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

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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.