The geometric error for homographies

We address the problem of finding optimal point correspondences between images related by a homography: given a homography and a pair of matching points, determine a pair of points that are exactly consistent with the homography and that minimize the geometric distance to the given points. This problem is tightly linked to the triangulation problem, i.e., the optimal 3D reconstruction of points from image pairs. Our problem is non-linear and iterative optimization methods may fall into local minima. In this paper, we show how the problem can be reduced to the solution of a polynomial of degree eight in a single variable, which can be computed numerically. Local minima are thus explicitly modeled and can be avoided. An application where this method significantly improves reconstruction accuracy is discussed. Besides the general case of homographies, we also examine the case of affine transformations, and closely study the relationships between the geometric error and the commonly used Sampsonʹs error, its first order approximation. Experimental results comparing the geometric error with its approximation by Sampsonʹs error are presented.