On triangulation algorithms in large scale camera network systems

Geometric triangulation is at the basis of the estimation of the 3D position of a target from a set of camera measurements. The problem of optimal estimation (minimizing the L
₂
norm) of the target position from multi-view perspective projective measurements is typically a hard problem to solve. In literature there are different types of algorithms for this purpose, based for example on the exhaustive check of all the local minima of a proper eigenvalue problem [2], or branch-and-bound techniques [3]. However, such methods typically become unfeasible for real time applications when the number of cameras and targets become large, calling for the definition of approximate procedures to solve the reconstruction problem. In the first part of this paper, linear (fast) algorithms, computing an approximate solution to such problems, are described and compared in simulation. Then, in the second part, a Gaussian approximation to the measurement error is used to express the reconstruction error's standard deviation as a function of the position of the reconstructed point. An upper bound, valid over all the target domain, to this expression is obtained for a case of interest. Such upper bound allows to compute a number of cameras sufficient to obtain a user defined level of position estimation accuracy.