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In this paper and its sequel, we will examine two fundamental problems related to object recognition for the generalized weak perspective model of image formation. We offer a complete solution to both problems for configurations of point features.
Geometric Constraints (Object/Image Equations): The problem of single view recognition is central to many image understanding and computer vision tasks. The results in this paper provide a way to understand
the relationship that exists between the 3D geometry and its "residual" in a 2D image. This relationship is completely characterized (for a particular combination of features) by a set of fundamental equations in the 3D and 2D shape coordinates. These equations are know as object/image relations. They can be used to
test for the geometric consistency between an object and an image. For example, by fixing point features on a known object, we get constraints on the 2D shape coordinates of possible images of those
features. Conversely, if we have specific 2D features in an image, we will get constraints on the 3D shape coordinates of objects capable of producing that image. This yields a test for which object is being
viewed. The object/image equations are thus fundamental tools for the identification/recognition problem. In this paper we give the complete solution, and find explicitly, the object/image equations for
point features in the generalized weak perspective case. This entails working on spaces known as Grasssmannians and exploiting certain incidence correspondences and the so-called Plucker relations.
Vision Metrics: We would like to know if two configurations of a fixed number of points in 2D or 3D are the same if we allow affine transformations. If they are, then we want a distance of zero, and if not, we want a distance that expresses their dissimilarity -
always recognizing that we can transform the points. The Procrustes metric, described in the shape theory literature [2], provides such a notion of distance for similarity transformmations. However, it does not allow for weak perspective or perspective transformations and is fixed in a particular dimension. By the later we mean that it cannot be regarded as giving us a notion of "distance" between, say, a 3D
configuration of points and a 2D configuration of points, where zero distance corresponds to the 2D points being, say, a generalized weak perspective projection of the 3D points. In the sequel to this
paper, we show that generalizations of the Procrustes metric exist in the above cases. Moreover these new metrics are quite natural in the context of the algebro-geometric formulation of the object/image
equations mentioned above. These metrics also provide a rigorous foundation for error and statistical analysis in the object recognition problem.
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