Electron cryo-microscopy (cryoEM) involves the estimation of orientations of projection images or three-dimensional (3D) volumes. However, the lack of statistical tools of rotations in cryoEM fails to answer the growing demands for adopting advanced statistical methods. In this study, we develop a comprehensive statistical tool specialized for cryoEM based on an unit quaternion description of spatial rotations. Some basic properties and definitions of the quaternion, as well as a way to use the unit quaternion to describe and perform rotations, are first recalled. Then, based on the unit quaternion, the distance and geodesic between rotations are designed for cryoEM to enable comparisons and interpolations between rotations, which are prerequisites of statistics of rotations in 3D cryoEM. Further, methods of directional statistics specialized for cryoEM are developed, including calculations of the average rotation, sampling, and inference with uniform and angular central Gaussian (ACG) distribution, as well as an estimation of the rotation precision. Finally, the method of handling molecular symmetry is introduced. Using the unit quaternion system for cryoEM, we provide comprehensive mathematical tools for the analysis of spatial rotations in cryoEM.
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