Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.
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