Object recognition via configurations of lines
Peter F. Stiller · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1998
In this paper we explore the application of several advanced mathematical techniques from algebraic geometry, notably the theory of correspondences and a novel 'equivariant' invariant theory, to the problem of recognizing 3D geometric configurations from a single 2D view. We specifically require the approach to be view independent. This forces us to characterize line configurations by their 3D or 2D geometric invariants. Recent work of R. Huang on 'invariants of sets of lines in projective 3-space' provides a first description of the necessary line invariances in 3D. In this paper, we simplify Huang's results and develop the algebro- geometric machinery needed to understand the relationship that exists between the 3D algebraic geometry. Exploiting this, we compute a set of fundamental equations in the combined set of 3D and 2D invariants, which generate the ideal of the correspondence, and which completely describe the mutual 3D/2D constraints. We have chosen to call these equations 'object/image equations'. They can be exploited in a number of ways. For example, from a given 2D configuration of lines, we can determine a set of non-linear constraints on the geometric invariants of the 3D line configurations capable of producing that given 2D configuration, and thus arrive at a test for determining the object being viewed. Conversely, given a 3D geometric configuration, we can derive a set of equations that constrain the images of that object. Methods to compute a complete set of generating object/image equations that constrain the images of that object. Methods to compute a compete set of generating object/image equations are mentioned. These include advanced geometric techniques like KSY resultants, sparse resultants, and Groebner bases. The calculations have been carried out using a mix of such advanced techniques in the specific case under consideration, namely line features. The resulting object/image equations have been sued in industrial and defense applications.