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We revise some mathematical morphological operators such as Dilation, Erosion, Opening and Closing. We show proofs of our theorems for the above operators when the structural elements are partitioned. Our results show that structural elements can be partitioned before carrying out morphological operations.

Mathematical morphology is the theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. We consider classical mathematical mor- phology as a field of nonlinear geometric image analysis, developed initially by Matheron [

In the development of mathematical morphology in the mid-1960s by Georges Matheron and Jean Serra, they heavily stressed the mathematical formalism on mathematical morphology, and in the work of Haralick, Sternberg and Zhuang [

In this paper, we outline in details the mathematical morphological operators and their algebraic structures when they are linked with union and intersection. We show that the partitioning of structural element before morphological operations is possible.

The following definitions are important for our purpose.

Let the image set X and the structuring element B be subsets of the discrete space

dilation of X by B is defined as

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,

Let the image set X and the structuring element B be subsets of the discrete space

erosion of X by B is defined as

image A by structure element

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

Let the image set X and the structuring element B be subsets of the discrete space

Opening of X by B is defined as

element B, is

Let the image set X and the structuring element B be subsets of the discrete space

Closing of X by B is defined as

B, is

We note that Dilation is commutative and associative, that is,

where as Erosion is non-commutative and non-associative, that is,

Furthermore, Dilation and Erosion are both translation invariant, that is, if

B (

Erosion are increasing in A, that is, if an image set

and

that

We have Opening and Closing transforms as duals of each other,

but Opening and Closing are not the inverse of each other,

Opening and Closing are translation invariant, if x is a vector belonging to A and B, then

Opening of A by a structuring element B is always contained in A, regardless of B

transform is extensive, the Closing of A by a structuring element B always contains A, regardless of B

Furthermore, Opening and Closing are both increasing in A. If an image set

element

both idempotent,

said to be open, whereas if X is unchanged by closing with B, X is said to be closed.

In this section we present unions and intersections of Dilation, Erosion, Opening and Closing of two different sets and their extensions. The following theorems and their proofs will help us to describe the various results.

The morphological operators with n distinct sets

Theorem 1 (The union of Dilation with n differents sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 2 (The intersection of Dilation with n different sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 3 (The union of Erosion with n distinct sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 4 (The intersection of Erosion with n distinct sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 5 (The union of Opening with n different sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 6 (The intersection of Opening with n different sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 7 (The union of Closing with n distinct sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

Theorem 8 (The intersection of Closing with n distinct sets)

If

Then

Proof.

If

Then

This implies

Assume that if

Then

Now we show that if

Then

We have shown that Dilation, Erosion, Opening and Closing of two or more sets with the same structural element and carrying out the union of the outcome is the same as taking the union of the two or more sets and operating the results with the structural element. The above result also holds for the intersection. These operators show ways of partitioning the structural element in order to carry out the morphological operation with ease. Further- more, the results above give a simplification of morphological operations when dealing with lots of sets with the same structural element.

We are grateful to the Almighty God and the Department of Mathematics, Kwame Nkrumah University of Science and Technology for providing us resources to help complete this research successfully.