# A Robust Dual-Platform GMTI Method against Nonuniform Clutter
## 2 The Dual-Platform GMTI Method against Nonuniform Environment
### High-Precision Clutter Covariance Matrix Estimation Based on Dual-Platform SAR System
#### 2.3.1 Image Registration of Former and Latter Platforms  
In our proposed STAP method using the dual-platform system, we need to accurately obtain the former platform image corresponding to the area where the latter platform target is located. Due to the errors in observation geometry between the two SAR images, there is a certain offset and distortion between the images, so it is necessary to register the images in the former and latter platforms.  
After obtaining the single-view complex images, the same name points in two or more images are often not on the same grid points, so it is necessary to register the images so that they are on the same range cell. The basic flowchart is shown in Figure 7.  
To achieve image registration, a candidate image block in the auxiliary SAR image and a reference image block in the main SAR image are selected. Due to the computational burden, the registration can be based on some control point pairs in the main and auxiliary images. The registration principle is that the evaluation criterion will reach its maximum value when the candidate image block shifts with a certain offset. This offset can be regarded as the image position difference between the matching window and search window. The registration method adopted here is the correlation method, which uses the coherence coefficient as the evaluation criterion. By calculating the coherence coefficient under different offsets, the offset can be found, which corresponds to the maximum coherence coefficient position. The calculation of the coherence coefficient is expressed as [35]:  
\[\hat{\gamma}=\frac{\left|\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}\mathbf{S}_{ 1}(m,n)\mathbf{S}_{2}^{*}(m,n)\exp[-j\varphi(m,n)]\right|}{\sqrt{\sum\limits_{m=0}^ {M-1}\sum\limits_{n=0}^{N-1}\left|\mathbf{S}_{1}(m,n)^{2}\right|\sum\limits_{m=0}^ {M-1}\sum\limits_{n=0}^{N-1}\left|\mathbf{S}_{2}(m,n)^{2}\right|}}, \tag{12}\]  
where \(m=0,1,\cdots,M-1\) and \(n=0,1,\cdots,N-1\) are the size of the main image, \(\mathbf{S}\) is the image amplitude, and \(\phi(m,n)\) is the image phase. To avoid calculating the phase, the fol  
Figure 6: The simplified flowchart of the proposed GMTI method.  
lowing frequency-independent coherence coefficient estimator can be used to calculate the coherence coefficient [36]:  
\[\hat{\rho}=\frac{\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}\left|\mathbf{S}_{ 1}(m,n)\right|^{2}\left|\mathbf{S}_{2}(m,n)\right|^{2}}{\sqrt{\sum\limits_{m=0 }^{M-1}\sum\limits_{n=0}^{N-1}\left|\mathbf{S}_{1}(m,n)\right|^{4}\sum\limits_ {m=0}^{M-1}\sum\limits_{n=0}^{N-1}\left|\mathbf{S}_{2}(m,n)\right|^{4}}}, \tag{13}\]  
\[\hat{\gamma}=\left\{\begin{array}{ll}\sqrt{2\hat{\rho}-1}&\hat{\rho}>1/2\\ 0&\hat{\rho}<1/2\end{array}\right.. \tag{14}\]  
The correlation registration is divided into two steps: pixel registration and subpixel registration. Pixel registration moves the main image by a whole pixel to obtain the pixel-level position offset and moves the main image by a subpixel size near the pixel-level registration position to obtain the subpixel-level position offset. When the subpixel registration is aligned, the data corresponding to the auxiliary image do not fall on the grid points; therefore, interpolation is needed. In order to complete the whole image registration, it is necessary to obtain the registration offset of other pixels by using the registration offset that corresponds to the control points, which can be obtained using the second-order polynomial:  
\[\left\{\begin{array}{ll}u=a_{0}+a_{1}m+a_{2}n+a_{3}m^{2}+a_{4}n^{2}+a_{5}mn \\ v=b_{0}+b_{1}m+b_{2}n+b_{3}m^{2}+b_{4}n^{2}+b_{5}mn\end{array}\right.. \tag{15}\]  
\(a_{0}\sim a_{5}\) and \(b_{0}\sim b_{5}\) are the fitting parameters, which need to be calculated by the least square method according to the control point offset. \((m,n)\) is the pixel coordinate in the main image, and \((u,v)\) is the offset corresponding to the registration position offset between the main and auxiliary images.