Point Triangulation through Polyhedron Collapse Using the l∞ Norm

Abstract

Multi-camera triangulation of feature points based on a minimisation of the overall 2 reprojection error can get stuck in suboptimal local minima or require slow global optimisation. For this reason, researchers have proposed optimising the ∞ norm of the 2 single view reprojection errors, which avoids the problem of local minima entirely. In this paper we present a novel method for ∞ triangulation that minimizes the ∞ norm of the ∞ 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.

Multi-View Triangulation: Systematic Comparison and an Improved Method

“…However, more recent works used the L ∞ norm in both aspects. The polyhedron collapse method proposed by Donne et al used the L ∞ norm in both the reprojection error and the aggregation error [18]. Its process only involves unary quadratic equations and some basic algebraic geometry, which is very simple and fast.…”

“…It shows that L ∞ optimization comes down to minimizing a cost function with a single minimum (local or global) on a convex domain [17]. Donné et al [18] proposed the Polyhedron collapse method based on the L ∞ norm. It adopts the L ∞ norm to quantify both the single view reprojection error and the aggregated reprojection error, and it is simpler and faster than previous methods [19], [20].…”

Coresets for Triangulation

“…For p = 1, the Dinkelbach method was used as the embedded solver for Algorithm 1. For p = ∞, the state-of-the-art polyhedron collapsed method [13] was used as the embedded solver. Evidently the results show that the coreset method is able to significantly speed up global convergence.…”

“…Unlike the sum of squared error function which contains multiple local minima, the maximum reprojection error function is quasiconvex and thus contains a single global minimum. Algorithms that take advantage of this property have been developed to solve such quasiconvex problems exactly [6], [7], [8], [9], [10], [11], [12], [13]. In particular, Agarwal et al [10] showed that some of the most effective algorithms belong to the class of generalised fractional programming (GFP) methods [14], [15].…”

Bound and Conquer: Improving Triangulation by Enforcing Consistency

“…Inspired by these difficulties, the ℓ ∞ -norm has recently been considered as a measure of the reprojection error [3], [4], [17]. Although this may not correspond to the best model of the underlying uncertainty, the resulting cost is a quasi-convex function and efficient algorithms can find the global optimum [5]. However, since the ℓ ∞norm implicitly assumes a bounded noise model, care must be taken to limit the potential catastrophic effect of outliers [18], [19].…”