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

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