Existence of Two View Chiral Reconstructions

Abstract

A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper we provide a complete classification of k point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the non-emptiness of certain semialgebraic sets. We describe these sets and develop tools to certify their non-emptiness. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schläfli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two non-generic combinatorial types, in which case they may or may not.

Nonlinear Algebra and Applications

“…It is also desirable to better understand semialgebraic constraints implicit in image formation, such as those which enforce that world points lie in front of the camera [105]. Two recent preprints that go in this direction are [7,187]. For other geometric problems in vision requiring robust estimation, methods based on convex relaxations are popular alternatives to RANSAC, since solutions can often be certified by computing duality gaps.…”