Retrieval of Euler rotation angles from 3D similarity transformation based on quaternions

Recently, it has been shown how quaternion-based representation of a rotation matrix has advantages over conventional Eulerian representation in 3D similarity transformations. The iterative estimation procedure in similarity transformations based on quaternions results in translations and (scaled) quaternion elements. One needs, therefore, an additional procedure for evaluating the rest of the transformation parameters (translation, scale factor and rotation angles) after this solution.This contribution shows how to evaluate the rotation angles and the full covariance matrix of the transformation parameters from the estimation results in asymmetric and symmetric 3D similarity transformations based on quaternions.

近年来，已有研究表明，在三维相似变换（3D similarity transformations）中，旋转矩阵的四元数（quaternion）表示法相较于传统欧拉角表示法（Eulerian representation）具有显著优势。基于四元数的相似变换迭代估计流程可得到平移量与（缩放后的）四元数元素，因此在获得该初步解后，还需通过额外的处理流程来求解变换的其余参数：平移、尺度因子与旋转角。本文提出了一种方法，可从基于四元数的非对称与对称三维相似变换的估计结果中，求解得到旋转角以及变换参数的完整协方差矩阵。