# Fine Recognition of MEO SAR Ship Targets Based on a Multi-Level Focusing-Classification Strategy
## 3 Coarse-Level Focusing and Classification of MEO SAR Ship Targets  
### Ship Target Coarse-Focusing Based on Doppler Parameter Estimation  
Coarse focusing compensates for motion errors caused by ship sailing. The input for this step is the range-compressed signals of a target, as shown in Equation (2). The Doppler centroid frequency of the target signals is first estimated, and imaging processing is carried out using the target's Doppler centroid frequency to compensate for the spectral shift induced by the target's radial velocity. Due to the uncompensated target azimuth velocity, the image still remains defocused. The target image is subsequently transformed into the range-Doppler domain via azimuth Fast Fourier Transform (FFT), where the quadratic phase errors caused by the azimuth velocity are estimated and corrected, yielding the coarsely focused SAR image. The Doppler centroid frequency error induced by the target's radial velocity \(\Delta v_{r}\) is expressed as  
\[\Delta f_{d}=\frac{2\Delta v_{r}}{\lambda} \tag{11}\]  
The projection of the ship's velocity in the direction of the radar line of sight is within 10 m/s. For an X-band MEO SAR satellite, the Doppler centroid frequency error caused by the ship's radial velocity is within 700 Hz, which usually does not exceed one Pulse Repetition Frequency (PRF). Therefore, it is only necessary to estimate the baseband Doppler centroid frequency error, and there is no need to consider the ambiguity number portion of the Doppler centroid frequency error. The average phase increment method is employed to estimate the Doppler centroid frequency. \(S_{p,rc}(\tau,\eta)\) is the ship signal after range compression. \(S_{p,rc}(\tau,\eta+1/PRF)\) denotes the signal at the adjacent azimuth time. Figure 7: Imaging results of rotating point targets: (**a**) MEO SAR; (**b**) LEO SAR. By performing average cross-correlation processing and extracting the phase, we can obtain [29]  
\[\phi_{a}=\angle\left(\sum_{\eta,\tau}{S_{p,rc}}^{*}\left(\tau,\eta+\frac{1}{PRF} \right)\cdot S_{p,rc}\left(\tau,\eta+\frac{1}{PRF}\right)\right) \tag{12}\]  
The Doppler centroid frequency can then be calculated by  
\[f_{d}=\frac{PRF}{2\pi}\phi_{a} \tag{13}\]  
The estimated Doppler centroid frequency is then utilized to conduct subsequent range cell migration correction (RCMC) and azimuth compression. In order to directly process range-compressed SAR data, an approximate wavenumber focusing algorithm is used to form the ship SAR image [29; 30]. In addition to Doppler centroid frequency, the Doppler FM rate should also be estimated when focusing a moving ship target. According to Section 2.2, the phase error in the azimuth frequency domain caused by \(\Delta K_{a}\) is  
\[\varphi(f)=\pi\frac{\Delta K_{a}}{K_{a}^{2}}f^{2},\ f\in\left[-\frac{PRF}{2},\ \frac{PRF}{2}\right] \tag{14}\]  
The azimuth FM rate error can lead to a mismatch in the azimuth-matched filter, causing defocusing along the azimuth direction in the SAR image. The degree of defocusing can be measured by the image entropy: the higher the image entropy, the more severe the image defocusing. The image entropy of a SAR image is  
\[IE=-\frac{1}{Q}\sum_{l=0}^{L-1}\sum_{n=0}^{N-1}{|I(l,n)|}^{2}\ln{|I(l,n)|}^{2}+ \ln{Q} \tag{15}\]  
where \(I\) represents the discrete-domain SAR image, \(L\) and \(N\) denote the number of azimuth pixels and range pixels, respectively, and \(Q\) is the image energy. Therefore, refocusing of the SAR image can be formulated as an optimization problem, with SAR image entropy serving as the objective function. The goal is to find the optimal azimuth FM rate error \(\Delta K_{a}\) that minimizes image entropy.