Conic epipolar constraints from affine correspondences

• Linear constraints on the fundamental matrix are derived from affine correspondences.
• It is shown that an affine correspondence is equivalent to 3 linear constraints.
• Two affine correspondences are shown to constrain the epipole location to a conic.
• A calculation method of the fundamental matrix by intersection of conics is proposed.

We derive an explicit relation between local affine approximations resulting from matching of affine invariant regions and the epipolar geometry in the case of a two view geometry. Most methods that employ the affine relations do so indirectly by generating pointwise correspondences from the affine relations. In this paper we derive an explicit relation between the local affine approximations and the epipolar geometry.We show that each affine approximation between images is equivalent to 3 linear constraints on the fundamental matrix and that the linear conditions guarantee the existence of an homography, compatible with the fundamental matrix. We further show that two affine relations constrain the location of the epipole to a conic section. Therefore, the location of the epipole can be extracted from 3 regions by intersecting conics.The result is further employed to derive a procedure for estimating the fundamental matrix, based on the estimated location of the epipole. It is shown to be more accurate and to require less iterations in LO-RANSAC based estimation, than the current point based approaches that employ the affine relation to generate pointwise correspondences and then calculate the fundamental matrix from the pointwise relations.