  The advent of the X-ray Free Electron Laser (XFEL) has made it possible to
record 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 reconstruct the three-dimensional structure of the
object. Existing approaches are limited in reconstruction resolution to at best
1/30th 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 1/100th of the object diameter, and thus reconstruct the structure of the
satellite tobacco necrosis virus to atomic resolution. Combined with the
previously demonstrated capability to operate at ultralow signal, 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.
