Boolean approximation of a phase-coded aperture diffraction pattern system for X-ray crystallography

The phase retrieval problem involves recovering the phase of a signal from the amplitude of its Fourier transform. Recently, a phase recovery approach for a coded aperture-based acquisition system was proposed. The modulation is realized before the signal being diffracted, such that the underlying signal is recovered from coded diffraction patterns. Moreover, the modulation effect before diffraction can be obtained by using a phase coded aperture located after the sample under study. However, the practical implementation of a phase coded aperture in an X-ray application is not feasible, since it results in a matrix with complex entries and it requires changing the phase of the diffracted beams. Hence, this paper describes a coded X-ray diffraction pattern system based on block-unblock (Boolean) coded apertures that, unlike the phase coded apertures are easily implementable. The proposed system approximates the phase coded aperture by a block-unblock coded aperture by using the detour-phase method. This work used the SAXS/WAXS X-ray crystallography software to simulate the diffraction patterns of a real crystal structure called Rhombic Dodecahedron. Several simulations were realized to obtain the performance of some Boolean approximations in recovering the phase using the simulated diffraction pattern images. The quality of the reconstructions attain up to 22dB in terms of the Peak-Signal-to-Noise-Ratio.