Geometry of the phase retrieval problem

One of the most powerful approaches to imaging at the nanometer length scale is coherent diffraction imaging using X-ray sources. For amorphous (non-crystalline) samples, raw data collected in the far-field can be interpreted as the modulus of the two-dimensional continuous Fourier transform of the unknown object. The goal is then to recover the phase through computational means by exploiting prior information about the sample (such as its support), 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.