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.

中文翻译：

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相位恢复问题的几何结构

在纳米或亚纳米长度尺度上成像的最强大方法之一是使用 X 射线源的相干衍射成像。对于非晶（非晶）样品，原始数据可以解释为未知物体的连续傅立叶变换的模量。利用有关样本的先验信息（例如其支持），自然目标是通过计算手段恢复相位，然后可以高分辨率可视化未知对象。虽然已经针对这个相位检索问题提出了许多算法，但对其适定性的仔细分析却很少受到关注。在本文中，我们展示了该问题通常不是适定的，并描述了导致不适定的一些潜在问题。