Signal reconstruction from partial spectral information

New results in signal representation and reconstruction from partial Fourier information are introduced. In particular, necessary and sufficient conditions for unique representation of two-dimensional signals (images) by spectral amplitude are introduced. It is shown that under mild conditions, only half of the spatial information is required compared to the one-dimensional case. An algorithm for image reconstruction from spectral amplitude is described and examples of reconstructed image are presented, Based on the analysis of the reconstruction algorithm, a theorem on optimal reconstruction from Fourier amplitude is introduced, and it is proven that images of geometric form are best reconstructed by the algorithm. The results are compared to the dual case of image representation by spectral phase and conclusions are presented and discussed