Computing three dimensional project invariants from a pair of images using the Grassmann-Cayley algebra

Abstract

The Grassmann-Cayley algebra, also called the double algebra, is an invariant algebraic formalism for expressing statements in synthetic projective geometry. It allows the translation of any incidence relation or incidence theorem in projective geometry into a conjunction of simple double algebraic statements involving only join and meet. In 1993, Carlsson made a first attempt to use this algebra to compute three-dimensional invariants of sets of points and lines from a pair of weakly calibrated stero images that is completed here, such that the results permit the calculation of projective invariants for any configurations of points and/or lines. In the second part of the paper it is shown how these invariants can be used to effect implicit projective reconstruction of points and lines. Some experimental results with real data are presented.