Morphology for Image Processing. Part I

Let E be a non empty RLS structure. A binary image of E is a subset of E. In the sequel E denotes a real linear space and A denotes a binary image of E. Let E be a real linear space and let A, B be binary images of E. The functor A B yielding a binary image of E is defined as follows: (Def. 1) A B = {z ∈ E: ∧ b : element of E (b ∈ B ⇒ z − b ∈ A)}. Let a be a real number, let E be a real linear space, and let A be a subset of E. We introduce a ·A as a synonym of a A. The following propositions are true: (1) Let E be a real linear space and A, B be subsets of E. If B = ∅, then A⊕B = B and B ⊕A = B and A B = the carrier of E. (2) For every real linear space E and for all subsets A, B of E such that A 6= ∅ and B = ∅ holds B A = B. The authors wants to thank Prof. Yasunari Shidama for his kind support during the course of this work.