A Stable and Accurate Algorithm for Computing Epipolar Geometry

Abstract

This paper addresses the problem of computing the fundamental matrix which describes a geometric relationship between a pair of stereo images: the epipolar geometry. In the uncalibrated case, epipolar geometry captures all the 3D information available from the scene. It is of central importance for problems such as 3D reconstruction, self-calibration and feature tracking. Hence, the computation of the fundamental matrix is of great interest. The existing classical methods14 use two steps: a linear step followed by a nonlinear one. However, in some cases, the linear step does not yield a close form solution for the fundamental matrix, resulting in more iterations for the nonlinear step which is not guaranteed to converge to the correct solution. In this paper, a novel method based on virtual parallax is proposed. The problem is formulated differently; instead of computing directly the 3 × 3 fundamental matrix, we compute a homography with one epipole position, and show that this is equivalent to computing the fundamental matrix. Simple equations are derived by reducing the number of parameters to estimate. As a consequence, we obtain an accurate fundamental matrix with a stable linear computation. Experiments with simulated and real images validate our method and clearly show the improvement over the classical 8-point method.