Existence of Two View Chiral Reconstructions

TL;DR

This paper classifies when a set of point pairs in computer vision can be reconstructed with two cameras respecting scene orientation, revealing conditions for existence based on the number and configuration of points.

Contribution

It provides a complete classification of point pairs for which a chiral reconstruction exists, including geometric and algebraic conditions, and describes the structure of the chiral region.

Findings

Chiral reconstructions always exist for up to three point pairs.

For five or more point pairs, non-existence is Zariski-dense.

The chiral region for five generic points is bounded by line segments on a cubic surface.

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. 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 Schl\"afli 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.