Line triangulation, as a classical problem in computer vision, is to determine the 3D coordinates of a line based on its 2D image projections from more than two views of cameras. Classical approaches for line triangulation are based on algebraic errors, which do not have any geometrical meaning. In addition, an effective metric to evaluate 3D errors of line triangulation is not available in the literature. In this paper, a comprehensive study of line triangulation is conducted using geometric cost functions. Compared to the algebraic error based approaches, geometric error based algorithm is more meaningful, and thus, yields better estimation results. The main contributions of this study include: (i) it is proved that the optimal solution to minimizing the geometric errors can be transformed to finding the real roots of algebraic equations; (ii) an effective iterative algorithm, ITEg, is proposed to minimizing the geometric errors; and (iii) an in-depth comparative evaluations on three metrics in 3D line space, the Euclidean metric, the orthogonal metric, and the quasi-Riemannian metric, are carried out. Extensive experiments on synthetic data and real images are carried out to validate and demonstrate the effectiveness of the proposed algorithms. 
Triangulation and metric of lines based on geometric error