Many experiments in the field of optical imaging are modelled as phase retrieval problems. Motivated by imaging experiments with biological specimens that need to be measured using a preferably low dose of illumination particles, we consider phase retrieval systems with small Poisson noisy measurements. In this talk, we discuss how to formulate a suitable optimization problem. We study reasonable loss functions adapted to the Poisson distribution, optimized for low-dose data. As a solver, we apply gradient descent algorithms with Wirtinger derivatives. For the proposed loss functions, we analyze the convergence of the respective Wirtinger flow type algorithms to stationary points. We present numerical reconstructions from phase retrieval measurements in a low-dose regime to corroborate our theoretical observations.

We present a systematic derivation and local convergence analysis for various ADMM algorithms in phase retrieval and ptychography.

We also discuss the extension of these algorithms to blind ptychography where the probe is unknown and compare their numerical performance.

The phase problem is a well known challenge in propagation-based phase-contrast X-ray imaging, describing the situation that whenever a detector measures a complex X-ray wavefield, the phase information is lost, i.e. only the magnitude of the measured wavefield remains as a usable data set. The resulting inverse problem is ill-posed and non-convex, requiring twice as many variables to be reconstructed to obtain the complex-valued image of the object under study.

In a recent development we have changed the representation of the reconstructed image [1]. The classical representation as amplitude and phase suffers from phase wrapping ambiguities. The representation as the projected refractive index of the object avoids these problems. However, this algorithm still suffers from slow convergence and convergence to local minima.

In our work, we have investigated the main causes of slow convergence and local minima for the Nesterov accelerated projected gradient descent type of algorithm that is currently used in practice. We propose a framework of different techniques to address these problems and show that by combining the proposed methods, the reconstruction result can be dramatically improved in terms of reconstruction speed and quality. We apply our proposed methods to several datasets obtained from the nano-imaging experiment at the Hereon-operated beamline P05 at DESY (Hamburg, Germany). We demonstrate that our proposed framework can cope with single-distance measurements, which is a requirement for in-situ/operando experiments, and without a compact support constraint, while maintaining robustness along a wide range of samples.

[1] F. Wittwer, J.Hagemann et al. Phase retrieval framework for direct reconstruction of the projected refractive index applied to ptychography and holography, Optica 9: 295-302, 2022