### Compressive Phase Retrieval for X-Ray Nanocrystallography

**
Jeffrey Donatelli, University of California, Berkeley
**

X-ray nanocrystallography is an emerging technique for imaging nanoscale objects that alleviates the large crystallization requirement for conventional crystallography by collecting diffraction patterns from a large ensemble of smaller and easier-to-build nanocrystals. In general, diffraction measures the magnitude of the Fourier transform of the objectâ€™s electron density, but phase information is missing and must be recovered in order to determine the structure of the object. While a number of experimental phasing techniques exist, they tend to introduce extra difficulties in the experimental setup or require knowledge of a similar structure. Alternatively, in principle one can almost always retrieve phase information computationally with only the Fourier magnitude information, if it is sampled at a sufficiently high rate. However, the periodicity of the crystal causes the measured signal to be focused on a discrete set of points, known as Bragg peaks, which undersample the Fourier magnitudes. While this often prevents the use of computational phase retrieval for large crystals, diffraction images from nanocrystals also contain a significant amount of signal in between Bragg peaks, which allows for a higher sampling rate. We demonstrate that this sampling is sufficient for recovering the phase information computationally if one seeks a solution with minimal support.

**Abstract Author(s): **Jeffrey Donatelli