Economical, Optimal and Uncertain Multiple-View L₂ Triangulation via LMIs

This paper proposes a novel approach for multiple-view $L_2$ triangulation, a key problem in computer vision which consists of estimating a scene point from its estimated image projections on two or more cameras and from the estimated projection matrices of the cameras by minimizing the reprojection error in the $L_2$ norm. In the proposed approach, the estimated image projections are allowed to be uncertain in admissible regions described by polynomial inequalities and equalities, and an estimate of the scene point is obtained by solving a linear matrix inequality (LMI) problem built with matrix decompositions, polynomial multipliers, and the Gram matrix method. It is proven that the optimal estimate can always be achieved by using multipliers with sufficiently large degree. Moreover, a simple test is provided in order to establish the optimality of the obtained estimate. As shown by some examples with real and synthetic data, the proposed approach presents key advantages with respect to several existing methods of a different nature, which may fail to find the optimal estimate, may not allow one to establish the optimality of the found estimate, or may require a larger computational burden.