The Eight-Point Algorithm

This note describes a method for computing estimates of the rigid transformation G and estimates of the coordinates of a set of n points P1, . . . ,Pn in the two camera reference frames from the n pairs (x1,y ′ 1), . . . (xn,y ′ n) of noisy measurements of their corresponding images. The transformation G is called camera motion, and the point coordinates Xi, Xi of the world points in the two reference systems are collectively called the scene structure. This classic method is called the eight-point algorithm and is was invented by Hugh Christopher Longuet-Higgins in 1981 [3]. The points xi and y′ i are in the canonical reference system of each camera, so their third coordinate is equal to 1. As such, they can be viewed as either the Euclidean coordinates of 3D points, or as homogeneous coordinates of 2D image points. However, we use Euclidean coordinates for other 3D points to describe the eight-point algorithm—a method devised before homogeneous coordinates became pervasive in computer vision. Nowadays, this algorithm is typically embedded in code that uses homogeneous coordinates everywhere else, for notational convenience. We also resort to homogeneous coordinates in Appendix B where we discuss triangulation, that is, the calculation of X, X′ from the image points and G. Since cameras fundamentally measure angles, both structure and motion can be estimated only up to a common nonzero multiplicative scale factor. The resulting degree of freedom is eliminated by assuming that