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are initialized and the value of the hidden variable
z
(
i
m
is calculated. The value of
b
m
and
[
ʱ
,
]
The parameter
b
m
m
ʱ
m
are then updated by using the new value of
z
(
i
m
using Eq. (
10.6
) and Eq. (
10.7
). This is done iteratively until the algorithm
converges.
10.2.3
Comparison of Gaussian Mixture Model and Laplacian
Mixture Model
In order to make a comparison between GMM and LMM, a study was performed
on the Brodatz image database. The images from the Brodatz image database
were decomposed to three levels using the Daubechies db4 wavelet kernel. The
wavelet coefficients in all the high-frequency subbands (9 for 3-level decomposi-
tion) were modeled with two components, GMM and LMM. The model accuracy
was measured using the Kullback-Leibler Divergence (KL). KL is a quantity
which measures the difference between two probability distributions. KL can be
considered as a kind of distance between the two probability densities. But it is not
a real distance measure because it is not symmetric. The likelihood of the model
can be measured in terms of the KL between the observed density and the density as
calculated by the model. The KL between two probability distributions of a discrete
variable is defined as,
p
(
x
)
)=
x
KL
(
p
,
q
p
(
x
)
log
(10.9)
q
(
x
)
where
p
)
is the normalized histogram of the wavelet coefficients. There are a total of 16,704
high-frequency subbands for 3-level decomposition of 1,856 images (1
(
x
)
is the estimated probability distribution from GMM or LMM, and
q
(
x
9).
The wavelet coefficients of high-frequency subbands at the first, second, and third
levels of decomposition are quantized into 256, 128 and 64 levels respectively, to
properly represent the resolutions of different decomposition levels. A normalized
histogram is constructed that represents the true distribution
q
,
856
×
of the coefficients.
The distributions using LMM and GMM models are then calculated. The KL
distance is calculated between normalized histograms and the distributions obtained
from the GMM and LMM models. It is observed that, out of 16,704 total cases,
LMM gives a lower value of KL in 16,494 cases. In the remaining 210 cases, KL
was lower in case of GMM. These results indicate that in 98.74 % of the test cases,
the LMM model with two components is closer to the true distribution in terms of
KL statistics. This evaluation indicates that LMM is a more appropriate model for
the wavelet coefficient distribution and can model it with only two components. The
fitting of the GMM and LMM models for a typical high-frequency wavelet subband
of an image from the Brodatz image database is illustrated in Figs.
10.2
and
10.3
.
(
x
)
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