  The routine atomic-resolution structure determination of single particles is
expected to have profound implications for probing the structure-function
relationship in systems ranging from energy materials to biological molecules.
Extremely-bright, ultrashort-pulse X-ray sources---X-ray Free Electron Lasers
(XFELs)---provide X-rays that can be used to probe ensembles of nearly
identical nano-scale particles. When combined with coherent diffractive
imaging, these objects can be imaged; however, as the resolution of the images
approaches the atomic scale, the measured data are increasingly difficult to
obtain and, during an X-ray pulse, the number of photons incident on the
two-dimensional detector is much smaller than the number of pixels. This latter
concern, the signal "sparsity," materially impedes the application of the
method. We demonstrate an experimental analog using a synchrotron X-ray source
that yields signal levels comparable to those expected from single biomolecules
illuminated by focused XFEL pulses. The analog experiment provides an
invaluable cross-check on the fidelity of the reconstructed data that is not
available during XFEL experiments. We establish---using this experimental
data---that a sparsity of order $1.3\times10^{-3}$ photons per pixel per frame
can be overcome, lending vital insight to the solution of the atomic-resolution
XFEL single particle imaging problem by experimentally demonstrating 3D
coherent diffractive imaging from photon-sparse random projections.
