Digital Signal Processing Reference
In-Depth Information
t
w
f
(a)
(b)
FIGURE 10.4
(a) HMT-2 model. (b) A simplified interpretation of HMT-2.
where #
B
occurring. From
our numerous simulation results, the above scale-independent initial values,
by averaging the probabilities across different scales, outperform the scale-
dependent ones without averaging. This averaging operation may not be
necessary when we have enough data during the model training, such as in
image processing.
16
The above scheme can efficiently characterize the initial
marginal and joint statistics of
w
,
(
A
=
B
)
denotes the number of the event
A
=
0
, which allows the efficient EM model
θ
training as shown by experiments.
10.2.2
Improved HMT-2 Model
To capture more cross-correlation of wavelet coefficients between two neigh-
boring scales, a new HMM, HMT-2, is developed and shown in Figure 10.4a,
where the state of the wavelet coefficient
depends not only on the state of
its parent node, but also on the state of the twin of its parent. This strategy is
very popular in the graphical modeling and Bayesian network literatures.
17
Tw oreasons for this consideration are (1) the local stationary property of
most signals and the correlation of the wavelet functions in two neighbor-
ing scales, and (2) the wavelet filter-bank decompositions, where the filter
length may be long enough for wavelet coefficients to have more interscale
dependencies. Usually, the analysis and training of more complicated HMMs
become more difficult.
18
A simplified interpretation of HMT-2 is illustrated
in Figure 10.4b, where two coefficients are integrated into one node. Actually,
HMT-2 operates in the same way as the HMT model in Reference 1 except for
the number of hidden states associated with each node. If we assume two hid-
den states for each coefficient, i.e., 0 and 1, each node of HMT-2 will have four
states: 00, 01, 10, and 11. We call our new model HMT-2 instead of four-state
HMT in order to distinguish it from the original
M
-state HMT when
M
w
=
4.
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