title: Algebraic Error Based Triangulation and Metric of Lines.
abstract: Line triangulation, a classical geometric 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 with known projection matrices. Compared to point features, line segments are more robust to matching errors, occlusions, and image uncertainties. In addition to line triangulation, a better metric is needed to evaluate 3D errors of line triangulation. In this paper, the line triangulation problem is investigated by using the Lagrange multipliers theory. The main contributions include: (i) Based on the Lagrange multipliers theory, a formula to compute the Plücker correction is provided, and from the formula, a new linear algorithm, LINa, is proposed for line triangulation; (ii) two optimal algorithms, OPTa-I and OPTa-II, are proposed by minimizing the algebraic error; and (iii) two metrics on 3D line space, the orthogonal metric and the quasi-Riemannian metric, are introduced for the evaluation of line triangulations. Extensive experiments on synthetic data and real images are carried out to validate and demonstrate the effectiveness of the proposed algorithms.
authors.affiliation: National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing, 100190, China.
authors.forename: Fuchao
authors.initials: F
authors.lastname: Wu
authors.affiliation: National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing, 100190, China.
authors.forename: Ming
authors.initials: M
authors.lastname: Zhang
authors.affiliation: Department of Electrical Engineering & Computer Science, University of Kansas, 1520 West 15th Street, Lawrence, KS, 66045-7608, United States of America.
authors.forename: Guanghui
authors.initials: G
authors.lastname: Wang
authors.affiliation: National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing, 100190, China.
authors.forename: Zhanyi
authors.initials: Z
authors.lastname: Hu
mesh_terms: D008433:Mathematics; D008962:Models, Theoretical