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

Yifeng Fan, Tingran Gao, Zhizhen Zhao

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)723-771
Number of pages49
JournalInformation and Inference
Issue number3
StatePublished - Sep 1 2021


  • Wigner matrices
  • cryo-electron microscopy
  • differential geometry
  • mathematical biology
  • representation theory
  • spectral theory

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Numerical Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics


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