  In single particle reconstruction (SPR) from cryo-electron microscopy
(cryo-EM), the 3D structure of a molecule needs to be determined from its 2D
projection images taken at unknown viewing directions. Zvi Kam showed already
in 1980 that the autocorrelation function of the 3D molecule over the rotation
group SO(3) can be estimated from 2D projection images whose viewing directions
are uniformly distributed over the sphere. The autocorrelation function
determines the expansion coefficients of the 3D molecule in spherical harmonics
up to an orthogonal matrix of size $(2l+1)\times (2l+1)$ for each
$l=0,1,2,...$. In this paper we show how techniques for solving the phase
retrieval problem in X-ray crystallography can be modified for the cryo-EM
setup for retrieving the missing orthogonal matrices. Specifically, we present
two new approaches that we term Orthogonal Extension and Orthogonal
Replacement, in which the main algorithmic components are the singular value
decomposition and semidefinite programming. We demonstrate the utility of these
approaches through numerical experiments on simulated data.
