A universally efficient algorithm and precision assessment for seamless 3D similarity transformation

Abstract

Three-dimensional (3D) similarity transformation is popularly applied for measurement datum transformation. In this study, a seamless partial errors-in-variables (EIV) model with equality constraints is established to describe the universal 3D similarity transformation problem (with arbitrary rotation angles and scale factor). Unlike the traditional transformation model, all of the random errors in the measured coordinates for common and non-common points, and their variance–covariance information can be considered. To obtain the least squares solution of this model, the constrained total least squares prediction (CTLSP) algorithm is derived using Gauss–Newton iteration and the Euler–Lagrange approach. Unnecessary matrix calculations in the application of the CTLSP algorithm for 3D datum transformation are avoided and the efficient iterative formulae are derived. Compared with the existing generalized total least squares prediction (GTLSP) algorithm, in which the transformation model is nonlinear with respect to the parameters, the CTLSP algorithm avoids the complex calculations related to the rotation matrix expressed by the trigonometric functions, and allows us to use the simple linear least squares to obtain a satisfactory initial value of the parameter vector. In addition, the linearly approximate cofactor propagation law is employed to assess the precision of the transformed coordinates of non-common points based on the CTLSP algorithm. Finally, the superiorities of CTLSP in transformation accuracy and computational efficiency are verified using an experiment. It should be noted that the new algorithm along with the precision evaluation formulae can easily be extended to the 2D/3D affine and rigid transformation cases as well, such as the map rectification, the point clouds registration, and the image matching.