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A method that treats linear neighborhood operators within a unified framework that enables linear combinations, concatenations, resolution changes, or rotations of operators to be treated in a canonical manner is presented. Various families of operators with special kinds of symmetries (such as translation, rotation, magnification) are explicitly constructed in 1-D, 2-D, and 3-D. A concept of'order'is defined, and finite orthonormal bases of functions closely connected with the operators of various orders are constructed. Linear transformations between the various representations are considered. The method is based on two fundamental assumptions: a decrease of resolution should not introduce spurious detail, and the local operators should be self-similar under changes of resolution. These assumptions merely sum up the even more general need for homogeneity isotropy, scale invariance, and separability of …