Camera self-calibration from bivariate polynomial equations and the coplanarity constraint

This paper presents a new approach for self-calibrating a moving camera with constant intrinsic parameters. Unlike existing methods, the proposed method turns the self-calibration problem into one of solving bivariate polynomial equations. In particular, we show that each pair of images partially identifies a pair of 3D points that lie on the plane at infinity. These points are parameterized in terms of the real eigenvalue of the homography of the plane at infinity. A triplet of images identifies six such points on which the coplanarity constraint is enforced leading to a set of quintic and sextic polynomial equations. These equations are solved using a homotopy continuation method. More images allow to isolate the real eigenvalue associated with each motion and thus, to fully identify the points at infinity. The method also presents inequality conditions that allow to eliminate spurious solutions. Degenerate motions, not allowing the calculation of the eigenvalues, are also presented here. Once the 3D points at infinity are localized, both the plane at infinity and the Kruppa's coefficients can be linearly estimated.