Homaloidal parametrization for detecting critical two-view configurations

We consider the problem of identifying degenerate configurations while estimating the fundamental matrix from (at least) 8 point correspondences. It is known that such configurations correspond to an ill-posed estimation of the fundamental matrix, so it is important to identify them in practice. So far, a practical degeneracy test is only available for the cases of planar scenes and pure rotation, while the case of the general critical surface (e.g., a hyperboloid/cone/cylinder containing 3D points and camera centres) is less studied, and the only available method is highly unstable, involving a pre-computed fundamental matrix. In this paper, we propose a novel degeneracy test for detecting points on the critical surface. By exploiting the geometry of the so-called ``homaloidal net of conics'', we are able to design a simple and very 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 closest competitor, as shown in our experiments on both synthetic and real-world degenerate configurations.