Morphological Structuring Elements

For image processing systems that have a limited size of region of support, say 3 x 3, direct implementation of morphological operations by a structuring element larger than the prefixed size is impossible. The decomposition of morphological operations by a large structuring element into a sequence of recursive operations, each using a smaller structuring element, enables the implementation of large morphological operations. In this paper, we present the decomposition of arbitrarily shaped (convex or concave) structuring elements into 3 x 3 elements, optimized with respect to the number of 3 x 3 elements. The decomposition is based on the concept of factorization of a structuring element into its prime factors. For a given structuring element, all its corresponding 3 x 3 prime concave factors are first determined. From the set of the prime factors, the decomposability of the structuring element is then established, and subsequently the structuring element is decomposed into a smallest possible set of 3 x 3 elements. Examples of optimal decomposition and structuring elements that are not decomposable are presented. Index Terms Mathematical morphology, stucturing element decomposition, concave boundary.