Digital Signal Processing Reference
In-Depth Information
10.3.2
Local Contextual Hidden Markov Model
In addition to GMF, we use a context model to capture intrascale dependen-
cies of wavelet coefficients as shown in Figure 10.7b. We define the random
context variable of
W
j,k,i
by
V
j,k,i
whose value is
v
=
λ
j,k,i
>δ
j
,or
1if
j,k,i
v
=
λ
j,k,i
is the local average energy of the eight near-
0 otherwise, where
j,k,i
j
the average energy in scale
j
.Byconditioning on
V
j,k,i
and using GMF, we develop the local contextual hidden Markov model
(LCHMM) for
est neighbors of
w
j,k,i
, and
δ
w
j,k,i
as
1
g
j,k,i,m
,
2
f
W
j,k,i
|
V
j,k,i
(w
|
v
=
v)
=
p
S
j,k,i
|
V
j,k,i
(
m
|
v
=
v)
w
|
0
,
σ
(10.12)
j,k,i
j,k,i
m
=
0
where
p
S
j,k,i
(
m
)
p
V
j,k,i
|
S
j,k,i
(v
|
m
)
p
S
j,k,i
|
V
j,k,i
(
m
|
v
=
v)
=
)
.
(10.13)
1
m
=
0
j,k,i
p
S
j,k,i
(
m
)
p
V
j,k,i
|
S
j,k,i
(v
|
m
LCHMM is specified by
j,k,i
=
p
S
j,k,i
(
0
,
1
,
2
m
)
,
σ
j,k,i,m
,p
V
j,k,i
|
S
j,k,i
(v
|
m
)
|
v
,m
=
where
j
1. In fact, LCHMM defines a
local density function for each wavelet coefficient conditioning on its context
value. The EM training algorithm can be developed from that in Reference 13.
Because
=
1
,
...
,J
and
k, i
=
0
,
1
,
...
,N
j
−
j,k,i
has a small number of data, one might be concerned that the
estimation of
j,k,i
may not be robust. In this work, we can solve this problem
by providing a good initial setting of
j,k,i
based on an idea similar to that in
the previous section. Given the AWGN of variance
2
η
σ
, the LCHMM training
is performed as follows, where
x
y
denotes
k
+
C
j
C
j
i
+
C
j
C
j
.
x
=
k
−
y
=
i
−
•
Step 1. Initialization:
1.0
j
j,
0
2
η
j,
1
j
2
={
p
S
j
(
0
)
=
p
S
j
(
1
)
=
0
.
5
,
σ
=
σ
,
σ
=
2
δ
−
σ
η
}
and set
=
0.
1.1 E step: Given
p
p
j
, calculate (Bayes rule)
g
w
j,m
p
S
j,k,i
j
=
p
S
j
(
m
)
j,k,i
;0
,
σ
p
=
m
|
w
j,k,i
,
g
w
j,k,i
;0
,
j,m
.
1
m
=
0
2
p
S
j
(
m
)
σ
(10.14)
p
+
1
1.2 M step: Compute the elements of
by
j
N
j
−
1
N
j
−
1
p
S
j,k,i
j
,
p
p
S
j
(
m
)
=
=
m
|
w
j,k,i
,
(10.15)
k
=
0
i
=
0
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