3 Speckle Noise Reduction and Enhancement for OCT Images
61
By substituting “mixture of bivariate Gaussian pdfs with local parameters” as the
prior distribution of 3D RCWT coefficients, i.e., p ¯
w(k) ( ¯
w(k)) for AWGN we would
have:
g i ( ¯
y(k))
exp
−
1
2
y
2 (k)
σ 2
n +σ
2
i1 (k)
+
y
2
p (k)
σ 2
n +σ
2
i2 (k)
2π
(σ 2
n + σ
2
i1 (k))(σ 2
n + σ
2
i2 (k))
, i 1, 2
(3.32)
Similarly, after some simplifications for two-sided Rayleigh noise g i ( ¯
y(k)) would
be:
g i ( ¯
y(k))
exp
−
y
2 (k)
2σ
2
i1 (k)
−
y
2
p (k)
2σ
2
i2 (k)
8π (1 +
σ
2
i1 (k)
α 2 )(1 +
σ
2
i2 (k)
α 2 )σ i1 (k)σ i2 (k)
× (2 + z i (k)
√
π er f cx(−z i (k))
− z i (k)
√
π er f cx(z i (k)))(2 + z i p (k)
√
π er f cx(−z i p (k))
− z i p (k)
√
π er f cx(z i p (k))),
i 1, 2
(3.33)
where
z i (k)
y(k)
σ
2
i1 (k)
1
2
α 2 +
2
σ
2
i1 (k)
, i 1, 2
(3.34)
z i p (k)
y p (k)
σ
2
i2 (k)
1
2
α 2 +
2
σ
2
i2 (k)
, i 1, 2
(3.35)
The numerators of (3.30) can be obtained as MMSE estimate of a single component model [69]. So, for AWGN, the shrinkage function (3.30), which is called
BiGaussMixShrinkL, can be written as:
ˆ
w(k)
σ
2
11 (k)
σ
2
11 (k)+σ 2
n
+ R( ¯
y(k))
σ
2
21 (k)
σ
2
21 (k)+σ 2
n
1 + R( ¯
y(k))
y(k)
(3.36)
where
R( ¯
y(k))
(1 − a(k))
exp
−
1
2
y 2 (k)
σ 2
n +σ 2
21 (k)
+
y 2
p (k)
σ 2
n +σ 2
22 (k)
√
(σ 2
n +σ
2
21 (k))(σ 2
n +σ
2
22 (k))
a(k)
exp
−
1
2
y 2 (k)
σ 2
n +σ 2
11 (k)
+
y 2
p (k)
σ 2
n +σ 2
12 (k)
√
(σ 2
n +σ
2
11 (k))(σ 2
n +σ
2
12 (k))
(3.37)