  Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an
increasingly popular technique for determining the 3-D structure of a molecule
from several noisy 2-D projections images taken at unknown viewing angles. Most
reconstruction algorithms require a low-resolution initialization for the 3-D
structure, which is the goal of ab initio modeling. Suggested by Zvi Kam in
1980, the method of moments (MoM) offers one approach, wherein low-order
statistics of the 2-D images are computed and a 3-D structure is estimated by
solving a system of polynomial equations. Unfortunately, Kam's method suffers
from restrictive assumptions, most notably that viewing angles should be
distributed uniformly. Often unrealistic, uniformity entails the computation of
higher-order correlations, as in this case first and second moments fail to
determine the 3-D structure. In the present paper, we remove this hypothesis,
by permitting an unknown, non-uniform distribution of viewing angles in MoM.
Perhaps surprisingly, we show that this case is statistically easier than the
uniform case, as now first and second moments generically suffice to determine
low-resolution expansions of the molecule. In the idealized setting of a known,
non-uniform distribution, we find an efficient provable algorithm inverting
first and second moments. For unknown, non-uniform distributions, we use
non-convex optimization methods to solve for both the molecule and
distribution.
