On the direct determination of epipoles: a case study in algebraic methods for geometric problems

Studies experimentally the problem of computing the position of the epipoles in a pair of uncalibrated images. The approach, which is based on the definition of the epipolar transformation, exploits algebraic constraints obtained from point correspondences and provides a direct solution in which only the epipoles are involved. This is in opposition with the methods based on the computation of the fundamental matrix. In order to obtain a robust solution, three families of methods are successively considered: the first one uses statistics on closed-form solutions provided by the so-called Sturm method, the second one finds the intersection of plane cubics by deterministic procedures, and the third one is based on non-linear minimizations of a difference of cross-ratios. The authors discuss the shortcomings of each of these and show, using numerous experimental comparisons, that a drastic improvement can be obtained