The paper proposes a geometric error based triangulation and metric study of lines. As one of the fundamental problems in computer vision, line triangulation is to determine the 3D coordinates of a line based on its 2D image projections from more than two views of cameras. Compared to point features, line segments are more robust to matching errors, occlusions, and image uncertainties. However, when the number of views is larger than two, the back-projection planes usually do not intersect at one line due to measurement errors and image noise. Thus, it is critical to find a 3D line that optimally fits the measured data. This paper, at first time, provides a comprehensive study on the triangulation and metric of lines based on geometric error.

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.