Fast algorithms for L<inf>&#x221E;</inf> problems in multiview geometry

Abstract

Many problems in multi-view geometry, when posed as minimization of the maximum reprojection error across observations, can be solved optimally in polynomial time. We show that these problems are instances of a convex-concave generalized fractional program. We survey the major solution methods for solving problems of this form and present them in a unified framework centered around a single parametric optimization problem. We propose two new algorithms and show that the algorithm proposed by Olsson et al. [21] is a special case of a classical algorithm for generalized fractional programming. The performance of all the algorithms is compared on a variety of datasets, and the algorithm proposed by Gugat [12] stands out as a clearwinner. An open source MATLAB toolbox thats implements all the algorithms presented here is made available.

Point Triangulation through Polyhedron Collapse Using the l&#x221E; Norm

“…Due to the nonlinearity of the camera projection, γ(r) can have multiple local minima when defined in terms of the 2 norm [2,6,9]. In contrast, aggregating the reprojection errors with the ∞ norm always results in a convex set of stationary points because of its pseudo-convex character [1,3,5,12,16,17].…”

“…This quasi-convexity implies that the set of stationary points is convex: no local optima exist. Yet even for many existing ∞ methods, single view reprojection errors are expressed in terms of the 2 norm; we will call them hybrid ∞ methods [1,3,12,17]. We note that the criterion optimized by the hybrid approaches cannot easily be related to the maximum-likelihood estimation of a noise model.…”

“…Olsson et al [17] also propose an approach based on a series of auxiliary problems, but not based on bisection: for their method, the auxiliary problems are local approximations to the original problem such that the KKT criteria are a simple approximation to the KKT criteria of the original problem. In [1], Agarwal et al present a survey of the ∞ norm for aggregating reprojection errors. They present the Gugat method which outperforms the techniques of Olsson et al [17] and Kahl [12] et al while still being based on a series of SOCP problems.…”

“…Some existing methods use a mixed-norm: the ∞ norm for aggregation but the 2 norm for the reprojection error [1,3,12,17]. Minimizing this aggregated error leads to a series of auxiliary second-order cone problems which are complex and time consuming to solve.…”

Globally Optimal Algorithms for Stratified Autocalibration

“…Triangulation and resectioning are solved with a certificate of optimality using convex relaxation techniques for fractional programs in (Agarwal et al 2006). Several geometric problems in computer vision, when posed in the L ∞ -norm, can be solved to their global optimum using techniques of quasiconvex optimization (Kahl 2005;Sim and Hartley 2006;Agarwal et al 2008). A survey of some of the recent work in developing optimal algorithms for multiview geometry can be found in (Hartley and Kahl 2007).…”