# PBR Clutter Suppression Algorithm Based on Dilation Morphology of Non-Uniform Grid
## 3 Separate False Alarm Clutter from Data
### Separate False Alarm Clutter from Data Based on the Dilation Morphology  
In general, the true points of the target track have the feature of location dependency among several consecutive frames, whereas the false alarm clutter points are relatively isolated. Assume \(\rm F_{n}\) is the object frame. \(\rm\Psi=[F_{n-k}\ldots F_{n-1},F_{n+1}\ldots F_{n+k}]\) is the group of the reference frames, before and after several frames of \(\rm F_{n}\). Where k is the half number of the reference frames. Firstly, mark the points of the frame in group \(\rm\Psi\) one by one. Obtain the mark-matrix \(A_{n-k}\ldots\ A_{n-1}\), \(A_{n+1}\ldots A_{n+k}\) respectively, and compose them into a new mark-matrix \(\rm\Gamma_{n}\), shown in Equation (13). Meanwhile, mark the points of the object frame on the matrix \(A_{n}\).  
\[\rm\Gamma_{n}=A_{n-k}\cup\ldots\cup A_{n-1}\cup A_{n+1}\cup\ldots\cup A_{n+k} \tag{13}\]  
To facilitate following iteration calculation, we replace (13) by another more specific operation, described in Equation (14). Where \(\rm\mathfrak{B}m(\cdot)\) is the symbol of binarization.  
\[\rm\Gamma_{n}=\rm\mathfrak{B}m(A_{n-k}+\cdots+A_{n-1}+A_{n+1}+\cdots+A_{n+k}) \tag{14}\]  
Next, to ensure the target points in the neighborhood of the points from reference frames, we do morphological dilation on \(\rm\Gamma_{n}\) with a rectangular structural element \(\rm B\). The size of \(\rm B\) depends on the coarse estimation of the target's move range. The dilation result matrix marks the neighborhood area of the points from reference frames. \(\rm M_{n}\) in Equation (15) is the dot product of the dilation result and the object mark-matrix \(A_{n}\).  
\[\rm M_{n}=(\rm\Gamma_{n}\oplus B)\cdot\star A_{n} \tag{15}\]where \(\oplus\) is the symbol of dilation. \(\star\) is the symbol of the dot product. The matrix \(\mathrm{M_{n}}\) stands for the final marked area of screened data for the frame \(\mathrm{F_{n}}\). To facilitate realization, the dilation of binary matrix can be expressed by binarization after convolution. So Equation (15) can be rewritten as  
\[\mathrm{M_{n}}=\mathfrak{P}\mathrm{m}(\Gamma_{\mathrm{n}}\otimes B)\cdot\star A _{\mathrm{n}} \tag{16}\]  
where \(\otimes\) is the symbol of convolution.  
The processing progress is illustrated in Figure 4.  
We choose a simple case to clearly illustrate how the proposed algorithm operates. Figure 4a indicates the general view of the operation. Figure 4b-f shows the processing result in each step. After the composition of reference frames, dilation and the point product with object frame, we can obtain the screened data, where the clutter points are filtered out. In this case, Structural element B is a \(3\times 3\) square matrix, as shown in Figure 4b. Figure 4c shows the reference mark-matrix \(\Gamma_{\mathrm{n}}\). Figure 4d shows the dilation result of the B and \(\Gamma_{\mathrm{n}}\). Figure 4e is the object mark-matrix \(A_{\mathrm{n}}\) with five suspected  
Figure 4: The sketch map of processing progress: (**a**) general view; (**b**) structural element B; (**c**) reference mark-matrix \(\Gamma_{\mathrm{n}}\); (**d**) dilation result; (**e**) object mark-matrix \(A_{\mathrm{n}}\); (**f**) screened data for \(A_{\mathrm{n}}\).  
areas. Figure 4f shows the result, the screened data for \(A_{n}\). Obviously, the points in two areas of yellow background are retained, yet rest of them are suspected as clutters and filtered out.  
It is noteworthy that since the transfer error changes with the point location, the real size and shape of the structural element for dilation is not fixed in fact. It changes along with the location of the suspected point. However, as the processing data has been abstracted through the non-uniform polar grid, the change of the structural element does not involve in this section. The left part of Figure 5 is the partial enlarged map of Figure 3. There are two structural elements, corresponding to point A and B, with different shape and size. Both are projected into the same element for convenience during processing.