Robust phase retrieval with complexity-guidance for coherent X-ray imaging

Abstract

Reconstruction of a stable and reliable solution from noisy incomplete Fourier intensity data recorded in a coherent X-ray imaging (CXI) experiment is a challenging problem for iterative phase retrieval methods. The Relaxed Averaged Alternating Reflections (RAAR) algorithm that is concluded with a number of Error Reduction (ER) iterations is a popular choice in CXI literature. The RAAR-ER algorithm is usually employed for several hundreds of times starting with independent random guesses to obtain trial solutions, that are then averaged. The resolution of the averaged reconstruction is then assessed by plotting the phase retrieval transfer function (PRTF). In this paper, we examine the phase retrieval solution obtained using the RAAR-ER methodology from perspective of the complexity parameter that was introduced by us in recent works. The complexity parameter is a measure of fluctuations in the desired phase retrieval solution and may be computed directly from the Fourier intensity data. We observe that a single run of the RAAR-ER algorithm produces a solution with higher complexity compared to what is expected based on the complexity parameter as manifested by high frequency grainy artifacts in the solution. These spurious features whose size is smaller than the predicted resolution (as per the PRTF) do not seem to go away completely even after a number of trial solutions are averaged. We then describe a CG-RAAR (Complexity Guided RAAR) phase retrieval method that can effectively address this inconsistency problem and provides artifact-free solutions by controlling the solution complexity in the RAAR iterations. The CG-RAAR methodology is first illustrated with simulated unblocked noisy Fourier intensity data and later applied to centrally blocked noisy cyanobacterium data which is available from the CXIDB database. Our simulation and experimental results using CG-RAAR suggest two important improvements over the popular RAAR-ER algorithm. First, the CG-RAAR solutions after the averaging procedure is more reliable in the sense that it contains smallest features consistent with the resolution estimated by the PRTF curve. Secondly, since the single run of the CG-RAAR solution does not have grainy artifacts, the number of trial solutions needed for the averaging process is reduced. The data domain error and PRTF based resolution estimated for the CG-RAAR solution are comparable (although slightly worse) to the traditional RAAR-ER method. This is expected due to regularizing nature of the CG-RAAR algorithm that prevents over-fitting of the solution with noise.