Point Triangulation through Polyhedron Collapse Using the l∞ Norm

Multi-camera triangulation of feature points based on a minimisation of the overall ℓ₂ reprojection error can get stuck in suboptimal local minima or require slow global optimisation. For this reason, researchers have proposed optimising the ℓ_∞ norm of the ℓ₂ single view reprojection errors, which avoids the problem of local minima entirely. In this paper we present a novel method for ℓ_∞ triangulation that minimizes the ℓ_∞ norm of the ℓ_∞ reprojection errors: this apparently small difference leads to a much faster but equally accurate solution which is related to the MLE under the assumption of uniform noise. The proposed method adopts a new optimisation strategy based on solving simple quadratic equations. This stands in contrast with the fastest existing methods, which solve a sequence of more complex auxiliary Linear Programming or Second Order Cone Problems. The proposed algorithm performs well: for triangulation, it achieves the same accuracy as existing techniques while executing faster and being straightforward to implement.