  X-ray crystallography (XC) is an experimental technique used to determine
three-dimensional crystalline structures. The acquired data in XC, called
diffraction patterns, is the Fourier magnitudes of the unknown crystalline
structure. To estimate the crystalline structure from its diffraction patterns,
we propose to modify the traditional system by including an optical element
called coded aperture which modulates the diffracted field to acquire coded
diffraction patterns (CDP). For the proposed coded system, in contrast with the
traditional, we derive exact reconstruction guarantees for the crystalline
structure from CDP (up to a global shift phase). Additionally, exploiting the
fact that the crystalline structure can be sparsely represented in the Fourier
domain, we develop an algorithm to estimate the crystal structure from CDP. We
show that this method requires 50% fewer measurements to estimate the crystal
structure in comparison with its competitive alternatives. Specifically, the
proposed method is able to reduce the exposition time of the crystal, implying
that under the proposed setup, its structural integrity is less affected in
comparison with the traditional. We discuss further implementation of imaging
devices that exploits this theoretical coded system.
