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Unit quaternion description of spatial rotation in 3D electron cryo-microscopy

Electron cryo-microscopy (cryoEM) involves the estimation of orientations of projection images or three-dimensional (3D) volumes, and it is therefore important to properly describe and operate spatial rotations. However, the current Euler angle system that is widely adopted in cryoEM has many drawbacks, which limit the applications of advanced statistical methods in cryoEM. In this study, we introduce and develop a unit quaternion system specialized for cryoEM to replace the conventional Euler angle system. We first introduce the basic definitions and properties of a unit quaternion and rotational operation, as well as a swing-twist decomposition method. Then, the concepts of distance and geodesic between rotations is introduced and developed for cryoEM to enable comparisons and interpolations between rotations. 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 unit quaternion description of molecular symmetry is introduced. Using the unit quaternion system for cryoEM, we provide a comprehensive mathematical tool for the operation and analysis of the spatial rotations in cryoEM.