Database Reference
In-Depth Information
2.4.5
Fuzzy RBF Network with Soft Constraint
ʾ
f
defined in Eq. (
2.65
) is based on the binary labeling or
hard-decision. The desired network output
y
j
is equal to 1 for positive samples,
and zero for negative samples. For a soft-decision, a third option, “fuzzy” is used
to characterize a vague description of the retrieved (image) samples [
39
]. Thus,
in a retrieval session, users have three choices for relevance feedback: relevant,
irrelevant, and fuzzy. The error function is then calculated by:
The error function
y
j
−
2
ʾ
f
=
N
j
=
1
N
i
=
1
ʻ
i
K
(
x
j
,
z
i
)
1
2
(2.72)
⊧
⊨
1
,
x
j
is relevant
y
j
=
0
,
x
j
is irrelevant
(2.73)
⊩
(
X
+
|
P
x
j
)
,
x
j
is fuzzy
(
X
+
|
where
P
x
j
)
is the probability that a fuzzy sample
x
j
belongs to the relevant
X
+
. This represents the degree of relevancy for the corresponding fuzzy
sample. The learning problem is the problem in estimating the desired output
y
j
class
(
X
+
|
of the fuzzy sample
x
j
by the
a posteriori
probability estimator.
Let
x
j
be defined by feature vector that is concatenated from
M
sub-vector, i.e.,
x
j
=
P
x
j
)
, where
v
ji
is a
d
i
-dimensional feature sub-vector such as a
color histogram, a set of wavelet moments or others. To deal with the uncertainly, the
probability estimator takes into account the multiple features, by using the following
estimation principle:
≡
[
v
j
1
,...,
v
ji
,...,
v
jM
]
P
X
+
|
x
j
=
i
=
1
P
X
+
|
v
ji
M
1
M
(2.74)
(
X
+
|
where
P
is the
a posteriori
probability for the
i
-th feature vector
v
ji
of the
fuzzy sample
x
j
. the Bayesian theory is applied to
P
v
ji
)
(
X
+
|
v
ji
)
,
P
X
+
|
v
ji
=
v
ji
|X
+
)
(
X
+
)
P
(
P
(2.75)
v
ji
|X
+
)
(
X
+
)+
v
ji
|X
−
)
(
X
−
)
P
(
P
P
(
P
(
X
+
)
(
X
−
)
where
P
are, respectively, the prior probabilities of the pos-
itive and negative classes, which can be estimated from the feedback samples;
P
and
P
v
ji
|X
+
)
v
ji
|X
−
)
are the class conditional probability density functions
of
v
ji
for the positive and negative classes, respectively. Assuming the Gaussian
distribution, the probability density function for the positive class is given by:
(
and
P
(
P
v
ji
|X
+
=
2
v
ji
−
μ
i
t
1
1
−
1
∑
v
ji
−
μ
i
)]
exp
[
−
(
(2.76)
d
2
i
2
|
∑
i
|
(
2
ˀ
)
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