On the determination of epipoles using cross-ratios

We study the problem of computing the position of the epipoles in a pair of uncalibrated images. The approach, which is based on the invariance of the cross-ratio by the epipolar transformation, exploits algebraic constraints obtained from point correspondences and provides a solution in which only the epipoles are involved. This is in opposition to the methods based on the computation of the fundamental matrix. These notions are first presented as well as the epipolar ordering constraint. These families of methods are successively considered: the first uses statistics on closed-form solutions provided by the so-called Sturm method, the second used intersect plane cubics through deterministic procedures, and the third is based on nonlinear minimizations of a difference of cross-ratios. We discuss the shortcomings of each and show, using numerous experimental comparisons, that there is a trade-off between elegance and robustness to noise. The cross-ratio based methods do not turn out to be a generally viable alternative to the method based on the fundamental matrix. (C) 1998 Academic Press.