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

Abstract

Recently, the 3D 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 3D 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 3D symmetric transformation was abstracted as a partial EIV model with equality constraints. Then the equality constrained partial EIV model was solved by a proposed constrained total least squares (CTLS) algorithm and the corresponding efficient formulae were derived. Finally, a robust CTLS (RCTLS) algorithm of universal 3D 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 3D 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 non-symmetric transformation algorithms (dual quaternion-based algorithm, linear least squares 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 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.