On the Geometry and Algebra of the Point and Line Correspondences between $N$ Images
Résumé
We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to 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 have a fairly complete picture of the situation in the case of points: there are only three types of algebraic relations which are satisfied by the coordinates of the images of a 3-D point: 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 that for a given triple of images, once the epipolar constraints are known, there is only one algebraically independent trilinear relation which 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). We also show that the trilinear relations imply the bilinear ones, i.e. the epipolar constraints. Finally, we show that the quadrilinear relations are algebraically dependent of the trilinear and bilinear ones, i.e. do not bring in any new information. In the case of lines, we show how the traditional perspective projection equation can be suitably generalized and that in the case of three images there exist two independent trilinear relations between the coordinates of the images of a 3-D line.