Mathematical and algorithmical analysis of geometric reconstruction tasks arising in refraction- and diffraction-based tomography

计算机断层扫描（CT），从多个X射线投影获得标本内的密度分布的过程，在过去的三十年里彻底改变了诊断放射学。虽然标准CT是基于吸收的原则，我们专注于三个新的实验技术，是基于折射和（晶体）衍射。目标是进一步提高以前基本上不可见的物体部分的分辨率。从技术方面来看，这些方法是由我们的项目合作伙伴开发的，并且仍然是他们的研究重点。它们被称为通过微分相衬的层析成像、利用同步加速器X射线的三维X射线衍射（3DXRD）和利用由X射线自由电子激光（XFEL）产生的X射线的3DXRD。我们的目标是发起和发展的一般理论的几何重建折射和衍射层析成像。在这里，几何意味着样本-如在本项目中讨论的应用-可以被视为几何对象，如多面体，凸体，或某些格集（或工会）。这种几何的观点应该使我们能够设计和探索上述三种实验技术的强大的专用重建算法。

Computerized tomography (CT), the process of obtaining the density distribution within a specimen from multiple X-ray projections, has revolutionized diagnostic radiology over the past three decades. While standard CT is based on the principle of absorption, we focus on three novel experimental techniques that are based on refraction and (crystal) diffraction. The goal is to further enhance the resolution of previously largely invisible parts of the objects. From the technical side, these methods were developed by and are still at the research focus of our project partners. They are known as tomography by differential phase contrast, 3-dimensional X-ray diffraction (3DXRD) with synchrotron X-rays, and 3DXRD with X-rays generated by X-ray free electron lasers (XFELs). Our aim is to initiate and develop a general theory of geometric reconstruction in refraction- and diffraction-based tomography. Here, geometric means that the specimen - as in the applications discussed in this project - can be considered as geometric objects, such as polytopes, convex bodies, or certain lattice sets (or unions of those). This geometric point of view should allow us to devise and explore robust special-purpose reconstruction algorithms for the three experimental techniques mentioned above.