A note on Park and Chin´s algorithm [structuring element decomposition]

A finite subset of Z² is called a structuring element. A decomposition of a structuring element A is a sequence of subsets of the elementary square (i.e., the 3×3 square centered at the origin) such that the Minkowski addition of them is equal to A. H. Park and R.T. Chin (see ibid., vol.17, no.1, p.2-15, 1995) developed an algorithm for finding the optimal decomposition of simply connected structuring elements (i.e., 8-connected structuring elements that contain no holes), imposing the restriction that all subsets in this decomposition are also simply connected. The authors show that there exist infinite families of simply connected structuring elements that have decompositions but are not decomposable according to Park and Chin´s definition