Polar Transformers: $SO(2)^K$-Equivariant Angular Attention for Cryo-EM Image-Set Processing

Many projection-based imaging modalities reconstruct unknown objects from collections of noisy projections related by unknown in-plane rotations, making effective estimation dependent on rotationally consistent alignment and aggregation. In cryogenic electron microscopy (cryo-EM), the extremely low signal-to-noise ratio (SNR) makes such multi-image integration essential. We introduce the *polar transformer*, a permutation-invariant architecture for *sets* of images built around an $\mathrm{SO}(2)^K$-equivariant *angular attention* mechanism that jointly aligns and aggregates information across images. This architecture leverages a stable polar representation in which we define polar CNN feature extractors and evaluate attention weights over all relative angular shifts efficiently via FFT-based correlations, guaranteeing equivariance to independent in-plane rotations of each image in the set. We evaluate the polar transformer on simulated cryo-EM images using denoising as a quantitative proxy. Across different regimes, polar transformers outperform strong single-image and image-set baselines, achieving a reduction of up to 40% in relative MSE at an SNR of 0.02 and improving downstream ab initio reconstruction resolution by 20%.