  One of the most powerful approaches to imaging at the nanometer or
subnanometer length scale is coherent diffraction imaging using X-ray sources.
For amorphous (non-crystalline) samples, the raw data can be interpreted as the
modulus of the continuous Fourier transform of the unknown object. Making use
of prior information about the sample (such as its support), a natural goal is
to recover the phase through computational means, after which the unknown
object can be visualized at high resolution. While many algorithms have been
proposed for this phase retrieval problem, careful analysis of its
well-posedness has received relatively little attention. In this paper, we show
that the problem is, in general, not well-posed and describe some of the
underlying issues that are responsible for the ill-posedness. We then show how
this analysis can be used to develop experimental protocols that lead to better
conditioned inverse problems.
