Digital Signal Processing Reference
In-Depth Information
the original training patterns (i.e., RGB triplets) is evaluated as follows:
1
card
x
(
k
)
(
2
,
E
(
k
)
=
)
∈X
t
x
(
i, j
)
−
i, j
)
(11.90)
(
X
)
t
x
(
i, j
where
X
t
denotes the training set, card
(
X
)
stands for the cardinality of the
t
T
represents the original train-
training set,
x
(
i, j
)
=
(
x
R
(
i, j
)
,
x
G
(
i, j
)
,x
B
(
i, j
))
ing pattern and
x
is the quantized pattern. The training patterns can be
obtained from the input color image, e.g., by subsampling. In steepest-descent
algorithms (e.g., SOM), the MSE is a nonmonotonically decreasing function
of the iteration index. Consequently, unifying terminating rules, such as
E
(
(
i, j
)
k
−
1
)
− E
(
k
)
≤
ϑ
(11.91)
E
(
k
)
may not work, because there is no guarantee that
E
(
k
)
is always less than
E
(
.Wehave decided to tolerate the same maximum number of violations
of the terminating rule (Equation 11.91) in all the algorithms to be compared.
We decide that the learning procedure has converged, if Equation 11.91 is
satisfied for
k
−
1
)
10
−
4
.Inour comparative study, we have used as figure
of merit the MSE at the end of the recall phase that is defined similarly to
Equation 11.90, i.e.,
ϑ
=
M
M
1
M
2
2
,
=
1
(
)
−
x
(
)
MSE
x
i, j
i, j
(11.92)
i
=
1
j
=
where
M
is the number of image rows/columns.
Let us consider the case of a codebook of 32 RGB triplets that is learned from
a training set of 4096 patterns, extracted from a noisy frame of color image
sequence “Trevor White,” shown in Figure 11.8a. It is applied to quantize
several original frames of the same color image sequence. Figure 11.8a shows
the first frame of “Trevor White” corrupted by adding mixed white zero-
mean Gaussian noise having standard deviation
20 and impulsive noise
with probability of impulses 7% independently to each R, G, B component.
The weight vectors determined at the end of the learning procedure on the
training set have been applied to quantize the 10th, 50th, 100th, 120th, and
129th frame. The MSE achieved at the end of the recall phase of both SOM and
MMSOM is shown in Table 11.1. It is seen that the color palette determined
by the MMSOM yields the smallest MSE in all cases. The quantized images
produced by the SOM and the MMSOM when frame 50 is considered are
shown in Figure 11.8b and c. For comparison purposes, the original frame
50 is shown in Figure 11.8d. It can be verified that the visual quality of the
quantized output of MMSOM is higher than that of the SOM.
σ
=
11.6.3.2 Document Organization and Retrieval
An architecture based on the SOM algorithm, which employs the vector space
model
58
and is capable of clustering documents according to their semantic
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