Phase contrast tomography aims at identifying phase shifts of a coherent x-ray beam passing through a sample, rather than absorption as in CT. This corresponds to reconstructing the real part rather than the imaginary part of the refractive index and is particularly useful for imaging small biological samples since the contrast in the real part is an order of magnitude larger.
Since only intensities of x-rays can be measured directly, phase retrieval problems must be addressed, leading to non-linear inverse problems in general. We study the stability of the linearized problem in dependence of the Fresnel number. Furthermore, we derive resolution estimates of this imaging technology. Finally, we describe efficient reconstruction algorithms and address new developments.