  In 3D single particle imaging with X-ray free-electron lasers, particle
orientation is not recorded during measurement but is instead recovered as a
necessary step in the reconstruction of a 3D image from the diffraction data.
Here we use harmonic analysis on the sphere to cleanly separate the angu- lar
and radial degrees of freedom of this problem, providing new opportunities to
efficiently use data and computational resources. We develop the
Expansion-Maximization-Compression algorithm into a shell-by-shell approach and
implement an angular bandwidth limit that can be gradually raised during the
reconstruction. We study the minimum number of patterns and minimum rotation
sampling required for a desired angular and radial resolution. These extensions
provide new av- enues to improve computational efficiency and speed of
convergence, which are critically important considering the very large datasets
expected from experiment.
