Digital Signal Processing Reference
In-Depth Information
t
0
1
0
scale
j+
1
0
1
2
3
scale
j
0
1
2
3
4
5
6
7
scale
j-
1
f
(a)
t
f
(b)
(c)
FIGURE 10.3
(a) Tiling of the time-frequency plane of DWT. The solid dot at the center corresponds to the
scaling coefficients or wavelet coefficients. The tree structure is shown by the link of the dotted
lines. (b) The histogram of the one-scale DWT (Daubechies-4) of the “fruit” image where a two-
state zero-mean Gaussian mixture model can closely fit the real DWT data.
1
(c) Wavelet-domain
HMT, where the white node represents the state variable
S
and the black node denotes the wavelet
coefficient
W
.
10.2
Wavelet-Domain Hidden Markov Models
In the following, we briefly review wavelet-domain HMMs in the one-
dimensional (1D) case. For more details, we refer the reader to Reference 1.
Given a bandpass wavelet function
ψ(
t
)
and a lowpass scaling function
φ(
t
)
, the DWT represents a signal
s
(
t
)
of size
N
in terms of shifted versions
φ(
)
ψ(
)
of
t
and shifted and dilated versions of
t
,asshown in Figure 10.3a.
ψ
(
)
≡
2
−
j
/
2
ψ(
2
−
j
t
−
)
φ
(
)
≡
2
−
j
/
2
φ(
2
−
j
t
−
)
The atoms of DWT are
t
i
,
t
i
,
j,i
j,i
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