About the correspondences of points between N images

Abstract

We analyze the correspondence of points between an arbitrary number of images, from an algebraic and geometric point of view. We use the formalism of the Grassmann-Cayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed). We propose a systematic way to describe the algebraic relations which are satisfied by the coordinates of the images of a 3D point. They are of three types: bilinear relations arising when we consider pairs of images among the N and which are the well-known epipolar constraints, trilinear relations arising when we consider triples of images among the N, and quadrilinear relations arising when we consider four-tuples of images among the N. Moreover, we show how two trilinear relations imply the bilinear ones (i.e. the epipolar constraints). We also show how these trilinear constraints can be used to predict the image coordinates of a point in a third image, given the coordinates of the images in the other two images, even in cases where the prediction by the epipolar constraints fails (points in the trifocal plane, or optical centers aligned). Finally, we show that the quadrilinear relations are in the ideal generated by the bilinearities and trilinearities, and do not bring in any new information. This completes the algebraic description of correspondence between any number of cameras.