Geometry of the Phase Retrieval Problem

TL;DR

This paper analyzes the mathematical structure of phase retrieval in coherent diffraction imaging, revealing its ill-posedness and proposing experimental strategies to improve problem conditioning for better image reconstruction.

Contribution

It provides a theoretical analysis of the phase retrieval problem's well-posedness and suggests experimental protocols to enhance reconstruction stability.

Findings

The phase retrieval problem is generally ill-posed.

Analysis of ill-posedness guides improved experimental design.

Proposed protocols lead to better-conditioned inverse problems.

Abstract

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.