The Geometry of the Phase Retrieval Problem

Phase retrieval is a problem that arises in a wide range of imaging
applications, including x-ray crystallography, x-ray diffraction imaging
and ptychography. The data in the phase retrieval problem are samples of
the modulus of the Fourier transform of an unknown function. To
reconstruct this function one must use auxiliary information to determine
the unmeasured Fourier transform phases. There are many algorithms to
accomplish task, but none work very well. In this talk we present an
analysis of the geometry that underlies these failures and points to new
approaches for solving this class of problems.