Study of statistical properties of gray-scale compound morphological operators using the basis matrix

In this paper, we expand the statistical properties of grayscale compound function processing (FP) morphological operators. This is achieved by utilizing the basis matrix representation which is an extension of the basis function theorem. It is shown that the basis matrix is skew symmetric and this fact is highly exploited in finding the output density functions of grayscale opening and closing. The proposed method is also applicable to function set processing operators since these operators are a special case of the FP operators.