Constrained Reconstructions in X-ray Phase Contrast Imaging: Uniqueness, Stability and Algorithms

This chapter considers the inverse problem of X-ray phase contrast imaging (XPCI), as introduced in Chap. 2. It is analyzed how physical a priori knowledge, e.g. of the approximate size of the imaged sample (support knowledge), affects the inverse problem: uniqueness and—for a linearized model—even well-posedness are shown to hold under support constraints, ensuring stability of reconstruction from real-world noisy data. In order to exploit these theoretical insights, regularized Newton methods are proposed as a class of reconstruction algorithms that flexibly incorporate constraints and account for the inherent nonlinearity of XPCI. A Kaczmarz-type variant of the approach is considered for 3D image-recovery in tomographic XPCI, which remains applicable for large-scale data. The relevance of constraints and the capabilities of the proposed algorithms are demonstrated by numerical reconstruction examples.