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Promoted by the advent of coherent synchrotron light sources, phase contrasttomography allows to resolve three-dimensional variations of an unknownsample's complex refractive index from scattering intensities recorded atdifferent incident angles of an X-ray beam. By diffractive free-spacepropagation of the transmitted wave field, this method is sensitive not only toabsorption but also to refractive phase shifts induced by the specimen,permitting three-dimensional nanoscale imaging of quasi-transparent samplessuch as biological cells. However, the reconstruction of the specimen structurefrom the observed data constitutes an algorithmically challenging nonlinearill-posed inverse problem, mainly due to the characteristic loss of phaseinformation in the detection of the wave field.  In this work, regularized Newton methods are developed for the solution ofthis tomographic phase retrieval problem, based on a detailed analysis of itsmathematical structure. We consider both the near-field- or Fresnel regimecharacterized by a moderate propagation length between sample and detector andthe far-field limit of large detector distances, where propagation is governedby the Fourier transform. In the former setting, excellent numericalreconstructions are obtained via the chosen Newton-type approach, supplementedby novel theoretical results stating that measurements from a single detectordistance are sufficient to uniquely recover both refraction and absorption of asample. The proposed algorithm simultaneously performs tomographic- and phasereconstruction, which is found to stabilize the latter by exploitingcorrelations between the diffraction patterns recorded under different incidentangles.

Regularized Newton methods for simultaneous Radon inversion and phase retrieval in phase contrast tomography