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Toward the development of an object recognition and positioning system, able to deal with arbitrary shaped objects in cluttered environments, methods for matching two arbitrarily shaped regions of different objects are introduced, and how to efficiently compute the coordinate transformation which makes two matching regions coincide is shown. In both cases, matching and positioning, the results are invariant with respect to viewer coordinate system, and these techniques apply to both 2-D and 3-D problems, under either Euclidean or affine coordinate transformations. The 3-D Euclidean case is useful for the recognition and positioning of solid objects from range data, and the 2-D affine case for the recognition and positioning of solid objects from projections, e.g., from curves in a single image, and in motion estimation. The matching of arbitrarily shaped regions is done by computing for each region a vector of centered moments. These vectors are viewpoint-dependent, but the dependence on the viewpoint is algebraic and well known. This paper presents a new family of computationally efficient algorithms based on matrix computations, for the evaluation of both Euclidean and affine algebraic moment invariants of data sets. The use of algebraic moment invariants greatly reduces the computation required for the matching and, hence, initial object recognition. The approach to determining and computing these moment invariants is different than those used by the vision community previously. The method for computing the coordinate transformation which makes the two matching regions coincide provides an estimate of object position. The estimation of the matching transformation is based on the same matrix computation techniques introduced for the computation of invariants. It involves simple manipulations of the moment vectors. It neither requires costly iterative methods, nor going back to the data set. These geometric invariant methods appear to be very important for dealing with the situation of a large number of different possible objects in the presence of occlusion and clutter, and the approach to computing these moment invariants is different than those used by the vision community previously.
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