The reconstruction of a plane surface from two perspective projections

Abstract

This paper addresses the mathematical problem of reconstructing a visually textured plane surface from a pair of photographs taken from finitely separated camera positions of unknown relative orientation, lying on the same side or on opposite sides of the visible plane. If the surface lies at infinity, or perpendicularly bisects the line joining the centres of projection O and O$^{\prime}$, the reconstruction fails; otherwise the two images permit either one or two three-dimensional interpretations, obtainable by diagonalizing a 3 $\times $ 3 matrix. If all the visible texture elements lie nearer to one viewpoint than to the other, then there are two interpretations, which coincide if the line OO$^{\prime}$ is perpendicular to the visible plane. Otherwise, only the veridical interpretation survives. The relevance of these results to human and computer vision is briefly discussed.