Efficient implementation of morphological operations requires the deco
mposition of structuring elements into the dilation of smaller structu
ring elements, Zhuang and Haralick presented a search algorithm to fin
d optimal decompositions of structuring elements in binary morphology.
In this paper, we use the concepts of Top of a set and Umbra of a sur
face to extend this algorithm to find an optimal decomposition of any
arbitrary gray-scale structuring element.