  Phase retrieval (PR), also sometimes referred to as quadratic sensing, is a
problem that occurs in numerous signal and image acquisition domains ranging
from optics, X-ray crystallography, Fourier ptychography, sub-diffraction
imaging, and astronomy. In each of these domains, the physics of the
acquisition system dictates that only the magnitude (intensity) of certain
linear projections of the signal or image can be measured. Without any
assumptions on the unknown signal, accurate recovery necessarily requires an
over-complete set of measurements. The only way to reduce the
measurements/sample complexity is to place extra assumptions on the unknown
signal/image. A simple and practically valid set of assumptions is obtained by
exploiting the structure inherently present in many natural signals or
sequences of signals. Two commonly used structural assumptions are (i) sparsity
of a given signal/image or (ii) a low rank model on the matrix formed by a set,
e.g., a time sequence, of signals/images. Both have been explored for solving
the PR problem in a sample-efficient fashion. This article describes this work,
with a focus on non-convex approaches that come with sample complexity
guarantees under simple assumptions. We also briefly describe other different
types of structural assumptions that have been used in recent literature.
