The Eight-Point Algorithm

This note describes a method for computing estimates of the rigid transformation Tb = (Rb, tb) between two cameras a and b and estimates of the coordinates P1, . . . , Pn of a set of n points in the reference system of one of the two cameras from the n pairs (pa,1, pb,1), . . . ( pa,n, pb,n) of noisy measurements of their corresponding images. The transformation Tb is called camera motion, and the point coordinates P1, . . . , Pn are collectively called the scene structure. The image points pa,i and pb,i are regarded as 3D points with their third coordinate equal to 1, the standard focal distance. The classic method described below is called the eight-point algorithm and is was invented by Hugh Christopher Longuet-Higgins in 1981 [3]. Its main goal is to find Tb. Triangulation, that is, the calculation of structure from the image points and Tb, is outlined in Appendix B. To simplify notation in the manipulations that follow, we again let