School of Physics and Astronomy, Monash University, Clayton, Victoria, 3800, Australia.
Chair of Biomedical Physics, Department of Physics, Munich School of Bioengineering, and Institute of Advanced Study, Technische Universität München, 85748, Garching, Germany.
Sci Rep. 2019 Nov 25;9(1):17465. doi: 10.1038/s41598-019-52283-6.
X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; (a) phase contrast, which can capture low-density samples that are difficult to see using attenuation, and (b) dark-field x-ray imaging, which reveals the presence of sub-pixel sample structures. These three modalities can be accessed using a crystal analyser, a grating interferometer or by looking at a directly-resolved grid, grating or speckle pattern. Grating and grid-based methods extract a differential phase signal by measuring how far a feature in the illumination has been shifted transversely due to the presence of a sample. The dark-field signal is extracted by measuring how the visibility of the structured illumination is decreased, typically due to the presence of sub-pixel structures in a sample. The strength of the dark-field signal may depend on the grating period, the pixel size and the set-up distances, and additional dark-field signal contributions may be seen as a result of strong phase effects or other factors. In this paper we show that the finite-difference form of the Fokker-Planck equation can be applied to describe the drift (phase signal) and diffusion (dark-field signal) of the periodic or structured illumination used in phase contrast x-ray imaging with gratings, in order to better understand any cross-talk between attenuation, phase and dark-field x-ray signals. In future work, this mathematical description could be used as a basis for new approaches to the inverse problem of recovering both phase and dark-field information.
X 射线成像是传统上依赖于衰减来提供对比度。近年来，增加了两种互补的模式：（a）相位对比，可以捕获使用衰减难以看到的低密度样本，以及（b）暗场 X 射线成像，可以揭示亚像素样品结构的存在。这三种模式可以使用晶体分析仪、光栅干涉仪或直接观察分辨率网格、光栅或斑点图案来实现。基于光栅和网格的方法通过测量由于样品存在而照明中的特征横向移动了多远来提取差分相位信号。通过测量结构化照明的可见度如何降低来提取暗场信号，这通常是由于样品中存在亚像素结构。暗场信号的强度可能取决于光栅周期、像素大小和设置距离，并且由于强相位效应或其他因素，可能会看到额外的暗场信号贡献。在本文中，我们表明，福克-普朗克方程的有限差分形式可以应用于描述在使用光栅的相位对比 X 射线成像中周期性或结构化照明的漂移（相位信号）和扩散（暗场信号），以便更好地理解衰减、相位和暗场 X 射线信号之间的任何串扰。在未来的工作中，这种数学描述可以作为恢复相位和暗场信息的逆问题的新方法的基础。
