Analytical dual quaternion algorithm of the weighted three-dimensional coordinate transformation

• Published in: Earth, planets, and space Vol. 74; no. 1; pp. 1 - 16
• Main Authors: Zeng, Huaien, Wang, Junjie, Wang, Zhihao, Li, Siyang, He, Haiqing, Chang, Guobin, Yang, Ronghua
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022; Springer; Springer Nature B.V; SpringerOpen
• Subjects: 6. Geodesy; Algorithms; Analytical dual quaternion algorithm (ADQA); Coordinate transformations; Dual quaternion; Earth and Environmental Science; Earth Sciences; Geology; Geophysics/Geodesy; Lidar; Mathematical models; Modified procrustes algorithm (MPA); Optical radar; Orthonormal Matrix Algorithm (OMA); Parameters; Quaternions; Remote sensing; Rotating matter; Rotation; Transformation accuracy; Weighted three-dimensional coordinate transformation; China; Transformation accuracy; Analytical dual quaternion algorithm (ADQA); Weighted three-dimensional coordinate transformation; Orthonormal Matrix Algorithm (OMA); Dual quaternion; Modified procrustes algorithm (MPA)
• ISSN: 1880-5981; 1343-8832; 1880-5981
• DOI: 10.1186/s40623-022-01731-1

Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. Graphical Abstract