A general solution to the P4P problem for camera with unknown focal length

This paper presents a general solution to the determination of the pose of a perspective camera with unknown focal length from images of four 3D reference points. Our problem is a generalization of the P3P and P4P problems previously developed for fully calibrated cameras. Given four 2D-to-3D correspondences, we estimate camera position, orientation and recover the camera focal length. We formulate the problem and provide a minimal solution from four points by solving a system of algebraic equations. We compare the Hidden variable resultant and Grobner basis techniques for solving the algebraic equations of our problem. By evaluating them on synthetic and on real-data, we show that the Grobner basis technique provides stable results.