Sequential L _∞ Norm Minimization for Triangulation

Abstract

It has been shown that various geometric vision problems such as triangulation and pose estimation can be solved optimally by minimizing L _∞ error norm. This paper proposes a novel algorithm for sequential estimation. When a measurement is given at a time instance, applying the original batch bi-section algorithm is very much inefficient because the number of seocnd order constraints increases as time goes on and hence the computational cost increases accordingly. This paper shows that, the upper and lower bounds, which are two input parameters of the bi-section method, can be updated through the time sequence so that the gap between the two bounds is kept as small as possible. Furthermore, we may use only a subset of all the given measurements for the L _∞ estimation. This reduces the number of constraints drastically. Finally, we do not have to re-estimate the parameter when the reprojection error of the measurement is smaller than the estimation error. These three provide a very fast L _∞ estimation through the sequence; our method is suitable for real-time or on-line sequential processing under L _∞ optimality. This paper particularly focuses on the triangulation problem, but the algorithm is general enough to be applied to any L _∞ problems.