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We develop in this paper a novel intrinsic classification algorithm—multi-frequency class averaging (MFCA)—for classifying noisy projection images obtained from three-dimensional cryo-electron microscopy by the similarity among their viewing directions. This new algorithm leverages multiple irreducible representations of the unitary group to introduce additional redundancy into the representation of the optimal in-plane rotational alignment, extending and outperforming the existing class averaging algorithm that uses only a single representation. The formal algebraic model and representation theoretic patterns of the proposed MFCA algorithm extend the framework of Hadani and Singer to arbitrary irreducible representations of the unitary group. We conceptually establish the consistency and stability of MFCA by inspecting the spectral properties of a generalized local parallel transport operator through the lens of Wigner $D$-matrices. We demonstrate the efficacy of the proposed algorithm with numerical experiments.

Representation theoretic patterns in multi-frequency class averaging for three-dimensional cryo-electron microscopy