A framework for a large class of invariant-pattern-recognition problems, is suggested. The proposed approach combines the properties of uniqueness, registration, and optimality with the ability to use natural symmetries of input patterns. The approach is based on cross correlation. The cross-correlation function with respect to a group of transformations is defined. The properties of this function and methods of its calculation are discussed. As a special case, it is shown formally that the cross correlation with respect to the motion group is the basis of such techniques as Radon transforms and the Hough technique. The roles of symmetry and parameters are examined. A technique for decomposition of pictures into sums of separable functions associated with groups of transformations is applied to the calculation of the cross-correlation function. Several examples of applications of the proposed approach are given.
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