Representation of three-dimensional object structure as cross-ratios of determinants of stereo image points

Abstract

A uniform algebraic procedure is presented for deriving both epipolar geometry and three-dimensional object structure from general stereo imagery. The procedure assumes central-projection cameras of unknown interior and exterior orientations. The ability to determine corresponding points in the stereo images is assumed, but no prior knowledge of the scene is required. Epipolar geometry and the fundamental matrix are derived by algebraic elimination of the object-variables from the imaging equations. This provides a transfer procedure to any other perspective as long as 8 or more corresponding points can be identified in the new perspective. Next, invariant coordinates of the scene-points are derived by algebraic elimination of the camera-parameters from the imaging equations. Identical coordinates are obtained from any stereo images of non-occluding scene points as long as the same set of 5 corresponding points can be identified in both stereo pairs. The procedure extends methods utilizing the cross-ratios of determinants and cyclopean vectors, presented in earlier work. A technique for reconstructing the 3-dimensional object from the invariant coordinates is also given.