The double algebra: An effective tool for computing invariants in computer vision

Abstract

The double algebra is a system for computations involving subspaces of a general finite dimensional vector space. If this vector space is taken as projective 3-space, the operations of the double algebra can be interpreted as joins and intersections of points, lines and planes. All computations are coordinate free and invariant over linear transformations. The double algebra is therefore a very effective tool for computation of linear invariants of geometric configurations. In this paper we show how to compute linear invariants of general configurations points and lines observed in two images and polyhedral configurations observed in one image. For these cases we derive directly explicit expression of the invariants without reconstructing individual points and lines.