Mathematical Morphological Distributive Concepts over Unions and Intersections

Mathematical Morphological concepts outline techniques for analysing and processing geometric structures based on set theory. In this paper, we present proofs of our theorems on morphological distributive properties over Unions and Intersections with respect to Dilation and Erosion. These results provide new realizations of Dilation, Erosion and conclude that they are distributive over Unions but non-distributive over Intersections.


Introduction
Mathematical Morphology is a tool for the extraction of components of images used to describe and represent skeletons, boundaries etc., which involves techniques like morphological thinning, pruning and filtering.Morphological concepts date back to works done by Matheron and Minkowski who used binary mathematical morphology on integral geometry [1], [2].Matheron and Serra also used the techniques in texture and image analysis [1], [3].In our recent paper titled "Revised Mathematical Morphological Concepts" [4], we outlined in details some mathematical morphological operators and their algebraic structures when they are linked with unions and intersections.We showed that the partitioning of structural elements before morphological operations is possible.In this paper, we present results on the distributive properties of Dilation and Erosion over unions and intersections.This paper is a continuation of the revised mathematical morphological concept [4] and hence most of the concepts that were developed and discussed in it will be used here without explaining.Therefore, we urge that you read it before going through this paper.

Definitions
The following definitions are important for our purpose.Definition 1 (Dilation) Let the image set X and the structuring element B be subsets of the discrete space 2 Z : or the Dilation of a binary image A by structure element B, is The dilation transform generally causes image objects to grow in size.From the definitions above, dilation is equivalent to a union of translates of the original image with respect to the structure element, that is, Definition 2 (Erosion) Let the image set X and the structuring element B be subsets of the discrete space 2 Z : The erosion of X by B is defined as

Algebraic Properties of Dilation and Erosion
We note that Dilation is commutative and associative, that is, Furthermore, Dilation and Erosion are both translation invariant, that is, if x is a vector belonging to A and B ( x A ∈ , x B ∈ ), then ( ) ( ) . Also both Dilation and Erosion are increasing in A, that is, Dilation and Erosion transforms are duals of each other, that is, ( ) . Dilation and Erosion are also not the inverse of each other, that is, ( ) ( ) . Both the dilation and erosion transforms have an identity set, I, such that A I A ⊕ = and A I A =  . Dilation transform has an empty set, that is, ∅ such that A ⊕ ∅ = ∅ .

Results
We present results of the distribution of morphological operators over set unions and intersections of two different sets and their extensions.The theorems and their proofs below will facilitate the understanding of the various results.

The Distribution of Morphological Operators over Set Union and Intersection
Theorem 1 (The distribution of Dilation over union with n distinct structural elements) Let assume that if The distribution of Erosion over union with n distinct structural elements) If Now we show that if The dilation of a set of two different structural elements and taking the union is the