Optimal Triangulation Method is Not Really Optimal

TL;DR

This paper challenges the common belief that the optimal triangulation method is best, showing that the simple mid-point method outperforms it under camera parameter uncertainty, especially in structure-from-motion tasks.

Contribution

It demonstrates that the widely used L2 optimal triangulation method is suboptimal when camera uncertainties exist and advocates for the mid-point method as a better alternative.

Findings

Mid-point method outperforms L2 in uncertain camera scenarios.

L2 method is complex for multiple cameras, mid-point has a closed form.

Mid-point method is recommended for structure-from-motion applications.

Abstract

Triangulation refers to the problem of finding a 3D point from its 2D projections on multiple camera images. For solving this problem, it is the common practice to use so-called optimal triangulation method, which we call the L2 method in this paper. But, the method can be optimal only if we assume no uncertainty in the camera parameters. Through extensive comparison on synthetic and real data, we observed that the L2 method is actually not the best choice when there is uncertainty in the camera parameters. Interestingly, it can be observed that the simple mid-point method outperforms other methods. Apart from its high performance, the mid-point method has a simple closed formed solution for multiple camera images while the L2 method is hard to be used for more than two camera images. Therefore, in contrast to the common practice, we argue that the simple mid-point method should be used in structure-from-motion applications where there is uncertainty in camera parameters.