Degeneracy of the Intersection of Three Quadrics

Abstract

We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix F independent on the camera positions, when its inputs are image point configurations that are perspective projections of the intersection of three quadrics in \(\mathbb {P}^3\). This generalizes a multiple results of the degeneracies of the 8-point algorithm. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally, we analyze the regions of focal point positions where a reconstruction of F is possible at all, when the world points are the vertices of a combinatorial cube in \(\mathbb {R}^3\).