Dark-field phase retrieval under the constraint of the Friedel symmetry in coherent X-ray diffraction imaging

Coherent X-ray diffraction imaging (CXDI) is a lensless imaging technique that is suitable for visualizing the structures of non-crystalline particles with micrometer to sub-micrometer dimensions from material science and biology. One of the difficulties inherent to CXDI structural analyses is the reconstruction of electron density maps of specimen particles from diffraction patterns because saturated detector pixels and a beam stopper result in missing data in small-angle regions. To overcome this difficulty, the dark-field phase-retrieval (DFPR) method has been proposed. The DFPR method reconstructs electron density maps from diffraction data, which are modified by multiplying Gaussian masks with an observed diffraction pattern in the high-angle regions. In this paper, we incorporated Friedel centrosymmetry for diffraction patterns into the DFPR method to provide a constraint for the phase-retrieval calculation. A set of model simulations demonstrated that this constraint dramatically improved the probability of reconstructing correct electron density maps from diffraction patterns that were missing data in the small-angle region. In addition, the DFPR method with the constraint was applied successfully to experimentally obtained diffraction patterns with significant quantities of missing data. We also discuss this method's limitations with respect to the level of Poisson noise in X-ray detection.