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In this paper we explore the general problem of recognizing 3D geometric configurations from a single 2D view.Of necessity the approach must be viewpoint independent, forcing us to characterize configurations by their 3D or 2D geometric invariants. Our results make use of several advanced mathematical techniques from algebraic geometry, notably the theory of correspondences and a novel 'equivariant' invariant theory, that clarify the relationship that exists between the 3D geometry and its 'residual' in a 2D image. This relationship has been shown to be a correspondence in the technical sense of algebraic geometry. Exploiting this, one can compute for a particular set of features a set of fundamental equations, 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 used in a number of ways. For example, from a given 2D configuration we can determine a set of non-linear polynomial constraints on the geometric invariants of those 3D configurations capable of producing that given 2D configuration as an image; thereby arriving at a test for determining which object is being viewed. Conversely, given a 3D geometric configuration, we can derive a set of equations which constrain the images of that object, telling us which images contain a view of that particular object configuration. Methods to compute a complete set of generating object/image equations will be discussed. These include symbolic computational techniques like resultants, sparse resultants, and KSY resultants. The calculations have been carried out in a number of important cases, and the resulting object/image equations used in industrial and defense applications.
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