Magnitude reconstruction of complex images from incomplete Fourier phase data

The problem of reconstructing the magnitude of a complex-valued image from incomplete phase data of its Fourier transform is investigated. This is done by formulating the problem as passing the image through a bandpass filter of bandwidth Δ corresponding to the data sample size and evaluating the magnitude of the resulting image. It is revealed that the reconstruction is essentially equivalent to the magnitude of the resulting image obtained by passing the original magnitude function through a set of two dimensional extended all-pass filters. The concept of an extended all-pass filter is introduced to describe a filter whose amplitude response spreads over the whole frequency space with random variations. It is shown that the quality of the reconstruction is content-dependent and improves with increase of the sample size