  The problem of phase retrieval is a classic one in optics and arises when one
is interested in recovering an unknown signal from the magnitude (intensity) of
its Fourier transform. While there have existed quite a few approaches to phase
retrieval, recent developments in compressed sensing and convex
optimization-based signal recovery have inspired a host of new ones. This work
presents an overview of these approaches.
  Since phase retrieval, by its very nature, is ill-posed, to make the problem
meaningful one needs to either assume prior structure on the signal (e.g.,
sparsity) or obtain additional measurements (e.g., masks, structured
illuminations). For both the cases, we review conditions for the
identifiability of the signal, as well as practical algorithms for signal
recovery. In particular, we demonstrate that it is possible to robustly and
efficiently identify an unknown signal solely from phaseless Fourier
measurements, a fact with potentially far-reaching implications.
