Combinatorial constraints on multiple projections of set points

Abstract

Multiple projections of a scene cannot be arbitrary, the allowed configurations being given by matching constraints. This paper presents new matching constraints on multiple projections of a rigid point set by uncalibrated cameras, obtained by formulation in the oriented projective rather than projective geometry. They follow from consistency of orientations of camera rays and from the fact that the scene is the affine rather that projective space. For their non-parametric nature, we call them combinatorial. The constraints are derived in a unified theoretical framework using the theory of oriented matroids. For example, we present constraints on 4 point correspondences for 2D camera resectioning, on 3 correspondences in two 1D cameras, and on 4 correspondences in two 2D cameras.