Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 68

Closure Let A be a subset of a topological space (X, τ ). Then the subset A∪A 0 con-
sisting of all its limit points is called the closure of A and is denoted by A.
Interior Let (X, τ ) be any topological space and A be any subset of X. The largest open
set contained in A is called the interior of A, A.
Recall that in algebra every vector is a linear combination of the basis. In topology, every
open set can be obtained by a union of members of the basis.

Basis of a Topology Let (X, τ ) be a topological space. A collection B of open sub-
sets of X is said to be a basis for the topology τ , if every open set is a union of members of
B. In other words, B generates the topology.

4.3 Mathematical Morphology

Objects or images in our application are considered as subsets of the euclidean space E n
or subsets of an a nely closed subspace X⊆E n . For digital objects(images) space is
considered to be Z n , where Z is the set of integer numbers. Dilation and erosion are two
primary mathematical morphology operators and can be defined by Minkowski sum and
Minkowski difference:

A⊕B ={x + y : x∈A, y∈B}. (4.1)

where A, B⊆X, and '+' is the sum in euclidean space E n .

A · B ={x∈X : x⊕B⊆A}. (4.2)

For simplicity, a set B is assumed to be symmetric about the origin, therefor B =−B =
{−x : x∈B}. Mathematical morphology operators are defined in different ways. For
example, consider two binary images A⊂Z 2 and B⊂Z 2 . The dilation of A by B is also
defined as

A⊕B ={x|( B x )∩A 6= φ}. (4.3)
where B x is obtained by first reflecting B about its origin, and then shifting it such that
its origin is located at point x. B is called a structuring element(SE) and it can have
any shape, size and connectivity. The characteristic of SE is application dependent. As
mentioned earlier, for simplicity, we consider B x = B x . Based on equation 4.3, dilation
of an image A by B, is the set of all points x such that B x and A have overlapping. The
erosion of binary images A by B is defined:

A · B ={x|(B x )⊆A} (4.4)

Erosion of A by B is the collection of all points x such that B x is contained in A.
To be consistent with Polkowski (Polkowski, 1999), we use d B (A) for dilation of A by
B and e B (A) for erosion of A by B. New morphological operations could be obtained by
composition of mappings. Opening (o B (A)) and closing (c B (A)) are two operators obtained
by the following compositions, respectively:

o B (A) = d B (e B (A)) ={x∈X :∃y, (x∈{y}⊕B⊆A)}. (4.5)

c B (A) = e B (d B (A)) ={x∈X :∀y, (x∈y⊕B⇒A∩({y}⊕B) 6= φ)}. (4.6)
By moving the structuring element B on the image A, we are gathering information about
the medium A in terms of B. The simplest relationships can be obtained by B moving on