Image Processing Reference
In-Depth Information
FIGURE 6.4: GK clusters projections and derived membership functions. h
(i)
are defined
in Eq.6.29 and the lower index indicates the cluster number.
where a = v
(i)
−
h
(i)
2
h
(i)
2
, b = c = v
(i)
, d = v
(i)
+
for i = 1, 2, . . . , n and h
(i)
defines the
width of the membership function
h
(i)
= 4·var
(i)
(v)
(6.29)
Since membership functions overlap each other, they are merged. In detail, two neigh-
boring membership functions Π(x, a
l−1
, b
l−1
, c
l−1
, d
l−1
) and Π(x, a
l
, b
l
, c
l
, d
l
) are merged if
the following condition is satisfied:
b
l
+ c
l
2
−
b
l−1
+ c
l−
2
≤l
t
(6.30)
where l
t
={
1
, . . . ,
n
}are pre-specified thresholds defined for each input. The resulting
membership function after the merging has the following form: Π(x, a = min(a
l
, a
l−1
), b =
min(b
l
, b
l−1
), c = min(c
l
, c
l−1
), d = min(d
l
, d
l−1
)). As a result of the merging process, some
membership functions have trapezoidal shapes instead of triangular ones.
After merging, pattern labelling is performed. Given the input feature vector from the
training data set (Eq. 6.22) the resulting class C
K
is assigned the label of a cluster with the
highest membership value.
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