## 2 Concepts and Methods
### Multiplicative Speckle Noise Behavioral Formulation and Modeling
On the other hand, the standard deviation divided by the mean is \(\sqrt{4/\pi-1}\). The observed intensity has a negative exponential distribution, as:  
\[\mathrm{P}_{\mathrm{I}}(\mathrm{y})=2\mathrm{y}\exp\left(-\mathrm{y}\right) \mathrm{y}\geq 0 \tag{28}\]  
where the mean value and standard deviation are both in unity. The decibel image as the logarithmic transformed intensity is derived as:  
\[\mathrm{P}_{\mathrm{I}}(\mathrm{y})=\mathrm{e}^{\mathrm{y}}\exp\left(- \mathrm{e}^{\mathrm{y}}\right) \tag{29}\]  
where the mean and variance are \(-0.57\) and \(\pi^{2}/6\). If the number of observations is K, the average intensity of the K-observation can be written as:  
\[\mathrm{Y}_{\mathrm{O}}=\frac{\sum_{i=1}^{K}\mathrm{Y}_{\mathrm{O}}^{i}}{K} \tag{30}\]  
where \(\mathrm{Y}_{\mathrm{O}}^{i}\) is the independent single-observation intensity with a negative exponential distribution. In the case of multi-observation of \(\mathrm{Y}_{\mathrm{O}}\), the model will be rewritten as:  
\[\mathrm{P}_{\mathrm{O}}(\mathrm{y})=\frac{K^{K}}{\Gamma(K)}\mathrm{y}^{K-1} \exp\left(-\mathrm{Ky}\right)\mathrm{y},K>0 \tag{31}\]  
where K is the multi-observation number of \(\mathrm{Y}_{\mathrm{O}}\) with the mean and variance of unity and \(1/\mathrm{K}\). For a single look image, the amplitude obeys Rayleigh PDF. This multiplicative behavior on the basis of signal characteristics is generated by the interaction of the reflected waves from various independent scatterers within a resolution cell. In total, the characteristics of the multiplicative speckle noise in SAR imagery are very different from those found in the other sensors, which might contain additive white Gaussian noise. Hence, the formulated SAR multiplicative speckle noise has an asymmetrical PDF and the Gamma distribution, which is considered completely signal dependent based on the phase distribution, the amplitude distribution, the intensity distribution, and the observation number. On the other hand, according to the characteristics of speckle noise, the coefficients are being modulated by the characteristic of the noise in a manner that is proportional to the target mean backscattering coefficients. In other words, extracting the coefficients of speckle noise can be evaluated based on their expectation and variance, despite their limitations while dealing with random entities within the terrain. Considering \(\mathrm{O}_{1}\) and \(\mathrm{O}_{2}\) as two adjacent pixels in a SAR observation:  
\[\mathrm{O}_{1}=\mathrm{c}_{1}Y_{1}\] \[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]0  
where \(\mathrm{c}_{1}\) and \(\mathrm{c}_{2}\) are constants and \(\mathrm{Y}_{1}\) and \(\mathrm{Y}_{2}\) are independent random variables with identical probability, according to the general Gamma distribution \(\Gamma(n,\lambda)\). Therefore, the PDF can be modeled as:  
\[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]1If a and b are considered as the backscattered intensities, the two adjacent pixels \(\mathrm{O_{1}}\) and \(\mathrm{O_{2}}\) belong to a homogenous region in case of a equals b. Otherwise, \(\mathrm{O_{1}}\) and \(\mathrm{O_{2}}\) are located in two different regions. If a \(\
eq 0\) and b \(\
eq 0,\mathrm{O_{1}}\) and \(\mathrm{O_{2}}\) follow the general Gamma distribution \(\Gamma(n,\lambda/a)\) and \(\Gamma(n,\lambda/b)\), with the form below:  
\[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]2  
\[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]3  
The property of \(\mathrm{O_{1}}-\mathrm{O_{2}}\) is the basis for the development of the multiplicative speckle property and its complex modeling. If the cumulative distribution function of \(\mathrm{O_{1}}-\mathrm{O_{2}}\) is defined as:  
\[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]4  
This will result in:  
\[\mathrm{P}(\mathrm{O_{1}}-\mathrm{O_{2}}<t)=\left\{