On the Existence of Epipolar Matrices

This paper considers the foundational question of the existence of a fundamental (resp. essential) matrix given m point correspondences in two views. We present a complete answer for the existence of fundamental matrices for any value of m. Using examples we disprove the widely held beliefs that fundamental matrices always exist whenever m ≤ 7. At the same time, we prove that they exist unconditionally when m ≤ 5. Under a mild genericity condition, we show that an essential matrix always exists when m ≤ 4. We also characterize the six and seven point configurations in two views for which all matrices satisfying the epipolar constraint have rank at most one.