Representation theoretic patterns in three dimensional cryo-electron microscopy II - The class averaging problem

TL;DR

This paper provides a rigorous mathematical foundation for an algebraic classification algorithm in cryo-EM, explaining its correctness and stability through spectral analysis of geometric operators on the sphere.

Contribution

It offers a formal algebraic and spectral analysis of the class averaging algorithm in cryo-EM, establishing its correctness and stability.

Findings

Proof of the algorithm's correctness

Spectral analysis of the localized parallel transport operator

Enhanced representation theoretic framework for cryo-EM

Abstract

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced by Hadani, Shkolnisky, Singer and Zhao, for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability, thus putting it on firm mathematical grounds. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two dimensional sphere, called the localized parallel transport operator. Along the way, we develop further the representation theoretic set-up for three-dimensional cryo-EM.