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

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.