  Fourier phase retrieval is a classical problem of restoring a signal only
from the measured magnitude of its Fourier transform. Although Fienup-type
algorithms, which use prior knowledge in both spatial and Fourier domains, have
been widely used in practice, they can often stall in local minima. Modern
methods such as PhaseLift and PhaseCut may offer performance guarantees with
the help of convex relaxation. However, these algorithms are usually
computationally intensive for practical use. To address this problem, we
propose a novel, unsupervised, feed-forward neural network for Fourier phase
retrieval which enables immediate high quality reconstruction. Unlike the
existing deep learning approaches that use a neural network as a regularization
term or an end-to-end blackbox model for supervised training, our algorithm is
a feed-forward neural network implementation of PhaseCut algorithm in an
unsupervised learning framework. Specifically, our network is composed of two
generators: one for the phase estimation using PhaseCut loss, followed by
another generator for image reconstruction, all of which are trained
simultaneously using a cycleGAN framework without matched data. The link to the
classical Fienup-type algorithms and the recent symmetry-breaking learning
approach is also revealed. Extensive experiments demonstrate that the proposed
method outperforms all existing approaches in Fourier phase retrieval problems.
