Certifying the Existence of Epipolar Matrices
This work addresses a foundational problem in computer vision for researchers and practitioners, providing a method to verify geometric consistency without computing the matrices, though it is incremental in nature.
The paper tackles the problem of certifying the existence of epipolar matrices (fundamental or essential matrices) from point correspondences in two-view geometry, presenting an efficient algorithm using exact linear algebra to test this existence and deriving polynomial conditions in image coordinates as new invariants.
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.