International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
quantitative and qualitative evaluation of the results. Finally, 
concluding remarks are given in section 4. 
2. SPECKLE NOISE REDUCTION MODEL 
A simple model for speckle noisy image has a multiplicative 
form [16], 
YG5 y) 2 505 y).NG, y) () 
where Y, S and N represent the noisy data, signal and speckle 
noise, respectively. In order to change the multiplicative nature 
of the noise to additive one, we apply a logarithmic 
transformation to the image data. Taking logarithm of the both 
sides of Eq. (1), we will have: 
F(x, y) = s(x, y) + e(x, y) 2) 
where f, s and e represent logarithms of the noisy data, signal 
and noise, respectively. The next step is the computation of 
wavelet transform of f(x, y) . One of the important issues to be 
considered in wavelet transform is the choice of the best 
wavelet function as well as the transformation algorithm. Since 
we are interested in isolating the speckle noise in the image, the 
most appropriate wavelet function is one, which its shape looks 
like the speckle pattern. For this purpose, we computed the 
average of x and y cross sections of several speckle samples in 
the logarithmically transformed data. According to this study, 
the 2D Gaussian function has been found to be the best model 
fitted to the speckle pattern cross-section. Figure (1) illustrates 
the shapes of the x and y cross sections of the averaged speckle 
noise and Gaussian curves fitted to them. 
  
  
54 
Figurel. The average of x and y cross sections (solid line) and 
the Gaussian curve (dotted line) fitted to the speckle pattern. 
The Laplacian of Gaussian (LOG) function is therefore 
considered as the best wavelet among other filters for wavelet 
decomposition. Unfortunately, complete reconstruction of the 
image using LOG, is not possible. Hence another wavelet basis 
called Coiflet (with the filter length of 6) whose shape is similar 
to LOG may be used. Using this wavelet and Mallat's algorithm 
for wavelet decomposition, the complete reconstruction of the 
image is possible. Further improvements may be achieved by 
using Gaussian low pass filter and a trous algorithm for 
decomposition. This algorithm is well-known for using non- 
decimated wavelet transform which minimizes the artifact in 
the denoised data [5]. Shift invariancy is one of the important 
properties of a trous algorithm. In speckle noise reduction this 
property can improve the performance of the algorithm. 
Wavelet coefficients of the logarithmically transformed image 
are best modeled by alpha-stable distribution, SoS ‚which is 
the family of heavy-tailed densities [2]. The alpha-stable 
distribution does not have a direct expression but it can be 
defined by its characteristic function as follows: 
Q(o) - exp(jóo — y | v |^) (3) 
where (0 < @¢ <2)is the characteristic exponent. Small 
values of this parameter reflect the non-Gaussianity of the 
distribution function. §(—w<§ <0) is the location 
parameter and y (y > 0)is the dispersion similar to variance 
used in the Gaussian distribution. 
The noise component, e, can be modeled as a zero mean 
Gaussian random variable [1]. The characteristic function of the 
Gaussian distribution is: 
A 
p, (w) = exp(Jôw — a o) (4) 
where ó , is the median value of the noise and ¢ is the variance 
or noise level. In the proposed method a Bayesian estimator is 
used for estimating the noise free signal. This estimator uses the 
wavelet coefficients distribution as a priori information. The 
goal is finding the estimator $, which minimizes the 
conditional risk, R($ | d): 
k 
R(31d)= EG |d)= 5 [LGls)P(|a) © 
VES 
In this equation, Li. , $ and $, represent the loss function, 
estimated noise-free signal and signal, respectively. The 
estimated signal $(g) is the loss averaged over the conditional 
distribution of s, given a set of wavelet coefficients, d [7, 8]. 
The above Bayes risk estimator under a quadratic cost function 
minimizes the mean-square error and is given by the conditional 
mean of s given d: 
Dw Spd) (6) 
The mean-square error is defined for random variables that have 
finite second order moments. Since alpha-stable distribution 
does not have finite second-order statistics, we use absolute 
error as the loss function. Using the Bayes' theorem, the 
estimator is then given by [11]: 
" N POP. 7) 
dd TOYS 
where P(e) and P,(s) are the PDFs of the noise and signal, 
respectively. In order to use Eq. (7), we have to estimate the 
28 
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