For decades, it has been widely accepted that the gold standard for two-view triangulation is to minimize the cost based on reprojection errors. In this work, we challenge this idea. We propose a novel alternative to the classic midpoint method that leads to significantly lower 2D errors and parallax errors. It provides a numerically stable closed-form solution based solely on a pair of backprojected rays. Since our solution is rotationally invariant, it can also be applied for fisheye and omnidirectional cameras. We show that for small parallax angles, our method outperforms the state-of-the-art in terms of combined 2D, 3D and parallax accuracy, while achieving comparable speed.
@cite_35: 
@cite_9: Linear subspace has many important applications in computer vision, such as structure from motion, motion estimation, layer extraction, object recognition, and object tracking. Singular Value Decomposition (SVD) algorithm is a standard technique to compute the subspace from the input data. The SVD algorithm, however, is sensitive to outliers as it uses L2 norm metric, and it can not handle missing data either. In this paper, we propose using L1 norm metric to compute the subspace. We show that it is robust to outliers and can handle missing data. We present two algorithms to optimize the L1 norm metric: the weighted median algorithm and the quadratic programming algorithm. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Carnegie Mellon University or the U.S. Government.
@cite_1: This thesis describes a system that completely automaticallybuilds a three-dimensional model of a scene given a sequence ofimages of the scene. The system also estimates the internalparameters of t ...
@cite_16: We investigate the use of the L∞ cost function in geometric vision problems. This cost function measures the maximum of a set of model-fitting errors, rather than the sumof-squares, or L2 cost function that is commonly used (in least-squares fitting). We investigate its use in two problems; multiview triangulation and motion recovery from omnidirectionalcameras, though the results may also apply to other related problems. It is shown that for these problems the L∞ cost function is significantly simpler than the L2 cost. In particular L∞ minimization involvesfinding the minimum of a cost function with a single local (and hence global)minimumon a convexparameter domain. Theproblem may be recast as a constrained minimization problem and solved using commonly available software. The optimal solution was reliably achieved on problems of small dimension.
@cite_27: A higher order scheme is presented for the optimal correction method of Kanatani [5] for triangulation from two views and is compared with the method of Hartley and Sturm [3]. It is pointed out that the epipole is a singularity of the Hartley-Sturm method, while the proposed method has no singularity. Numerical simulation confirms that both compute identical solutions at other points. However, the proposed method is significantly faster.
@cite_23: Abstract An explicit three-dimensional (3D) representation is constructed from feature points extracted from a sequence of images taken by a moving camera. The points are tracked through the sequence, and their 3D locations are accurately determined by use of Kalman filters. The egomotion of the camera is also determined.
@cite_34: This is a survey paper summarizing recent research aimed at finding guaranteed optimal algorithms for solving problems in Multiview Geometry. Many of the traditional problems in Multiview Geometry now have optimal solutions in terms of minimizing residual imageplane error. Success has been achieved in minimizing L2 (least-squares) or L∞ (smallest maximum error) norm. The main methods involve Second Order Cone Programming, or quasi-convex optimization, and Branch-andbound. The paper gives an overview of the subject while avoiding as far as possible the mathematical details, which can be found in the original papers.
refer to those triangulation methods that minimize the cost based on reprojection errors. Assuming that the image measurements are independently perturbed by the noise in the same distribution of certain types, the optimal methods find the maximum likelihood (ML) solution. For Gaussian and Laplacian distribution, the ML solution is to minimize the @math norm or @math norm of the reprojection errors, respectively @cite_9 . This can be found in closed form by solving a polynomial of degree six or eight @cite_23 . Alternatively, the @math solution can be obtained using iterative correction methods @cite_27 @cite_35 . While these iterative methods do not guarantee global optimality, they were shown to be faster and more stable. For a uniform distribution, minimizing the @math norm leads to the ML estimate for the lower bound of the noise @cite_34 , and the solution is obtained by solving a quartic polynomial @cite_1 . Unlike the @math and @math cost, the @math cost has a simple shape with a single minimum, but it is relatively more sensitive to noise and outliers @cite_16 .