Simple, Fast and Accurate Estimation of the Fundamental Matrix Using the Extended Eight-Point Schemes

The eight-point scheme is the simplest and fastest scheme for estimating the fundamental matrix (FM) from a number of noisy correspondences. As it ignores the fact that the FM must be singular, the resulting FM estimate is often inaccurate. Existing schemes that take the singularity constraint into consideration are several times slower and significantly more difficult to implement and understand. This paper describes extended versions of the eight-point (8P) and the weighted eight-point (W8P) schemes that effectively take the singularity constraint into consideration without sacrificing the efficiency and the simplicity of both schemes. The proposed schemes are respectively called the extended eight-point scheme (E8P) and the extended weighted eight-point scheme (EW8P). The E8P scheme was experimentally found to give exactly the same results as Hartley’s algebraic distance minimization scheme while being almost as fast as the simplest scheme (i.e., the 8P scheme). At the expense of extra calculations per iteration, the EW8P scheme permits the use of geometric cost functions and, more importantly, robust weighting functions. It was experimentally found to give near-optimal results while being 8-16 times faster than the more complicated schemes such as Levenberg-Marquardt schemes. The FM estimates obtained by the E8P and the EW8P schemes perfectly satisfy the singularity constraint, eliminating the need to enforce the rank-2 constraint in a post-processing step.