8-point algorithm revisited: factorized 8-point algorithm

In this paper, a novel algorithm for the fundamental matrix estimation, called factorized 8-point algorithm, is presented. The factorized 8-point algorithm is composed of three steps: (1) The measurement matrix in the traditional 8-point algorithm is decomposed into two factor matrices; (2) By introducing some auxiliary variables, a new linear minimization problem is formed, where every element of its associated measurement matrix is simply either a measurement datum or a constant; (3) The fundamental matrix is determined by solving this minimization problem by a least squares method. Like the traditional 8-point algorithm and Hartley's normalized 8-point algorithm, the factorized 8-point algorithm is also completely linear. But unlike the normalized 8-point algorithm, the factorized 8-point algorithm does not need any pre-normalization step. Since every element of the measurement matrix in the factorized 8-point algorithm is a measurement datum or a constant, no amplification of measurement error is involved; the factorized 8-point algorithm can boost effectively the robustness of the estimation. Large numbers of experiments show that the factorized 8-point algorithm consistently outperforms the traditional 8-point algorithm. In addition, although the factorized 8-point algorithm is specially designed for fundamental matrix estimation, its basic principle can be generalized to other estimation problems in computer vision, such as camera projection matrix estimation, homography estimation, focus of expansion estimation, and trifocal tensor estimation.