On the Estimation of the Fundamental Matrix: A Convex Approach to Constrained Least-Squares

In this paper we consider the problem of estimating the fundamental matrix from point correspondences. It is well known that the most accurate estimates of this matrix are obtained by criteria minimizing geometric errors when the data are aaected by noise. It is also well known that these criteria amount to solving non-convex optimization problems and, hence, their solution is aaected by the optimization starting point. Generally, the starting point is chosen as the fundamental matrix estimated by a linear criterion but this estimate can be very inaccurate and, therefore, inadequate to initialize methods with other error criteria. Here we present a method for obtaining a more accurate estimate of the fundamental matrix with respect to the linear criterion. It consists of the minimization of the algebraic error taking into account the rank 2 constraint of the matrix. Our aim is twofold. First, we show how this non-convex optimization problem can be solved avoiding local minima using recently developed convexiication techniques. Second, we show that the estimate of the fundamental matrix obtained using our method is more accurate than the one obtained from the linear criterion, where the rank constraint of the matrix is imposed after its computation by setting the smallest singular value to zero. This suggests that our estimate can be used to initialize non-linear criteria such as the distance to epipolar lines and the gradient criterion, in order to obtain a more accurate estimate of the fundamental matrix. As a measure of the accuracy, the obtained estimates of the epipolar geometry are compared in experiments with synthetic and real data.