Image Processing Reference
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and
A =
\
B∈
˚
B
c
B
(A) =
\
B∈B
c
B
(A).
(4.10)
The following equations also relate the interior and closure to erosion and dilation, (Serra,
1983):
A =
[
B∈
˚
B
e
B
(A) =
[
B∈B
e
B
(A).
(4.11)
and
A =
\
B∈
˚
B
d
B
(A) =
\
B∈B
d
B
(A).
(4.12)
In words, the interior of a set A is the union of the opening or erosions of the set A with
open or closed balls of different sizes. The closure of a set A is the union of the closing or
dilations of A with open/closed balls of different sizes.
4.4
Rough Sets
In rough set theory, objects in a universe X, are perceived by means of their attributes(features).
Let ϕ
i
denote a real-valued function that represents an object feature. Each element
x∈A⊆X is defined by its feature vector, ϕ(x) = (ϕ
1
(x), ϕ
2
(x), . . . , ϕ
n
(x)) Peters and
Wasilewski (2009). An equivalence (called an indiscernibility relation (Pawlak and Skowron,
2007c)) can be defined on X. Let∼be an equivalence relation defined on X, i.e.,∼is re-
flexive, symmetric and transitive. An equivalent relation(∼) on X classifies objects (x∈X)
into classes called equivalence classes. Objects in each class have the same feature-value
vectors and are treated as one generalized item. The indiscernibility(equivalence) relation
∼
X,ϕ
is defined in (4.13).
∼
X,ϕ
={(x, y)∈X×X : ϕ(x) = ϕ(y)}.
(4.13)
∼
X,ϕ
partitions universe X into non-overlapping equivalence classes denoted by X
/∼
ϕ
or
simply X
/∼
. Let x
/∼
denote a class containing an element x as in (4.14).
x
/∼
={y∈X|x∼
ϕ
y}.
(4.14)
Let X
/∼
denote the quotient set as defined in (4.15).
X
/∼
={x
/∼
|x∈X}.
(4.15)
The relation∼holds for all members of each class x
/∼
in a partition.
In a rough set model, elements of the universe are described based on the available infor-
mation about them. For each subset A⊆X, rough set theory defines two approximations
based on equivalence classes, lower approximation A
−
and upper approximation, A
−
:
A
−
={x∈X : x
/∼
⊆A}
(4.16)
A
−
={x∈X : x
/∼
∩A 6= φ}
(4.17)
The set A⊂X is called a rough set if A
−
6= A
−
, otherwise it is called exact set, (Pawlak,
1991). Figure 4.1 shows the set A and lower and upper approximation of it in terms of the
partitioned space. The space X is partitioned in to squares of the form (j, j + 1]
2
.
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