Digital Signal Processing Reference
In-Depth Information
The influence functions of the MMSOM weights are
1
IF
(
x
;
w
M
1
,F
)
=
sgn
(
x
−
w
)
[1
−
u
s
(
x
−
T
MMSOM
)
]
( 1.85)
M
1
f
1
2
(w
)
M
1
1
(
w
)
=
(
−
w
)
(
−
)
IF
x
;
M
2
,F
sgn
x
u
s
x
T
MMSOM
,
(11.86)
M
2
f
2
2
(w
)
M
2
where sgn
(
x
)
is the signum function. Accordingly,
1
V
(w
Mi
,F
)
=
,
i
=
1
,
2
.
(11.87)
f
i
(w
Mi
)
4
The modified ARE (Equation 11.78) for the Gaussian mixture model are plot-
ted in Figure 11.7a for several values of
1, the
SOM outperforms MMSOM with respect to the mean-squared error. The per-
formance of MMSOM is improved, as
and
σ
.Itisseen that for
n
=
increases. However, even in this case,
the SOM is still better than the MMSOM. If the mixture comprised Laplacian
distributions, i.e.,
σ
f
(
x
)
=
f
1
(
x
)
+
(
1
−
)
f
2
(
x
)
,
(11.88)
where
σ
i
√
2
exp
√
2
|
1
x
−
m
i
|
f
i
(
x
)
=
−
i
=
1
,
2
,
(11.89)
σ
i
with
m
1
10, the same analysis would reveal that the MMSOM
always outperforms the SOM.
44
The corresponding modified ARE is plotted
in Figure 11.7b.
=
5 and
m
2
=
11.6.3
Applications
Tw o applications of the variants of SOM that are based on robust statistics are
described. The first application is in color image quantization and the second
is in document organization and retrieval.
11.6.3.1 Color Image Quantization
In practice, 8 bits are used for representing each of the R, G, B components in
color images. That is, 24 bits are needed for each pixel in total. Color image
quantization aims at reducing the number of RGB triplets (which are 2
24
at
most) to a predefined number (e.g., 16 to 256) of codevectors so that 8 bits
or less suffice to encode all the RGB.
56
Color image quantization is a useful
preprocessing step in histogram-based algorithms for color image retrieval.
57
A set of experiments has been conducted to assess the performance of
MMSOM in color image quantization and to compare it to that of well-known
vector quantization (VQ) methods such as the LBG algorithm
28
and the SOM.
We have also included noisy color images as inputs to the learning phase.
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