Optimal decomposition of convex morphological structuring elements for 4-connected parallel array processors

A morphological operation using a large structuring element can be decomposed equivalently into a sequence of recursive operations, each using a smaller structuring element. However, an optimal decomposition of arbitrarily shaped structuring elements is yet to be found. In this paper, we have derived an optimal decomposition of a specific class of structuring elements-convex sets-for a specific type of machine-4-connected parallel array processors. The cost of morphological operation on 4-connected parallel array processors is the total number of 4-connected shifts required by the set of structuring elements. First, the original structuring element is decomposed into a set of prime factors, and then their locations are determined while minimizing the cost function. Proofs are presented to show the optimality of the decomposition. Examples of optimal decomposition are given and compared to an existing decomposition reported by Xu (1991)