Propagation-based phase-contrast X-ray imaging is by now a well established imaging technique, which – as a full-field technique – is particularly useful for tomography applications. Since it can be implemented with synchrotron radiation and at laboratory micro-focus sources, it covers a wide range of applications. A limiting factor in its development has been the phase-retrieval step, which was often performed using methods with a limited regime of applicability, typically based on linearization. In this work, a much larger set of algorithms, which covers a wide range of cases (experimental parameters, objects and constraints), is compiled into a single toolbox – the HoloTomoToolbox – which is made publicly available. Importantly, the unified structure of the implemented phase-retrieval functions facilitates their use and performance test on different experimental data.

Non-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor-Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers Re the optimal transient energy growth always follows a Re 2/3 scaling, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations we show that linear stability and transient growth are independent of the cylinder rotation ratio and we derive a universal Re 2/3 scaling of optimal energy growth using WentzelKramersBrillouin theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.

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

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 95nm isotropic resolution.

Abstract. Phase contrast imaging seeks to reconstruct the complex refractive index of an unknown sample from scattering intensities, measured for example under illumination with coherent X-rays. By incorporating refraction, this method yields improved contrast compared to purely absorption-based radiography but involves a phase retrieval problem which, in general, allows for ambiguous reconstructions. In this paper, we show uniqueness of propagation-based phase contrast imaging for compactly supported objects in the near-field regime, based on a description by the projectionand paraxial approximations. In this setting, propagation is governed by the Fresnel propagator and the unscattered part of the illumination function provides a known reference wave at the detector which facilitates phase reconstruction. The uniqueness theorem is derived using the theory of entire functions. Unlike previous results based on exact solution formulae, it is valid for arbitrary complex objects and requires intensity measurements only at a single detector distance and illumination wavelength. We also deduce a uniqueness criterion for phase contrast tomography, which may be applied to resolve the three-dimensional structure of micro-and nano-scale samples. Moreover, our results may have some significance to electronic imaging methods due to the equivalence of paraxial wave propagation and Schrödinger's equation.