Unified Modulation Pattern Analysis (UMPA) algorithm for 1D sensitive X-ray phase contrast imaging techniques

X-ray phase contrast imaging (XPCI) techniques are sensitive to refraction (differential-phase) and small-angle X-ray scattering (dark-field) signals, not measurable with conventional absorption imaging techniques. Among XPCI techniques, edge illumination (EI), grating interferometry (GI), and speckle-based imaging (SBI) make use of wavefront markers, such as absorbing masks with periodical apertures or random diffusers, to encode refraction and dark-field signals induced by the sample. The Unified Modulated Pattern Analysis (UMPA) provides an algorithmic solution to extract the transmission, refraction, and dark-field images from EI, GI, and SBI datasets where the wavefront marker is directly resolved by the employed detection system. In its original implementation, UMPA has been designed for XPCI techniques sensitive to refractions along two axes. This work presents a modified version of the algorithm to extend its applicability to all the existing XPCI techniques that use wavefront markers with sensitivity to refraction limited along one direction (UMPA-1D). The algorithm, written in C++ and Cython and parallelized with OpenMP, enables fast reconstruction times that are particularly convenient for large tomographic datasets. The validity of the UMPA-1D has been demonstrated using both simulated images and real acquisitions with an EI setup in beam-tracking mode.