  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 (cryo-EM) 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.
