Image Processing Reference

In-Depth Information

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

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