anbul 2004 
  
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's that have 
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se absolute 
eorem, the 
and signal, 
stimate the 
PDFs parameters which are a, , y, andg,,. For this purpose, it 
is preferable to use the characteristic functions rather than the 
PDFs. Since the wavelet coefficient of the noisy data d is equal 
to the sum of the wavelet coefficients of the signal s and noise 
e, we have [3]: 
P,(d)= P,(s)* F,(e) (8) 
where * represents the convolution operation. The 
corresponding characteristic function ¢@,(w) will be the 
product of the signal and noise characteristic functions: 
$,(0)  $, ().4, (m) (9) 
For estimating the unknown parameters, @, andy, à; (œ) is 
fitted to the Fourier transform of the wavelet coefficients 
histogram. For this purpose, the least square (LS) method is 
used which gives a robust solution. Moreover, the noise 
level, 0, can be estimated using 
à, 21.3* MAD(d,) (10) 
where the operator MAD represents the mean absolute deviation 
and d; is the wavelet coefficients of the highest level. After 
estimating the unknown parameters, s(d ) in each level of the 
wavelet transform can be computed using Eq. (7). 
At the final stage, by applying the inverse wavelet and 
exponential transformations, the denoised image is 
reconstructed. In the next section, experimental results of the 
proposed technique are presented and compared to other noise 
reduction methods. 
3. EXPERIMENTAL RESULTS 
In order to compare different noise reduction techniques, 
speckle noise with the variance of 0.005 was added to an aerial 
photo of size 64 x 64. In order to evaluate selected wavelet 
basis, different wavelet functions included coifletl and 
symmlet4 were used while applying Mallat’s algorithm. 
Furthermore, in order to decrease the artifacts in the results, the 
a trous algorithm with the Gaussian lowpass filter was used to 
decompose the image. 
The wavelet coefficients were obtained from the logarithmically 
transformed image using a frous algorithm. The maximum 
number of wavelet decomposition levels was 2. Then, these 
wavelet coefficients were modeled using alpha-stable and 
Gaussian distribution functions. Figure (2) illustrates the 
modeling of the coefficients of the second 
decomposition level. 
To evaluate the results, two criteria including the signal to noise 
ratio and correlation were used. The signal to noise ratio (S/N) 
is defined by: 
wavelet 
k k 
SIN=~S/mse=10log,,} 521). G6, -s)?) D 
i=l i=l 
29 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
where S and $ represent the original (before adding noise) and 
denoised images, respectively. The correlation measure ( f) is 
computed using: 
3 (As, - AS(AS, — A5) (12) 
BE 
  
where As,A$ are high pass filtered of 8,8 using Laplacian 
filter. The correlation value is close to one when the edges are 
optimally preserved in the image. 
ir 
09- 
04 
07} 
0.87 
os 
94r 
  
03} 
22r 
  
  
04 
  
  
= 3s 4 2 
Figure 2. Modeling of the second level of wavelet transforms 
using SaS and Gaussian distributions. The estimated 
parameters Qr, , y, and C, are 0.6469, 0.0774 and 0.0707, 
respectively. 
Table (3) presents the quantitative results of different noise 
reduction methods. It can be observed that the proposed method 
using a trous algorithm has the best results from the signal to 
noise ratio and edge preservation points of view. 
Table 3. Results of different speckle noise reduction methods. 
a trous ithm 
Mallat i 
Mallat 
Wiener Filter 
Hard-Thresholding 
Garrote-Thresholding 
Soft-Thresholding 
Frost Filter 
Median Filter 
Lee Filter 
Gamma Filter 
Mean Filter