The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
The multiplicative noise can be translated to the additional noise 
with logarithmic transform, which can be formulated as in the 
wavelet domain 
y = w+ n 
(2) 
Here n is denoted as wavelet coefficient of noise and different 
with n in (1). The definition of n in the following sections is 
same as that in (2) without special instructions. 
In (Levent S.endur et al., 2002a) the wavelet coefficient y ', 
w ', n ' of current scale and y i, w 2, at the same position of 
parent scale are denoted as 
y = w + n 
where , w = (w 1 ,w 2 )^ n = (« I ,H 2 ) r . 
(3) 
The problem of speckle denoising is equivalent to finding an 
SUp II w - w || 
optimal estimation w to make minimum. Then 
the MAP estimator w is 
w(y) = argmax p(w | y) 
w (4) 
After some manipulations and logarithmic transform, (4) can be 
written as 
w(y) = arg max{log[/? n (y - w)] + log[/? w (w)]} 
(5) 
From this equation, in order to use this equation to estimate the 
original signal, we must know both pdfs of noise and original 
signal. We assume the joint pdf of noise as 
PM 
ln x n 2 
2 T 
cr.o-, 
' exp[-( 
(6) 
P (w) 
The same problem of wV ’ appears. Levent S.endur and Ivan 
W. Selesnick (Levent S.endur et al., 2002a) have used the joint 
empirical coefficient-parent histogram to observe P" ^. Since 
SAR image is a description of natural scene, we hypothesize 
P (w) 
w v ' of SAR image is according to this distribution. Levent 
S.endur and Ivan W. Selesnick (Levent S.endur et al., 2002a) 
have proposed four models. The model 2, 3, 4 are better than 
model 1 but too complicated to be solved, so that we choose 
model 1 to estimate w v ’. 
Model 1 in (Levent S.endur et al., 2002a) is denoted as 
AvO) 
3 
Itzo 1 
•VOi) 2 +o 2 ) 2 ) 
(7) 
\og[2{y x -w x \y 2 -W2)\- 
—-^(w,) 2 +(w 2 ) 2 
a (8) 
where = log3-log(cr„ 2 ,cr^ 2 )-log(2^a 2 ) 
(8) is equivalent to solving the following equations together, if 
P (w) 
w v ’ is assumed to be strictly convex and differentiable: 
2(y, -w,) 
1 
A 
*1 _n 
<£“ 
1 
cr 
/ a 2 * 2 
V W 1 + W 2 
2 (y 2 ~w 2 ) 
1 
S 
^2 _n 
<4 
1 
<N 
A 
CJ 
yjw 2 +w 2 2 
This is a bivariate equation about w > and , and it is hard to 
be solved in program and have to be deduced. 
Here a new joint pdf of noise similar with (6) is denoted as 
cr ,cr 
exp[-(- 
)] 
(10) 
The difference between (6) and (10) is X. Many experiments 
show shat when ”• and "2 are changing in [0, 1], ”'"2 in (6) 
can be substituted by X. Figure 1 shows the surface of (6) and 
(10) with a: = 33 i n [0, 1]. And we find that AC is a constant 
while the variance of w in [0, 255]. When the wavelet 
coefficients are normalized in [0, 1], (10) can be substituted for 
(6). Then (9) can be written as 
2{y x -w x ) V3 w, 
cr 2 , cr 
2 * 2 
W, + W 0 
= 0 
2OV-W2) ^2 
cr 
n 2 
V 
2 a 2 
W y + 
= 0 
After some manipulations, this equation can be written as 
(H) 
W, 
W, 
1 + 
2 o -Y 
= y\ 
1 + 
V3 • cr 2 ^ 
2 cr-r 
=y 2 
(12) 
where r = 
By using logarithmic transforms of (6) and (7), (5) can be 
written as 
We hypothesize cr ” 1 CT " 2 CT " and r can be written as 
w(y) = arg max{- 
(a-^) 2 (y 2 -û 2 y 
V3cr 
2cr 
2 
-)h