Efficient Three-view Triangulation Based on 3D Optimization

Abstract

Triangulation of a 3D point from two or more views can be solved in several ways depending on how perturbations in the image coordinates are dealt with. A common approach is optimal triangulation which minimizes the total L2 reprojection error in the images, corresponding to finding a maximum likelihood estimate of the 3D point assuming independent Gaussian noise in the image spaces. Computational approaches for optimal triangulation have been published for the stereo case and, recently, also for the three-view case. In short, they solve an independent optimization problem for each 3D point, using relatively complex computations such as finding roots of high order polynomials or matrix decompositions. This paper discuss three-view triangulation and reports the following results: (1) the 3D point can be computed as multi-linear mapping (tensor) applied on the homogeneous image coordinates, (2) the set of triangulation tensors forms a 7-dimensional space determined by the camera matrices, (3) given a set of corresponding 3D/2D calibration points, the 3D residual L1 errors can be optimized over the elements in the 7-dimensional space, (4) using the resulting tensor as initial value, the error can be further reduced by tuning the tensor in a two-step iterative process, (5) the 3D residual L1 error for a set of evaluation points which lie close to the calibration set is comparable to the three-view optimal method. In summary, three-view triangulation can be done by first performing an optimization of the triangulation tensor and once this is done, triangulation can be made with 3D residual error at the same level as the optimal method, but at a much lower computational cost. This makes the proposed method attractive for real-time three-view triangulation of large data sets provided that the necessary calibration process can be performed.