^{1}

^{*}

^{1}

^{1}

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.

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 [

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

The dilation transform generally causes image objects to grow in size. From the defi- nitions 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

Similarly erosion transform allows image objects to shrink in size, that is,

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 (

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 IntersectionTheorem 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