One of the most powerful approaches to imaging at the nanom
eter 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 unk
nown object. Making use of prior information about the sample (such as its
support)\, a natural goal is to recover the phase through computational m
eans\, 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 relativel
y little attention. In fact the problem is\, in general\, not well-posed a
nd describe some of the underlying geometric issues that are responsible f
or the ill-posedness. We then show how this analysis can be used to develo
p experimental protocols that lead to better conditioned inverse problems.
*