How hard is 3-view triangulation really?

Stewénius, H.; Schaffalitzky, Frederik; Nistér, D.

doi:10.1109/iccv.2005.115

Cited by 112 publications

(152 citation statements)

References 16 publications

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“…Gröbner basis methods have been used to achieve optimal solutions for several geometric reconstruction problems, such as triangulation (Stewénius et al 2005), but they do not scale well for more than a very few number of views.…”

Section: Previous Workmentioning

confidence: 99%

Globally Optimal Algorithms for Stratified Autocalibration

Chandraker,

Agarwal,

Kriegman

et al. 2009

Int J Comput Vis

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We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step.For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the inherent problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data.

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Section: Previous Workmentioning

confidence: 99%

Globally Optimal Algorithms for Stratified Autocalibration

Chandraker,

Agarwal,

Kriegman

et al. 2009

Int J Comput Vis

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show abstract

“…In parallel to this work, some recent contributions have also appeared. In [26], the three-view triangulation problem is solved with a Gröbner basis technique. In [1], fractional programming using branch-and-bound is proposed to solve a subset of the problems considered in this paper.…”

Section: Related Workmentioning

confidence: 99%

Globally Optimal Estimates for Geometric Reconstruction Problems

Kahl

Henrion

2006

Int J Comput Vision

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We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or non-optimality -or a combination of both -we pursue the goal of achieving global solutions of the statistically optimal cost-function. Our approach is based on a hierarchy of convex relaxations to solve non-convex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems

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“…In fact studies indicate (e.g. [1]) that it might not be possible to find an algorithm that is guaranteed to always work. On the other hand, under the assumption of Gaussian noise the L 2 -norm is known to give the statistically optimal solution.…”

Section: Introductionmentioning

confidence: 99%

Globally Optimal Least Squares Solutions for Quasiconvex 1D Vision Problems

Byröd

Kahl

2009

Image Analysis

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Abstract. Solutions to non-linear least squares problems play an essential role in structure and motion problems in computer vision. The predominant approach for solving these problems is a Newton like scheme which uses the hessian of the function to iteratively find a local solution. Although fast, this strategy inevitably leeds to issues with poor local minima and missed global minima.In this paper rather than trying to develop an algorithm that is guaranteed to always work, we show that it is often possible to verify that a local solution is in fact also global. We present a simple test that verifies optimality of a solution using only a few linear programs. We show on both synthetic and real data that for the vast majority of cases we are able to verify optimality. Further more we show even if the above test fails it is still often possible to verify that the local solution is global with high probability.

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How hard is 3-view triangulation really?

Cited by 112 publications

References 16 publications

Globally Optimal Algorithms for Stratified Autocalibration

Globally Optimal Algorithms for Stratified Autocalibration

Globally Optimal Estimates for Geometric Reconstruction Problems

Globally Optimal Least Squares Solutions for Quasiconvex 1D Vision Problems

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