3d Aspects of 2d Epipolar Geometry

Relative orientation in a stereo pair (establishing 3D epipolar geometry) is generally described as a rigid body transformation, with one arbitrary translation component, between two formed bundles of rays. In the uncalibrated case, however, only the 2D projective pencils of epipolar lines can be established from simple image point homologies. These may be related to each other in infinite variations of perspective positions in space, each defining different camera geometries and relative orientation of image bundles. It is of interest in photogrammetry to also approach the 3D image configurations embedded in 2D epipolar geometry in a Euclidean (rather than a projective-algebraic) framework. This contribution attempts such an approach initially in 2D to propose a parameterization of epipolar geometry; when fixing some of the parameters, the remaining ones correspond to a ‘circular locus’ for the second epipole. Every point on this circle is related to a specific direction on the plane representing the intersection line of image planes. Each of these points defines, in turn, a circle as locus of the epipole in space (to accommodate all possible angles of intersection of the image planes). It is further seen that knowledge of the lines joining the epipoles with the respective principal points suffices for establishing the relative position of image planes and the direction of the base line in model space; knowledge of the actual position of the principal points allows full relative orientation and camera calibration of central perspective cameras. Issues of critical configuration are also addressed. Possible future tasks include study of different a priori knowledge as well as the case of the image triplet.