On the determination of epipoles using cross-ratios

Abstract

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 new epipolar ordering constraint. Three families of methods are successively considered: the first uses statistics on closed form solutions provided by the so-called Sturm method, the second intersect plane cubics through deterministic procedures, and the third is based on non-linear minimizations of a difference of cross-ratios. We discuss the shortcomings of each and show, using numerous experimental comparisons, that there is a trade-o 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.