On the probabilistic epipolar geometry

In this paper, we are going to answer the following question: assuming that we have estimates for the epipolar geometry and its uncertainty between two views, how probable is it that a new, independent point pair will satisfy the true epipolar geometry and be, in this sense, a feasible candidate correspondence pair? If we knew the true fundamental matrix, the answer would be trivial but in reality we do not know it because of estimation errors. So, as an independent point in the first view is given, we will show we may compute the point-probability-density function, termed as the epipolar pdf, for the epipolar line points in the second view that describes the current level of knowledge of the epipolar geometry between the views. This point–point-probability-density relation is a probabilistic form of the epipolar constraint that also approaches the true point–line relation as the number of training correspondences tends to infinity. In this paper, we will also show that the eigenvectors of the epipolar line covariance matrix have certain interpretations on the image plane, of which one is the previously observed, narrowest point of the epipolar envelope. The results of this paper are important since the uncertainty of the epipolar constraint can be now taken into account in a sound way in applications.