Triangulation made easy

We describe a simple and efficient algorithm for two-view triangulation of 3D points from approximate 2D matches based on minimizing the L₂ reprojection error. Our iterative algorithm improves on the one by Kanatani et al. by ensuring that in each iteration the epipolar constraint is satisfied. In the case where the two cameras are pointed in the same direction, the method provably converges to an optimal solution in exactly two iterations. For more general camera poses, two iterations are sufficient to achieve convergence to machine precision, which we exploit to devise a fast, non-iterative method. The resulting algorithm amounts to little more than solving a quadratic equation, and involves a fixed, small number of simple matrix-vector operations and no conditional branches. We demonstrate that the method computes solutions that agree to very high precision with those of Hartley and Sturm´s original polynomial method, though achieves higher numerical stability and 1-4 orders of magnitude greater speed.