  Recovering a signal from its Fourier magnitude is referred to as phase
retrieval, which occurs in different fields of engineering and applied physics.
This paper gives a new characterization of the phase retrieval problem.
Particularly useful is the analysis revealing that the common gradient-based
regularization does not restrict the set of solutions to a smaller set.
Specifically focusing on binary signals, we show that a box relaxation is
equivalent to the binary constraint for Fourier-types of phase retrieval. We
further prove that binary signals can be recovered uniquely up to trivial
ambiguities under certain conditions. Finally, we use the characterization
theorem to develop an efficient denoising algorithm.
