Homaloidal parametrization for detecting critical two-view configurations

We consider the problem of identifying critical configurations while estimating the fundamental matrix from (at least) eight point matches. So far, practical degeneracy tests are only available for planar scenes and pure rotations, whereas the case of a general critical surface (e.g., a hyperboloid, cone, or cylinder containing 3D points and camera centers) remains less explored. The only existing degeneracy test for the general case is highly unstable, as it relies on a precomputed fundamental matrix. We propose a novel test for detecting points on the critical surface. By exploiting the geometry of the so-called "homaloidal net of conics", we design a simple and practical test that requires the linear estimation of a quadratic transformation from image correspondences. Our test does not require a fundamental matrix in advance and turns out to be more stable than its competitor, as shown in our experiments on both synthetic and real-world configurations.