Electron cryo-microscopy (cryoEM) involves the estimation of spatial rotations, or saying orientations, of projection images or three-dimensional (3D) volumes. Euler angle system is widely used to describe spatial rotations in most cryoEM algorithms and software. In this review, we introduce unit quaternion as an alternate to Euler angles for describing spatial rotations, customize and develop corresponding tools for increasing demands of statistical analysis of spatial rotations in cryoEM. Some basic properties and definitions of quaternion are first recalled. Thereafter, distance and geodesic between rotations are introduced to aid comparisons and interpolations between rotations, which are prerequisites of statistics of rotations in 3D cryoEM. Furthermore, statistics of rotations are reviewed. Techniques potentially useful in cryoEM, such as calculations of the average rotation, generation of quasi-regular grids, sampling, inference with uniform distribution and angular central Gaussian (ACG) distribution, and estimation of rotation precision, are reviewed and developed. Finally, molecular symmetry presented in unit quaternion form is discussed. Unit quaternion system is shown as a convenient and comprehensive mathematical tool for cryoEM.

电子冷冻显微镜 (cryoEM) 涉及对投影图像或三维 (3D) 体积的空间旋转或方向的估计。在大多数冷冻电镜算法和软件中，欧拉角系统被广泛用于描述空间旋转。在这篇综述中，我们引入了单位四元数作为欧拉角的替代品来描述空间旋转，定制和开发相应的工具，以增加冷冻电镜中空间旋转统计分析的需求。首先回顾一下四元数的一些基本性质和定义。此后，引入了旋转之间的距离和测地线，以帮助旋转之间的比较和插值，这是 3D 冷冻电镜中旋转统计的先决条件。此外，还审查了轮换的统计数据。在冷冻电镜中可能有用的技术，回顾和发展了诸如平均旋转的计算、准规则网格的生成、采样、均匀分布和角中心高斯(ACG)分布的推理以及旋转精度的估计。最后，讨论了以单位四元数形式表示的分子对称性。单位四元数系统是一种方便且全面的冷冻电镜数学工具。