Statistically Testing Uncertain Geometric Relations

This paper integrates statistical reasoning and Grassmann-Cayley algebra for making 2D and 3D geometric reasoning practical. The multi-linearity of the forms allows rigorous error propagation and statistical testing of geometric relations. This is achieved by representing all objects in homogeneous coordinates and expressing all relations using standard matrix calculus. 1 Motivation Many Computer Vision tasks involve grouping of geometric elements within one image or in 3D space. This requires testing geometric relations such as identity , incidence, parallelity or orthogonality. Due to uncertainty of the elements, checking these relation requires thresholds which in general are diicult to set. The goal of this paper is to integrate statistical and geometric reasoning by integrating statistical testing theory and Grassmann-Cayley algebra. Grassmann-Cayley algebra has been introduced by 1] and 2] and showed to be useful for analyzing the geometry of image triplets 3]. Representing geometric entities in projective space, thus using homogeneous coordinates, leads to less singular cases, includes entities at innnity and in most cases leads to multi-linear relations , which itself allows to perform error propagation rigorously. On the other hand there is a profound knowledge about optimal hypothesis testing 9, 7] which is appropriate for checking the validity of geometric relations. The use of statistical testing theory reduces the choice of thresholds to the choice of a single value, the signiicance level. What is lacking, is the integration of both concepts. This paper integrates statistical reasoning and Grassmann-Cayley algebra for making 2D and 3D geometric reasoning practical. The multi-linearity of the forms allows rigorous error propagation and statistical testing of geometric relations. This is achieved by representing all objects in homogeneous coordinates and expressing all relations using standard matrix calculus. The goal is to derive a simple rule for testing the basic relations between 2D points and lines, and 3D points, lines and planes, namely identity, incidence, parallelity and orthogonality. The solution proposed have been developed parallel to the one given in 6]. They are equivalent to those but much more transparent.