Sendur and Selesnick [5] proposed a bivariate shrinkage estimator by estimating the marginal variance of the wavelet coefficients via small neighborhoods as well as from the the corresponding neighborhoods of the parent coefficients. The developed method maintains the simplicity and intuition of softthresholding.
We can write
y
k
=
w
k
+
n
k
,
y
k
=
w
k
+
n
k
,
(1)where wkwk are the parent and child wavelet coefficients of the true, noisefree image and nknk is the noise. We have for our variance, then, that
σ
y
2
=
σ
k
2
+
σ
n
2
.
σ
y
2
=
σ
k
2
+
σ
n
2
.
(2)Noting that we will always be working with one coefficient at a time, we will suppress the kk.
In [4], Sendur and Selesnick proposed a bivariate pdf for the wavelet coefficient w1w1 and the parent w2w2 to be
p
w
(
w
)
=
3
2
π
σ
2
exp
(

3
σ
(
w
1
2
+
w
2
2
)
,
p
w
(
w
)
=
3
2
π
σ
2
exp
(

3
σ
(
w
1
2
+
w
2
2
)
,
(3)where the marginal variance σ2σ2 is dependent upon the coefficient index kk. They derived their MAP estimator to be
w
^
1
=
(
y
1
2
+
y
2
2

3
σ
n
2
σ
)
+
y
1
2
+
y
2
2
y
1
w
^
1
=
(
y
1
2
+
y
2
2

3
σ
n
2
σ
)
+
y
1
2
+
y
2
2
y
1
(4)To estimate the noise variance σn2σn2 from the noisy wavelet coefficients, they used the median absolute deviance (MAD) estimator
σ
^
n
2
=
m
e
d
i
a
n
(

y
i

)
0
.
6745
,
y
i
∈
s
u
b
b
a
n
d
H
H
,
σ
^
n
2
=
m
e
d
i
a
n
(

y
i

)
0
.
6745
,
y
i
∈
s
u
b
b
a
n
d
H
H
,
(5)where the estimator uses the wavelet coeffiecients from the finest scale.
The marginal variance σy2σy2 was estimated using neighborhoods around each wavelet coefficient as well as the corresponding neighborhood of the parent wavelet coefficient. For instance, for a 7x7 window, we take the neighborhood around y1,(4,4)y1,(4,4) to be the wavelet coefficients located in the square (1, 1), (1, 7), (7, 7), (7, 1) as well as the coefficients in the second level located in the same square; this square is denoted N(k)N(k). The estimate used for σy2σy2 is given by
σ
^
y
2
=
1
M
∑
y
i
∈
N
(
k
)
y
i
2
,
σ
^
y
2
=
1
M
∑
y
i
∈
N
(
k
)
y
i
2
,
(6)where MM is the size of the neighborhood N(k)N(k). We can then estimate the standard deviation of the true wavelet coefficients through Equation 2:
σ
^
=
(
σ
^
y
2

σ
^
n
2
)
+
.
σ
^
=
(
σ
^
y
2

σ
^
n
2
)
+
.
(7)We then have the information we need to use equation Equation 4.