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.
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