Analysis of Exact Recovery using Spherical Harmonic Transform Methods for Cryo-Electron Microscopy

Abstract

In cryo-electron microscopy, the main problem is to use noisy 2D projection images to reconstruct the 3D structure of the molecule. Recent methods involve computing the autocorrelation matrix from the projections, using Kam's theory, and then estimating the coefficients in the spherical harmonic expansion of the Fourier transform of the original volume. These methods, however, require the knowledge of the exact 3D structure of some subset of the molecule. In this problem, we investigate the accuracy of methods for exact recovery involving two different sampling schemes of the spherical harmonic transform. We propose and study an invariant for optimal recovery of non-negative data - the fraction of negative part of reconstruction. By studying the recovery error of using random orthogonal matrix perturbations of varying sizes, we have found that this is indeed an invariant of exact recovery.