We present a new method to estimate the trajectories of particle motion and the amount of cumulative beam damage in electron cryo-microscopy (cryo-EM) single particle analysis. We model the motion within the sample through the use of Gaussian Process regression. This allows us to associate with each hypothetical set of particle trajectories a prior likelihood that favours spatially coherent and temporally smooth motion without imposing hard constraints. This formulation enables us to express the a-posteriori likelihood of a set of particle trajectories as a product of that prior likelihood and an observation likelihood given by the data, and to then maximise this a-posteriori likelihood. Since our smoothness prior requires three parameters that describe the statistics of the observed motion, we also propose an efficient stochastic method to estimate those parameters. Finally, we propose a practical means of estimating the average amount of cumulative radiation damage as a function of radiation dose and spatial frequency, and a robust method of fitting relative B-factors to it. We evaluate our method on three publicly available datasets, and illustrate its usefulness by comparison with state-of-the-art methods and previously published results. The new method has been implemented as Bayesian polishing in RELION-3, where it replaces the existing particle polishing method, as it outperforms the latter in all tests performed.
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