Efficient and Robust Solution to Universal Symmetric Transformation for 3-D Point Sets

Recently, the 3-D symmetric transformation, which can simultaneously consider the measurement errors of the source and target coordinates, has become a hot research topic especially in geodetic datum transformation and point clouds registration. However, there are rare studies on outlier resisting for 3-D symmetric transformation and most of them only focused on similarity transformation. In this study, to unify affine, similarity, and rigid transformations, a universal expression of 3-D symmetric transformation was abstracted as a partial errors-in-variable (EIV) model with equality constraints. Then, the equality-constrained partial EIV model was solved by the proposed constrained total least-squares (CTLS) algorithm, and the corresponding efficient formulas were derived. Finally, a robust CTLS (RCTLS) algorithm of universal 3-D symmetric transformation was presented by introducing the equivalent weight principle of robust estimation. Unlike the existing algorithms, the RCTLS can efficiently deal with all kinds of 3-D symmetric transformation problems influenced by outliers with a unified formula framework. To verify the performance of the RCTLS, comparative experiments were employed in two simulated scenarios of similarity and affine transformation and a real-world rigid transformation scenario for point clouds registration, respectively. The proposed RCTLS was compared with three nonsymmetric transformation algorithms (dual quaternion-based algorithm, linear LS, and singular value decomposition) and two symmetric transformation algorithms (outlier detected TLS (OD-TLS) and CTLS) in terms of accuracy, robustness, and computational efficiency. The results indicated that both OD-TLS and RCTLS are robust to outliers with higher transformation accuracy, but the proposed RCTLS is more universal and efficient than OD-TLS.