Existence of Two View Chiral Reconstructions
Andrew Pryhuber, Rainer Sinn, Rekha R. Thomas · SIAM Journal on Applied Algebra and Geometry · 2022
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 nonemptiness of certain semialgebraic sets. We describe these sets and develop tools to certify their nonemptiness. 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 nongeneric combinatorial types, in which case they may or may not.