Root signal sets of morphological filters

The theory of state description for the binary root signal set of morphological filters is presented. Some properties of root signals are discussed. For a signal of length n and structuring element of size k, the number of binary roots for morphological opening and closing is represented by the difference equation R(n+1)=2R(n)-R(n-1)+R(n-k). For open-closing and close-opening, the difference equation is R(n+1)=R(n)+R(n+1-k).