Iterative approach of 3D datum transformation with a non-isotropic weight

作者：Zeng, Huaien^*; Yi, Qinglin; Wu, Yue

来源：Acta Geodaetica et Geophysica, 2016, 51(3): 557-570.

DOI：10.1007/s40328-015-0144-2

• 3D datum transformation
• Lagrangian extremum
• Non-isotropic weight
• Iterative approach
• Procrustes algorithm

The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a nonisotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a nonisotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

• 出版日期2016-9
• 单位三峡大学