponding to the measured data. The problem therefore becomes a blind deconvolution, in which the unknowns are both the image to be reconstructed and the exact frequencies. The availability of information on the image and the frequencies allows to reformulate the problem as a constrained minimization of the least squares functional. A regularized solution of this optimization problem is achieved by early stopping an alternating minimization scheme. In particular, a gradient projection method is employed at each step to compute an inexact solution of the minimization subproblems. The resulting algorithm is applied on some numerical examples arising in a real-world astronomical application.