A Revisit of Methods for Determining the Fundamental Matrix with Planes

Determining the fundamental matrix from a collection of inter-frame homographies (more than two) is a classical problem. The compatibility relationship between the fundamental matrix and any of the ideally consistent homographies can be used to compute the fundamental matrix. Using the direct linear transformation (DLT), the compatibility equation can be translated into a least squares problem and can be easily solved via SVD decomposition. However, this solution is extremely susceptible to noise and motion inconsistencies, hence rarely used. Inspired by the normalized eight-point algorithm, we show that a relatively simple but non-Trivial two-step normalization of the input homographies achieves the desired effect, and the results are at last comparable to the less attractive hallucinated points method. The algorithm is theoretically justified and verified by experiments on both synthetic and real data.