Hon-leung Lee

Hon-Leung Lee

In this note we study the connection between the existence of a projective reconstruction and the existence of a fundamental matrix satisfying the epipolar constraints.

Hon-Leung Lee

Two-view triangulation is a problem of minimizing a quadratic polynomial under an equality constraint. We derive a polynomial that encodes the local minimizers of this problem using the theory of Lagrange multipliers. This offers a simpler derivation of the critical points that are given in Hartley-Sturm [6].

Sameer Agarwal, Hon-Leung Lee, Bernd Sturmfels et al.

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 $m \leq 7$. At the same time, we prove that they exist unconditionally when $m \leq 5$. Under a mild genericity condition, we show that an essential matrix always exists when $m \leq 4$. 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.

Sameer Agarwal, Hon-leung Lee, Bernd Sturmfels et al.

Given a set of point correspondences in two images, the existence of a fundamental matrix is a necessary condition for the points to be the images of a 3-dimensional scene imaged with two pinhole cameras. If the camera calibration is known then one requires the existence of an essential matrix. We present an efficient algorithm, using exact linear algebra, for testing the existence of a fundamental matrix. The input is any number of point correspondences. For essential matrices, we characterize the solvability of the Demazure polynomials. In both scenarios, we determine which linear subspaces intersect a fixed set defined by non-linear polynomials. The conditions we derive are polynomials stated purely in terms of image coordinates. They represent a new class of two-view invariants, free of fundamental (resp.~essential)~matrices.