3×3 decomposition of circular structuring elements

This paper presents some results to decompose circular structuring elements into 3×3 elements. Decomposition allows one to improve the expended time in computing morphological operations. Generally, the shape of the structuring element determines the image transformation. Morphological operations with disks can be used as shape and size descriptors. The optimal discrete approximation of a disk can not be decomposed into 3×3 factors. Therefore, for a given radius, we give a hexadecagon that can be decomposed, and which optimally fits a disk. Afterwards, we present the decomposition of the disk in terms of the hexadecagon parameters. The decomposition prime factors can be given by different families of basic structuring elements