Rank-Constrained Fundamental Matrix Estimation by Polynomial Global Optimization Versus the Eight-Point Algorithm

Abstract

The fundamental matrix can be estimated from point matches. The current gold standard is to bootstrap the eight-point algorithm and two-view projective bundle adjustment. The eight-point algorithm first computes a simple linear least squares solution by minimizing an algebraic cost and then projects the result to the closest rank-deficient matrix. We propose a single-step method that solves both steps of the eight-point algorithm. Using recent results from polynomial global optimization, our method finds the rank-deficient matrix that exactly minimizes the algebraic cost. In this special case, the optimization method is reduced to the resolution of very short sequences of convex linear problems which are computationally efficient and numerically stable. The current gold standard is known to be extremely effective but is nonetheless outperformed by our rank-constrained method for bootstrapping bundle adjustment. This is here demonstrated on simulated and standard real datasets. With our initialization, bundle adjustment consistently finds a better local minimum (achieves a lower reprojection error) and takes less iterations to converge.

A Practical Rank-Constrained Eight-Point Algorithm for Fundamental Matrix Estimation

“…More seriously, it suffers from some singularities caused by improper parametrization 2 , thus inapplicable to some otherwise well-posed camera configurations (provided nondegenerate parametrization), such as pure camera translation along the horizontal axis. In a very recent online manuscript, Bugarin et al [3] avoided these singularities by using the determinant constraint and advocated moment relaxation to solve the resulting polynomial optimization problem. The cost is to solving a hierarchy of SDP relaxation problems of increasing size, whose computational burden is even higher than the SOS relaxation in [5].…”

“…Similar to [3,5], it is based on the algebraic error, and copes with the rank-2 constraint directly. By carefully investigating the linear dependence between the three columns of a fundamental matrix, we solve seven subproblems so as to avoid improper singularities and keep the resulting optimization problems tractable.…”

“…Sec. 3 shows the details of solving polynomial systems, with emphasis on the generalized eigenvalue factorization method for multivariate systems. We show extensive experiment results in Sec.4 and conclude this work in Sec.5.…”

“…In order to avoid the drawback of posterior rank-2 correction in [8,12], it is mandatory to incorporate the rank constraint in the minimization process. To this end, we can add the determinant constraint [3,24], which leads to a challenging polynomial optimization problem. On the contrary, it is possible to parameterize the fundamental matrix properly such that the rank-2 constraint is interiorly satisfied.…”

A convex optimization approach to robust fundamental matrix estimation

“…In [16] a Levenberg-Marquard (LM) approach is proposed to optimize the singular value decomposition (SVD) of the fundamental matrix. In [20] and [1] the rank constraint is imposed by setting its determinant to 0, leading to a 3rd-order polynomial constraint. Alternatively, in [2] and [21] the estimation problem is reduced to one or several constrained polynomial optimization problems by imposing the constraint that the null space of the solution must contain a non-zero vector.…”