Efficient triangulation based on 3D Euclidean optimization

Abstract

This paper presents a method for triangulation of 3D points given their projections in two images. Recent results show that the triangulation mapping can be represented as a linear operator K applied to the outer product of corresponding homogeneous image coordinates, leading to a triangulation of very low computational complexity. K can be determined from the camera matrices, together with a so-called blind plane, but we show here that it can be further refined by a process similar to gold standard methods for camera matrix estimation. In particular, it is demonstrated that K can be adjusted to minimize the Euclidean L1 residual 3D error, bringing it down to the same level as the optimal triangulation by Hartley and Sturm. The resulting K optimally fits a set of 2D+2D+3D data where the error is measured in the 3D space. Assuming that this calibration set is representative for a particular application, where later only the 2D points are known, this K can be used for triangulation of 3D points in an optimal way, which in addition is very efficient since the optimization need only be made once for the point set. The refinement of K is made by iteratively reducing errors in the 3D and 2D domains, respectively. Experiments on real data suggests that very few iterations are needed to accomplish useful results.