# PBR Clutter Suppression Algorithm Based on Dilation Morphology of Non-Uniform Grid
## 2 Non-Uniform Polar Grid Construction for PBR
\[\Delta\rm R_{r}(\Delta r,\Delta\theta)|_{R_{s}=R_{s0},\theta=\theta_{0}} \approx\rho_{1}(R_{s0},\theta_{0})\Delta r+\rho_{2}(R_{s0},\theta_{0})\Delta\theta \tag{6}\]  
From Equation (6), we can find that when \(\Delta r\) and \(\Delta\theta\) are fixed, the transfer error changes with the position \((\rm R_{s0},\theta_{0})\). To facilitate following operation, we divide the detection coverage into grids based on the transfer error expansion and project the processing data into grids. Assume the bi-static range error \(\Delta r\) and the azimuth angle error \(\Delta\theta\) obey the Gaussian distribution with zero-mean, as shown below. \[\Delta\rm r\sim N(0,\sigma_{r}^{2});\Delta\theta\sim N(0,\sigma_{\theta}^{2}) \tag{7}\]  
In general, \(\sigma_{r}\), the standard deviation of \(\Delta r\), mostly relates to the range resolution. In addition \(\sigma_{\theta}\), the standard deviation of \(\Delta\theta\), mostly relates to the space synchronization accuracy and the array error. We suppose them as known constant. Further discussion about them is not included in this paper. Since \(\Delta\rm R_{r}\) is the linear function of \(\Delta r\) and \(\Delta\theta\), \(\Delta\rm R_{r}\) is also obey the Gaussian distribution with zero-mean, as shown below. \[\Delta\rm R_{r}\sim N(0,\rho_{1}^{2}(R_{s0},\theta_{0})\sigma_{r}^{2}+\rho_{2} ^{2}(R_{s0},\theta_{0})\sigma_{\theta}^{2}) \tag{8}\]  
There is a \(3\sigma\) principle of Gaussian distribution in the probability theory. In Gaussian distribution, the probability of the data distributing in the range \((\mu-3\sigma,\mu+3\sigma)\) is \(99.74\%\). Where \(\mu\) is the mean value of sample set. \(\sigma\) is the standard deviation of database. Based on the analysis above, the grid spacing is designed shown in Equations (9) and (10). \[\delta_{r}(R_{s0},\theta_{0})=3\sqrt{\rho_{1}^{2}(R_{s0},\theta_{0})\sigma_{r}^ {2}+\rho_{2}^{2}(R_{s0},\theta_{0})\sigma_{\theta}^{2}} \tag{9}\]  
\[\delta_{\theta}(R_{s0},\theta_{0})=3\sigma_{\theta} \tag{10}\]where \(\delta_{r}(R_{s0},\theta_{0})\) and \(\delta_{\theta}(R_{s0},\theta_{0})\) are the grid spacing at the position \((R_{s0},\theta_{0})\) in range dimension and angular dimension respectively. As the \(\delta_{\theta}\) is uncorrelated with the position \((R_{s0},\theta_{0})\), the grid is divided evenly in the angular dimension. However, in range dimension, the pace will be iteratively calculated based on the present position. Assume the observing scope is from \(\theta_{0}\) to \(\theta_{max}\) and from \(r_{0}\) to \(r_{max}\). The polar mesh grid calculation step is shown in Table 1.  
Referring to the proposed polar grid calculation algorithm, we calculate and make an example map of the polar grid in Figure 3. Set the observing scope \(100^{\circ}\) to \(160^{\circ}\) in azimuth and \(20\) km to \(100\) km in range. Blue lines denote the grid division in angular dimension, while red lines stand for the grid division in range dimension. It is obvious that closer to the baseline angle \(180^{\circ}\), the grid size is bigger.