A complete solution of an improved universal 3D coordinate similarity transformation model

Abstract When linearizing three-dimensional (3D) coordinate similarity transformation model with large rotations, we usually encounter the ill-posed normal matrix and this problem may aggravate the instability of solutions. To alleviate the problem, a series of conversions are contributed to the 3D coordinate similarity transformation model in this paper. We deduced a complete solution for the 3D coordinate similarity transformation at any rotation with the nonlinear adjustment methodology, which involves the errors of the common and the non-common points. Furthermore, as the large condition number of the normal matrix resulted in an intractable form, we introduced the bary-centralization technique and a surrogate process for deterministic element of the normal matrix, and proved its benefit for alleviating the condition number. The experimental results show that our approach can obtain the smaller condition number to stabilize the convergence of the interested parameters. Especially, our approach can be implemented for considering the errors of the common and the non-common points, thus the accuracy of the transformed coordinates improves.