SET OPERATIONS ON CLOSED INTERVALS AND THEIR APPLICATIONS TO THE AUTOMATIC PROGRAMMING OF MORPHOLOGICAL MACHINES

Mathematical morphology on sets can be understood as a formal language , whose vocabulary comprises erosions, dilations, complementation, int ersection and union. This language is complete, that is, it is enough to perform any set operation. Since the sixties special machines, call ed morphological machines (MMachs), have been built to implement this language. In the literature, we find hundreds of MMach programs that a re used to solve image analysis problems. However, the design of these programs is not an elementary task. Thus, recently much research effo rt has been addressed to automating the programming of MMachs. A Very promising approach to this problem is the description of the target op erator by input-output pairs of images and the translation of these da ta into efficient MMach programs. This approach can be decomposed into two equally important steps: (1) learning of the target operator from pairs of images; (2) search for economical representations for the op erators learned. The theory presented in this paper is useful in the s econd step of this procedure. We present some set operations on collec tions of closed intervals and give efficient algorithms to perform the m. These operations are used to parallelize MMach programs and to prov e the equivalence between distinct MMach programs. (C) 1996 SPIE and I S&T.