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Propagation-based X-ray phase contrast enables nanoscale imaging ofbiological tissue by probing not only the attenuation, but also the real partof the refractive index of the sample. Since only intensities of diffractedwaves can be measured, the main mathematical challenge consists in aphase-retrieval problem in the near-field regime. We treat an often usedlinearized version of this problem known as contract transfer function model.Surprisingly, this inverse problem turns out to be well-posed assuming only acompact support of the imaged object. Moreover, we establish bounds on theLipschitz stability constant. In general this constant grows exponentially withthe Fresnel number of the imaging setup. However, both for homogeneous objects,characterized by a fixed ratio of the induced refractive phase shifts andattenuation, and in the case of measurements at two distances, a much morefavorable algebraic dependence on the Fresnel number can be shown. In somecases we establish order optimality of our estimates.

Stability estimates for linearized near-field phase retrieval in X-ray phase contrast imaging