Joint Orientation of Epipoles

It is known that epipolar constraint can be augmented with orientation by formulating it in the oriented projective geometry. This oriented epipolar constraint requires knowing the orientations (signs of overall scales) of epipoles and fundamental matrix. The current belief is that these orientations cannot be obtained from the fundamental matrix only and that additional information is needed, typically, a single correct point correspondence. In contrary to this, we show that fundamental matrix alone encodes orientation of epipoles up to their common scale sign. We present two formulations of this fact. The algebraic formulation gives a closed formula to compute the second epipole from fundamental matrix and the first epipole. The geometric formulation is in terms of the conic formed by intersections of corresponding epipolar lines in the common image plane; we show that the epipoles always lie on different antipodal components of the spherical interpretation of this conic. Further, we show that, under mild assumptions, fundamental matrix can discriminate between two classes of mutual position of a pair of directional cameras.