Morphological decomposition of restricted domains: a vector space solution

Restricted domains, which are a restricted class of 2-D shapes, are defined. It is proved that any restricted domain can be decomposed as n-fold dilations of thirteen basis structuring elements and hence can be represented in a thirteen-dimensional space. This thirteen-dimensional space is spanned by the thirteen basis structuring elements comprising of lines, triangles, and a rhombus. It is shown that there is a linear transformation from this thirteen-dimensional space to an eight-dimensional space wherein a restricted domain is represented in terms of its side lengths. Furthermore, the decomposition in general is not unique, and all the decompositions can be constructed by finding the homogeneous solutions of the transformation and adding it to a particular solution. An algorithm for finding all possible decompositions is provided