Image Reconstruction from Nonuniform Fourier Data

In many scientific frameworks (e.g., radio and high energy astronomy, medical imaging) the data at one's disposal are encoded in the form of sparse and nonuniform samples of the desired unknown object's Fourier Transform. From the numerical point of view, reconstructing an image from sparse Fourier data is an ill-posed inverse problem in the sense of Hadamard, since there are infinite possible images which match the available Fourier samples. Moreover, the irregular distribution of such samples in the frequency space makes the use of any FFT-based reconstruction algorithm impossible, unless an interpolation and resampling (also known as gridding) procedure is previously applied to the original data. However, if the distribution of the Fourier samples in the frequency space is particularly irregular and/or the signal-to-noise ratio is poor, then the gridding step might either distort the information enclosed in the data or amplify the noise level on the re-sampled data with the result of artefacts formation and undesirable effects in the corresponding reconstructed image.This talk will deal with a different approach to the reconstruction of an image from a nonuniform sampling of its Fourier transform which acts straightly on the data without interpolation and re-sampling operations, exploiting in this way the real nature of the data themselves. In particular, we show that the minimization of the data discrepancy is equivalent to a deconvolution problem with a suitable kernel and we address its solution by means of a gradient projection method with an adaptive steplength parameter, chosen via an alternation of the two Barzilai\u2013Borwein rules. Since the objective function involves a convolution operator, the algorithm can be effectively implemented exploiting the Fast Fourier Transform. The proposed algorithm is tested in a real-world problem, namely the restoration of X-ray images of the Sun during the solar flares by means of the datasets provided by the NASA RHESSI satellite