## 2 Concepts and Methods
### Multiplicative Speckle Noise Behavioral Formulation and Modeling
\begin{array}{l}\exp \left(-\frac{\lambda}{b}|t|\right)\frac{\lambda^{n}}{a^{n}(n-1)!}\sum\limits_{ i=0}^{n-1}\frac{\lambda^{i}}{b!}\left[\sum\limits_{j=0}^{i}\frac{|t|^{-j}}{ \left(\frac{n+i-1}{2}\right)!}\frac{\left(n+i-1\right)!}{\left(\frac{\lambda} {a}+\frac{b}{b}\right)^{n+i}}\right]t\leq 0\\ 1-\exp\left(-\frac{\lambda}{b}|t|\right)\frac{\lambda^{n}}{a^{n}(n-1)!}\sum \limits_{i=0}^{n-1}\frac{\lambda^{i}}{b!}\left[\sum\limits_{j=0}^{i}\frac{|t|^ {-j}}{\left(\frac{n+i-1}{2}\right)!}\frac{\left(n+i-1\right)!}{\left(\frac{ \lambda}{a}+\frac{b}{b}\right)^{n+i}}\right]t>0\end{array} \right.\right. \tag{36}\]  
The PDF of \(\mathrm{O_{1}}-\mathrm{O_{2}}\) can be derived as:  
\[\mathrm{P_{O_{1}}-\mathrm{O_{2}}}(y)=\frac{\exp\left(-\frac{\lambda}{b}|y| \right)}{(n-1)!}\left(\frac{\lambda}{a}\right)^{n}\left(\frac{\lambda}{b} \right)^{n}\left[\sum\limits_{j=0}^{n-1}\frac{\left(n+j-1\right)!}{j!(n-j-1)! \left(\frac{\lambda}{a}+\frac{\lambda}{b}\right)^{n+j}}|y|^{n-j-1}\right] \tag{37}\]  
Hence, the moments of \(\mathrm{O_{1}}-\mathrm{O_{2}}\) can be driven by:  
\[ \begin{array}{l}\left[x^{k}\right]=\frac{2}{(n-1)!}\left(\frac{\lambda}{a} \right)^{n}\left(\frac{\lambda}{b}\right)^{n}\left[\sum\limits_{j=0}^{n-1} \frac{\left(n+j-1\right)!\left(i+n-j-1\right)!}{j!(n-j-1)!\left(\frac{\lambda }{a}+\frac{j}{b}\right)^{n+j}}\left|\frac{k}{\left|i+n-j\right|}\right]\text{ if i is even}\\ \end{array}  \tag{38}\]  
\[ \begin{array}{l}\mathrm{E}\left[x^{k}\right]=0\text{ if i is odd}\end{array} \]  
To model the behavior of the speckle noise based on complex modeling, let \(a=b=1\). Hence, Equation (37) will be rewritten as:  
\[\mathrm{P_{O_{1}}-\mathrm{O_{2}}}(y)=\frac{\exp\left(-\lambda|y|\right)}{(n-1)! }\left(\lambda\right)^{2n}\left[\sum\limits_{j=0}^{n-1}\frac{\left(n+j-1 \right)!}{j!(n-j-1)!\left(2\lambda\right)^{n+j}}|y|^{n-j-1}\right] \tag{39}\]  
Equations (37) and (39), which are a combination of the exponential function and polynomial, are very important for behavioral modeling of multiplicative speckle noise. The aforementioned distribution and its corresponding statistical features on the basis of the phase distribution, the amplitude distribution, the intensity distribution, and the observation number are used for complex modeling. According to the above distribution and complex modeling of the multiplicative speckle noise, power spectral density function (PSD) modeling is considered the best possible option to investigate the effects of the speckle presence within SAR images. The PSD, as an objective quality assessment metric, measures the intensity of the joint pixels as a function of frequency and shows at which frequencies the intensity variations are stronger, and at which frequencies variations are weak. According to the above modeling, pixels and their statistical distribution based on amplitude, phase, and intensity play a key role for complex speckle PSD modeling. Figure 1 presents the reference SAR image sample and the resulting PSD function modeling. The PSD modeling is based on the pixel intensities and their distribution statistics, which are assigned  
and presented in the frequency domain. The results show that, in the case of the reference SAR image consisting of non-overlapping pixels that are not reduced to the replications of a specific statistics, lower frequencies with significant variance in the distribution of pixels have peaks in PSD modeling that can be distinguished in Figure 2b. The peaks in the PSD function modeling present the statistics of the main entity of the texture, and are also presented in Figure 2c,d. The difference between Figure 2b-d is based on their pixel statistics estimation and distribution modeling. Figure 2c shows the case of the speckle reduction technique on the basis of filter application, and 2d shows the case of noise-free supposition. The results show that speckle presence can directly affect the pixel statistics and distribution, such as the phase, the amplitude, and the intensity.