Bayesian perspective for orientation determination in cryo-EM with application to structural heterogeneity analysis

Xu, S.; Balanov, A.; Singer, A.; Bendory, T.

doi:10.1107/S2059798326001415

Figure 2
Impact of incorporating the prior rotation distribution on estimation accuracy. Simulations are performed for the cryo-ET model (equation 2

), excluding the projection step. The true rotation distribution is modeled as an isotropic Gaussian $[{\cal IG}_{{\sf{SO}}(3)}(\eta = 0.1)]$ . Estimation performance is measured using the geodesic distance defined in equation (10)

. Here, g denotes the true rotation, $[\hat{g}_{\rm {MLE}}]$ is the maximum-likelihood estimator from equation (24)

, $[\hat{g}_{\rm {MAP}}]$ is the maximum a posteriori estimator from equation (13)

and $[{\hat g}_{\rm {MMSE}}]$ denotes the Bayesian minimum mean-square error estimator from equation (21)

. The MMSE estimators are computed assuming isotropic Gaussian priors on $[{\sf{SO}}(3)]$ with different concentration parameters η ∈ {0.5, 0.1} (see Appendix E

). As η decreases, the prior becomes more concentrated and closer to the true underlying distribution, leading to improved accuracy of both the MAP and MMSE estimators. Each data point in the plot is averaged over 3000 Monte Carlo trials using a rotation grid of size L = 2976. Bottom 2 × 2 panel: the four images compare denoised 3D volumes obtained using different rotation estimators at two representative noise regimes marked on the curve plot. Rows correspond to the estimator: the top row uses $[{\hat g}_{\rm {MLE}}]$ and the bottom row uses $[{\hat g}_{\rm {MMSE}}]$ with the true prior $[{\cal IG}_{{\sf{SO}}(3)}(\eta = 0.1)]$ . Columns correspond to SNR: the left column is a low-SNR example and the right column is a high-SNR example (where the true rotation is correctly classified).

STRUCTURAL
BIOLOGY

ISSN: 2059-7983

Volume 82| Part 4| April 2026| Pages 274-294

https://doi.org/10.1107/S2059798326001415

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