In the last decade algebraic projective geometry have been successfully used to solve various computer vision problems. Projective Geometry has the advantage of a simple and consistent representation of geometric objects and transformations in 2D and 3D.
In particular, projective geometry can be used to do "geometrical reasoning", which means: (i) construction of geometric entities using join and intersection of given entities and (ii) Testing spatial relationships between the entities. However, observed entities such as image points or image lines are uncertain, since they are observations or measurements with some inaccuracy. The inaccuracy of entities has to taken into account when doing geometrical reasoning, therefore projective geometry has to be extended to include notions of uncertainty, enabling "statistical geometric reasoning".
We present a calculus for projective points, lines and planes in 2D and 3D with the following properties: (i) simple representation of the uncertainty of geometric entities, (ii) rules for direct and over-constrained constructions of new entities, (iii) statistical hypotheses tests for spatial relationships. An application of this calculus is the reconstruction of polyhedral objects from multiple images, for which we solve the following subtasks: (a) matching of 2D image line segments and points from different images, (b) optimal reconstruction of 3D segments using matched image features, (c) grouping of 3D edges to 3D corners. An important property of this algorithm is the fact that no data-dependent thresholds are used other than significance values for hypotheses tests.