Image Processing Reference
In-Depth Information
The roughness of X, denoted by ρ
R
(X), is defined by subtracting the accuracy from 1:
ρ
R
(X) = 1−α
R
(X) = 1−
|R(X)|
|R(X)|
Note that the lower the roughness of a subset, the better is its approximation. Further,
the following observations are easily obtained:
1. As R(X)⊆X⊆R(X), 0≤ρ
R
(X)≤1.
2. By convention, when X =∅, R(X) = R(X) =∅and ρ
R
(X) = 0.
3. ρ
R
(X) = 0 if and only if X is definable in < U, R >.
2.3
Rough-Fuzzy C-Means Algorithm
Incorporating both fuzzy and rough sets, next a newly introduced c-means algorithm,
termed as rough-fuzzy c-means (RFCM) (Maji and Pal, 2007a,c), is described. The RFCM
algorithm adds the concept of fuzzy membership of fuzzy sets, and lower and upper approx-
imations of rough sets into c-means algorithm. While the membership of fuzzy sets enables
e
cient handling of overlapping partitions, the rough sets deal with uncertainty, vagueness,
and incompleteness in class definition.
2.3.1 Objective Function
Let A(β
i
) and A(β
i
) be the lower and upper approximations of cluster β
i
, and B(β
i
) =
{A(β
i
)−A(β
i
)}denote the boundary region of cluster β
i
. The RFCM partitions a set of n
objects into c clusters by minimizing the objective function
8
<
w×A
+ w×B
1
if A(β
i
) =∅, B(β
i
) =∅
1
A
if A(β
i
) =∅, B(β
i
) =∅
J
RF
=
(2.4)
1
:
B
1
if A(β
i
) =∅, B(β
i
) =∅
c
c
||x
j
−v
i
||
2
B
1
=
(
ij
)
´m
||x
j
−v
i
||
2
A
1
=
i
=1
i
=1
x
j
∈A(β
i
)
x
j
∈B(β
i
)
v
i
represents the centroid of the ith cluster β
i
, the parameter w and w correspond to the
relative importance of lower bound and boundary region, and w + w = 1. Note that,
ij
has the same meaning of membership as that in fuzzy c-means.
In the RFCM, each cluster is represented by a centroid, a crisp lower approximation,
and a fuzzy boundary
(Fig. 2.1)
.
The lower approximation influences the fuzziness of final
partition. According to the definitions of lower approximations and boundary of rough sets,
if an object x
j
∈A(β
i
), then x
j
/∈A(β
k
),∀k = i, and x
j
/∈B(β
i
),∀i. That is, the object
x
j
is contained in β
i
definitely. Thus, the weights of the objects in lower approximation of
a cluster should be independent of other centroids and clusters, and should not be coupled
with their similarity with respect to other centroids. Also, the objects in lower approxima-
tion of a cluster should have similar influence on the corresponding centroid and cluster.
Whereas, if x
j
∈B(β
i
), then the object x
j
possibly belongs to β
i
and potentially belongs
to another cluster. Hence, the objects in boundary regions should have different influence
on the centroids and clusters. So, in the RFCM, the membership values of objects in lower
approximation are
ij
= 1, while those in boundary region are the same as fuzzy c-means
(Equation 2.3). In other word, the RFCM algorithm first partitions the data into two classes
- lower approximation and boundary. Only the objects in boundary are fuzzified.
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