Coded aperture design for solving the phase retrieval problem in X-ray crystallography

Abstract

X-ray crystallography is an experimental technique used in material analysis that allows to measure the atomic positions of the elements present in a crystal. This technique is based on the X-ray diffraction patterns that provide electronic and elastic properties of the crystal of interest. Thus, the crystal can be uniquely identified by means of the phase of its diffraction patterns that are also used to analyze the material of interest. The phase of the X-ray cannot be directly measured; however, it can be recovered from the intensity of diffraction patterns. A recent work has shown that the phase signal can be recovered more efficiently when the acquisition architecture includes an optical element, known as coded aperture, such that the underlying signal is recovered from coded diffraction patterns. A coded aperture is an element that modulates the X-ray diffraction patterns by blocking some X-ray beams. The structure and the number of coded projections are crucial inasmuch they determine the quality and the acquisition time of the X-ray signal. This paper presents the analysis of a coded X-ray Crystallography system, and the design of the spatial structure of the coded aperture, such that the images are recovered with high PSNR (Peak Signal to Noise Ratio) using the minimum number of coded projections. The simulations indicate that the designed coded apertures obtain a reduction of up to 50% in the number of coded projections and an increase in the PSNR of up to 2 dB when the results are compared with the reconstructed images by using random non-designed coded aperture structures. All simulations were carried out on a set of diffraction pattern images, obtained by using the SAXS/WAXS X-ray crystallography software to simulated the diffraction patterns of a real crystal structure, called Rhombic Dodecahedron.