Shot-to-shot flat-field correction at X-ray free-electron lasers

X-ray free-electron lasers (XFELs) provide high-brilliance pulses, which offer unique opportunities for coherent X-ray imaging techniques, such as in-line holography. One of the fundamental steps to process in-line holographic data is flat-field correction, which mitigates imaging artifacts and, in turn, enables phase reconstructions. However, conventional flat-field correction approaches cannot correct single XFEL pulses due to the stochastic nature of the self-amplified spontaneous emission (SASE), the mechanism responsible for the high brilliance of XFELs. Here, we demonstrate on simulated and megahertz imaging data, measured at the European XFEL, the possibility of overcoming such a limitation by using two different methods based on principal component analysis and deep learning. These methods retrieve flat-field corrected images from individual frames by separating the sample and flat-field signal contributions; thus, enabling advanced phase-retrieval reconstructions. We anticipate that the proposed methods can be implemented in a real-time processing pipeline, which will enable online data analysis and phase reconstructions of coherent full-field imaging techniques such as in-line holography at XFELs.