Nonnegative image reconstruction from sparse Fourier data: a new deconvolution algorithm

This paper deals with image restoration problems where the data are nonuniform samples of the Fourier transform of the unknown object. We study the inverse problem in both semidiscrete and fully discrete formulations, and our analysis leads to an optimization problem involving the minimization of the data discrepancy under nonnegativity constraints. In particular we show that such problem is equivalent to a deconvolution problem in the image space. We propose a practical algorithm, based on the gradient projection method, to compute a regularized solution in the discrete case. The key point in our deconvolution-based approach is that the Fast Fourier Transform can be employed in the algorithm implementation without the need of preprocessing the data. A numerical experimentation on simulated and real datafrom the NASA RHESSI mission is also performed.