Image Processing Reference
InDepth Information
TABLE 1 . 2
Pawlak Indiscernibility Relation and Partition Symbols
Symbol
Interpretation
∼
B
=
{
(
x
,
y
)
∈
X
×
X

f
(
x
)=
f
(
y
)
∀
f
∈ B}
, indiscernibility,
cf.
(Pawlak, 1981a),
x
/∼
B
x
/∼
B
=
{
y
∈
X

y
∼
B
x
}
, elementary set (class),
U
/∼
B
U
/∼
B
=
{
x
/∼
B

x
∈
U
}
, quotient set.
B
∗
(
X
)
B
∗
(
X
) =
x
/
∼
B
(lower approximation of X),
x
/
∼
B
⊆
X
B
∗
(
X
)
B
∗
(
X
) =
x
/
∼
B
(upper approximation of X).
x
/
∼
B
∩
X
=0
Set Theory Law 3 Rough Sets
Any nonempty set X is a rough set if, and only if the approximation boundary of X is not empty.
Rough sets were introduced by Z. Pawlak in (Pawlak, 1981a) and elaborated in (Pawlak, 1981b;
Pawlak and Skowron, 2007c,b,a). In a rough set approach to classifying sets of objects
X
, one
considers the size of the boundary region in the approximation of
X
. By contrast, in a near set
approach to classification, one does not consider the boundary region of a set. In particular, assume
that
X
is a nonempty set belonging to a universe
U
and that
is a set of features defined either by
total or partial functions. The lower approximation of
X
relative to
F
B ⊆
F
is denoted by
B
∗
(
X
) and
B
∗
(
X
), where
B
∗
(
X
)=
x
the upper approximation of
X
is denoted by
x
/
∼
B
,
/∼
B
⊆
X
B
∗
(
X
)=
x
x
/
∼
B
.
/
∼
B
∩
X
=0
The
B
boundary region of an approximation of a set
X
is denoted by
Bnd
B
(
X
), where
B
∗
(
X
)
∈ B
∗
(
X
) and
x
Bnd
B
(
X
)=
\ B
∗
(
X
)=
{
x

x
∈ B
∗
(
X
)
}.
Definition 3 Rough Set
(Pawlak, 1981a)
A nonempty, finite set
X
is a rough set if, and only if
B
∗
(
X
)
−B
∗
(
X
)

= 0.
Aset
X
is roughly classified whenever
Bnd
B
(
X
) is not empty. In other words,
X
is a rough set
whenever the boundary region
Bnd
B
(
X
)
= 0. In sum, a rough set is a Cantor set if, and only if its
approximation boundary is nonempty. It should also be noted that rough sets differ from near sets,
since near sets are defined without reference to an approximation boundary region. This means, for
example, with near sets the image correspondence problem can be solved without resorting to set
approximation.
Method 2 Rough Set Approach
1. Let (
U
,B
) denote a sample space (universe)
U
and set of object features
B
,
2. Using relation
∼
B
, partition the universe
U
,
3. Determine the size of the boundary of a set
X
.
∈
U
.
x
/∼
B
(any elementary set) is a nonrough set.
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