  The advent of the X-ray Free Electron Laser (XFEL) has made it possible to
record diffraction snapshots of biological entities injected into the X-ray
beam before the onset of radiation damage. Algorithmic means must then be used
to determine the snapshot orientations and thence the three-dimensional
structure of the object. Existing Bayesian approaches are limited in
reconstruction resolution typically to 1/10 of the object diameter, with the
computational expense increasing as the eighth power of the ratio of diameter
to resolution. We present an approach capable of exploiting object symmetries
to recover three-dimensional structure to high resolution, and thus reconstruct
the structure of the satellite tobacco necrosis virus to atomic level. Our
approach offers the highest reconstruction resolution for XFEL snapshots to
date, and provides a potentially powerful alternative route for analysis of
data from crystalline and nanocrystalline objects.
