Computing invariants using elimination methods

Geometric invariants appear to play an important role in object recognition as an aid to building model libraries of objects. Useful invariants are often found by extensive experience and they are based on geometric invariant properties studied by algebraists over many years. Given a geometric configuration, there is however a need to systematically generate and search for its invariants. In this paper we give a complete solution, in principle, to computing a single invariant for a geometric configuration, if it exists. The algorithm works in three steps: (i) the problem formulation step in which algebraic relations are established between object parameters and image parameters (or equivalently, parameters of two different images) using an imaging transformation, (ii) elimination of transformation parameters resulting in an invariant relation between object and image parameters, and (iii) finally, extraction of a single invariant from the algebraic relation. The main contribution of the paper is to give a complete solution for the third step, called the separability problem. It is shown that there is a simple and efficient solution to the separability problem provided the resultant obtained from the second step is explicitly known. The efficiency of the algorithm is linked to the efficiency of computing the resultant. Dixon´s method is proposed to be used for computing the resultant