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.

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.

Theorem 1 (The distribution of Dilation over union with n distinct structural ele- ments)

If

Then

Proof:

If

Then

This implies

Let assume that if

Then

Now we show that if

Then

Theorem 2 (The distribution of Erosion over union with n distinct structural ele- ments)

If

Then

Proof:

If

Then

This implies

Let assume that if

Then

Now we show that if

Then

The dilation of a set of two different structural elements and taking the union is the same as taking the union of the structural element and dilating with the set. This shows that morphological dilation distributes over set unions. It also leads to the fact that; if any structural element can be partitioned into n distinct parts then the union of each of the partitions dilation with the set is the same as the set’s dilation with the structural element. We note also that provided any structural element can be partitioned into n distinct parts, then the union of each of the partition’s erosion with the set is equal to the set’s erosion with the structural element.

Theorem 3 (Non-distribution of Erosion over intersection)

If

Then

Proof:

If

Then

This implies

Let assume that if

Then

Now we show that if

Then

Theorem 4 (Non-distribution of Dilation over intersection)

If

Then

Proof:

If

Then

This implies

Let assume that if

Then,

Now, we show that if

Then,

The intersection of the erosion of a set with structural elements is equivalent to the union of the structural elements on the erosion of the set. We note that since we are supposed to take the union instead of the intersection, it shows that morphological erosion is non-distributed over set intersection. Similar arguments hold for dilation which leads to the non-distributive property of dilation over intersection.

We have presented theorems and their proofs on morphological distribution properties over unions and intersections. Our results show that Dilation and Erosion are distri- butive over unions but non-distributive over intersections. In addition, our theorems facilitate the partitioning of structural elements in order to implement morphological operations.

Ackora-Prah, J., Acquah, R.K. and Ayekple, Y.E. (2016) Mathe- matical Morphological Distributive Concepts over Unions and Intersections. Advances in Pure Mathematics, 6, 633-637. http://dx.doi.org/10.4236/apm.2016.610052