Parameterizing Homographies CMU-RI-TR-06-11

The motion of a plane can be described by a homography. We study how to parameterize homographies to maximize plane estimation performance. We compare the usual 3 × 3 matrix parameterization with a parameterization that combines 4 fixed points in one of the images with 4 variable points in the other image. We empirically show that this 4pt parameterization is far superior. We also compare both parameterizations with a variety of direct parameterizations. In the case of unknown relative orientation, we compare with a direct parameterization of the plane equation, and the rotation and translation of the camera(s). We show that the direct parameteri-zation is both less accurate and far less robust than the 4-point parameterization. We explain the poor performance using a measure of independence of the Jacobian images. In the fully calibrated setting, the direct parameterization just consists of 3 parameters of the plane equation. We show that this parameterization is far more robust than the 4-point parameterization, but only approximately as accurate. In the case of a moving stereo rig we find that the direct parameterization of plane equation, camera rotation and translation performs very well, both in terms of accuracy and robustness. This is in contrast to the corresponding direct parameterization in the case of unknown relative orientation. Finally, we illustrate the use of plane estimation in 2 automotive applications.