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Proceedings of the SPIE, Vol. 435, 1983, pp. 60-67. 3 E. Horowitz and S. Sahni, Fundamentals of Data Structures, Computer Science Press, Rockville, MD, 1982. Y. Kong and A. Rosenfeld, "Digital Topology: Introduction and Survey", Computer Vision, Graphics, and Image Processing, Vol. 48, 1989, pp. 357-393. 5 R. G. Shapiro, and 0 . Zuniga, "A New Connected Components Algorithm for Virtual Memory Computers," Computer Vision, Graphics, and Image Processing, Vol. 22, 1983, pp. 287-300. 6 R. Lumia, "A New Three-Dimensional Connected Components Algorithm", Computer Vision, Graphics, and Image Processing, Vol.

P(,+2)2-i be the pixels in iV(,+2)x(»+2)(p) - ^»xt(p) in clockwise order starting with the pixel in the top left-hand corner. Now we can give our general definition of an operator. An operator is a Boolean function of a finite number of Boolean arguments 'Po,'^ij - -Vk (where k can be any non-negative integer), with the property that the function ha,s value 0 when all of its arguments have value 0. As in the 3 x 3 case discussed above, for any image / and pixel p we define the value of the operator at p in I to be the value of the Boolean function when each argument Vi is given the value of the pixel p, in / .

D is called the deleted residue set of (IjD). I's, O's and m-residues in / are also referred to as I's, O's and m-residues •m{I,D). Intuitively, we think of D as the set of all m-residues that have so far been deleted by an image algorithm. The restored residues image of an r-image (/,-D), denoted by {I^D)^, is the image whose set of Ts consists of the I's of / and all the pixels in D. An T'image transformation is a function which maps r-images to r-images. For any r-image transformation T, the restored residues image transform,ation associated with T, written T^, is the image transformation such that, for every image / , T""(/) = T((/, 0))"".

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