Morphological decomposition of restricted domains: a vector space solution

We deene a restricted domains, a class of discrete, 2D, convex shapes. We prove that there is a set of thirteen restricted domains fK 1 ; K 2 ; : : :; K 13 g such that any given restricted domain, K; is expressible as K = K 0 (k1 K 1) (k2 K 2) (k13 K 13) where (ki K i) represents the k i-fold dilation of K i and K 0 is a translation. We show that this entails a linear transformation from a thirteen dimensional space in which restricted domains are represented in terms of n-fold dilations of the thirteen basis structuring elements, to an eight dimensional space in which restricted domains are represented in terms of their eight side lengths. Furthermore , we show that any particular decomposition is a particular solution of this transformation and nding all possible dilation decompositions of a restricted domain is equivalent to nding the general solution of this transformation. Finally, we give an algorithm for nd-ing all possible decompositions.