## 2 Concepts and Methods
### De-Speckled Image HMS Noise Behavioral Modeling  
In total, the non-uniform antenna pattern illumination, possible receiver gain changes, loss effects, scattering abnormalities phenomenon, and so forth are the source of HMS noise, which will affect the pixel statistics and their inherent distribution, such as their amplitude, phase and the intensity, in a destructive or constructive manner [21; 24; 25]. According to these patterns of constructive and destructive interference, HMS noise can also be suppressed with the help of de-speckling filters. During the past decades, by considering the variance of backscatter amplitude, many de-speckling filters have been proposed for HMS noise reduction [25]. These filters can be divided into two groups--transform domain and spatial domain--which are usually designed to cope with the variation of pixel amplitude. In the case of spatial filters, they use local pixel intensity statistics to adjust the amount of noise removal in a certain threshold, and perform acceptably in homogeneous regions, but in heterogeneous regions they cause loss of detail [25]. Due to the lack of investigation on spatial domain filter application and suitable selection, this section will present the PSD function modeling of the de-speckled images and their embedded HMS noise for further comparison. Therefore, taking into consideration the implementation of seven different spatial de-speckling filters--Frost, Gamma, Kuan, Lee, Lee sigma, Local adaptive median and Local sigma--on the same reference SAR image and modeling their PSD function may be a good technique to evaluate their noise suppression ability while dealing with HMS noise. The results are shown in Figure 3.  
As can be deduced from the PSD modeling results in Figure 3b,d,f,h,j,l,n, the simulated PSD functions are not only different in the uniformity of the peak distribution, but also vary in terms of their magnitude. The results show that, despite using the same SAR image with a similar texture, different PSD models under different spatial de-speckling filters might result. In this case, the minimum quality and lowest magnitude belongs to the Lee sigma filter with 12.2 dB magnitude, while the best performance belongs to the Gamma filter with more than 25.5 dB magnitude. This shows that, if the distribution of the peaks within the PSD modeling is minor and their concentration has a large uniform magnitude, this is a sign of acceptable de-speckling. According to Figure 3, Table 1 presents the rankingof the aforementioned filters based on their peak values. Due to limitations of the filters in terms of the complete removal of the HMS, the presence of fluctuation in the PSD modeling is normal. Figure 4 illustrates the difference between the PSD modeling of the reference SAR image in Figure 2b, and their counterparts in Figure 3. The results verify that de-speckled images have different HMS pattern within the texture, and each spatial filter has its own specifications to reduce the level of the HMS. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Column** & **De-Speckling Filter** & **Value** \\ \hline
1 & Gamma & 25.5 dB \\
2 & Lee & 22 dB \\
3 & Local sigma & 20.5 dB \\
4 & Frost & 18.4 dB \\
5 & Local adaptive median & 14.4 dB \\
6 & Kuan & 14.1 dB \\
7 & Lee sigma & 12.2 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PSD modeling results after HMS suppression. Figure 2: Reference SAR image and its PSD modeling (**a**) Reference SAR image; (**b**) PSD of reference SAR image; (**c**) PSD of reference SAR image in presence of little noise; (**d**) PSD of reference SAR image in noise-free medium; (**e**) PSD of high-frequency noise. Figure 3: _Cont_. According to Figures 3 and 4, besides the ability of spatial filters to achieve HMS suppression, the effect of the spatial filter application on the texture is also a key concept. Hence, similar to the PSD modeling of the HMS suppression, PSD modeling of the pure HMS presence within the texture is helpful.