Method of moments for 3D single particle modeling with non-uniform distribution of viewing angles.

Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an increasingly popular technique for determining the 3D structure of a molecule from several noisy 2D projections images taken at unknown viewing angles. Most reconstruction algorithms require a low-resolution initialization for the 3D structure, which is the goal of modeling. Suggested by Zvi Kam in 1980, the method of moments (MoM) offers one approach, wherein low-order statistics of the 2D images are computed and a 3D 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 3D 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 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.