A Deconvolution Algorithm for Imaging Problems from Fourier Data

Prato, Marco

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 389-404

Résumé

In this paper we address the problem of reconstructing a two-dimensional image starting from the knowledge on nonuniform samples of its Fourier Transform. Such inverse problem has a natural semidiscrete formulation, that is analyzed together with its fully discrete counterpart. In particular, the image restoration problem in this case can be reformulated as the minimization of the data discrepancy under nonnegativity constraints, possibly with the addition of a further equality constraint on the total flux of the image. Moreover, we show that such problem is equivalent to a deconvolution in the image space, that represents a key property allowing the desing of a computationally efficient algorithm based on Fast Fourier Transforms to address its solution. Our proposal to compute a regularized solution in the discrete case involves a gradient projection method, with an adaptive choice for the steplength parameter that improves the convergence rate. A numerical experimentation on simulated data from the NASA RHESSI mission is also performed.