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Fast Space-variant Phase Error Compensation and Geometric Correction for Bistatic ISAR Imaging Using a Modified Newton’s Method
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摘要: 双基逆合成孔径雷达(Bi-ISAR)因其卓越的反隐身与抗干扰性能，受到广泛关注。然而，Bi-ISAR成像过程中双基角的变化会导致图像出现空变散焦和几何畸变，严重影响后续信息提取与目标识别的精度。为解决上述问题，该文提出了一种基于修正牛顿法的Bi-ISAR空变补偿快速成像与几何校正图像定标方法。该方法以Bi-ISAR成像结果的图像熵为代价函数，以空变系数和转动参数为优化变量，构建优化方程。通过对传统牛顿法进行修正，确保海森矩阵的正定性，从而保证代价函数在每次迭代中沿下降方向优化。通过求解该优化方程最小化图像熵，同时估计得到转动参数，进而构建几何校正函数并计算分辨率因子，实现对最终成像结果的几何校正与定标。所提方法可同步校正空变散焦误差与几何畸变，且为数据驱动模式(无需先验参数)，对初始图像质量要求较低。此外，受益于牛顿法的二次收敛特性，相较于其他方法，该方法具有更高的计算效率。最后，通过对点目标仿真、电磁计算以及地面等效实验数据的处理与对比分析，验证了所提方法的有效性。Abstract: Bistatic Inverse Synthetic Aperture Radar (Bi-ISAR) has garnered significant attention in the military and civilian domains due to its superior stealth and antijamming capabilities. However, the changing bistatic angle during Bi-ISAR imaging causes space-variant defocusing and geometric distortion in the resulting images, thereby severely compromising the accuracy of subsequent information extraction and target recognition. To address these issues, this study proposes a fast space-variant phase error compensation and geometric correction method for Bi-ISAR imaging based on a modified Newton’s method. This method uses the image entropy of the Bi-ISAR imaging result as the cost function and introduces space-variant coefficients and rotation parameters as optimization variables to formulate an optimization equation. By modifying the traditional Newton’s method to ensure the positive definiteness of the Hessian matrix, the cost function is guaranteed to be optimized along the descent direction in each iteration. Solving this optimization equation to minimize image entropy simultaneously estimates the rotation parameters, which are then used to construct a geometric correction function and calculate the scaling factor, that is, the actual size of each grid in the image, enabling geometric correction and scaling of the final imaging result. The proposed method simultaneously corrects space-variant phase errors and geometric distortion and operates in a data-driven manner, requiring only low initial image quality. Furthermore, due to the quadratic convergence property of Newton’s method, the proposed method offers higher computational efficiency compared with other methods. Finally, the effectiveness of the proposed method is validated through the processing and comparative analysis of the point target simulation, electromagnetic calculation, and ground real target experimental data.
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图 6 基于修正牛顿法的双基ISAR空变补偿流程图
Figure 6. Flowchart of two-dimensional spatial variation correction for Bi-ISAR based on modified Newton’s method
图 7 基于修正牛顿法的双基ISAR空变补偿快速成像与几何校正图像定标流程图
Figure 7. Flow chart of space-variant phase error compensation and geometric correction image scaling for Bi-ISAR imaging based on modified Newton’s method
图 11 参数初始值补偿后成像结果
Figure 11. Imaging results after initial parameter value compensation
图 12 二维空变校正后的RD成像结果
Figure 12. Imaging result after 2D space-variant phase error compensation
图 15 补偿后8倍插值点P1等高线图
Figure 15. Contour map of 8 times interpolated point P1 after compensation
图 16 补偿后8倍插值点P2等高线图
Figure 16. Contour map of 8 times interpolated point P2 after compensation
图 17 P1, P2点方位向切片图和距离向切片图
Figure 17. Azimuth and Range slice diagram of points P1 and P2
图 19 所提方法与对比方法(BFGS方法)图像熵变化曲线
Figure 19. Image entropy change curves of the proposed method and the contrast method (BFGS)
图 20 所提方法图像熵变化曲线放大图
Figure 20. Enlarged image of the entropy change curve of the proposed method
图 23 交叉极化电磁仿真信号包络和RD成像结果
Figure 23. Cross-polarized electromagnetic simulation signal’s envelope and RD imaging results
图 24 交叉极化所提方法校正结果
Figure 24. Correction results of the proposed method under cross-polarization
图 26 所提方法与对比方法图像熵变化曲线(视角1)
Figure 26. Image entropy change curves of the proposed method and the contrast method (view 1)
图 27 所提方法图像熵变化曲线放大图(视角1)
Figure 27. Enlarged image of the entropy change curve of the proposed method (view 1)
图 29 水平极化和交叉极化电磁仿真信号包络和RD成像结果
Figure 29. Horizontally polarized and cross-polarized electromagnetic simulation signal’s envelope and RD imaging results
图 30 所提方法二维空变校正前后结果对比
Figure 30. Comparison of the results before and after two-dimensional space-variant correction for the proposed method
图 31 水平极化下对比方法的成像结果图
Figure 31. Imaging results of the contrast method under horizontal polarization
图 32 所提方法与对比方法图像熵变化曲线(视角2)
Figure 32. Image entropy change curves of the proposed method and the contrast method (view 2)
图 33 所提方法图像熵变化曲线放大图(视角2)
Figure 33. Enlarged image of the entropy change curve of the proposed method (view 2)
图 36 Keystone变换后的RD成像结果及其与光学照片的对应情况
Figure 36. Imaging result after Keystone transform and its corresponding part in the optical picture
图 37 参数初始值补偿后实测数据成像结果
Figure 37. Imaging results of measured data after initial parameter value compensation
图 38 二维空变校正后的实测数据RD成像结果
Figure 38. RD imaging results of measured data after two-dimensional space-variation correction
图 39 二维空变校正后实测数据定标结果
Figure 39. Calibration results of measured data after two-dimensional space-variable correction
图 40 几何畸变校正后实测数据定标结果
Figure 40. Calibration results of measured data after geometric distortion correction
图 41 所提方法成像结果标记点选择位置
Figure 41. The location map for the selection of marking points in the imaging results of the proposed method
图 42 对比方法二维空变校正后实测数据定标结果
Figure 42. Calibration results of measured data after two-dimensional space-variable correction by contrast method
图 43 对比方法成像结果及标记点选择位置
Figure 43. Comparison method imaging results and marker point selection location map
图 44 实测数据所提方法与对比方法图像熵变化曲线
Figure 44. Image entropy change curve of the proposed method and the comparison method for measured data
图 45 实测数据所提方法图像熵变化曲线放大图
Figure 45. Enlarged image of the entropy change curve of the proposed method for measured data
表 1 仿真雷达参数表
Table 1. Parameters of simulated radar
参数 数值 载频 35 GHz 带宽 1 GHz 脉冲重复频率 300 Hz 脉宽 1 μs 脉冲数 2048 
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表 2 各点补偿前后方位向峰值旁瓣比和积分旁瓣比
Table 2. Azimuth’s PSLR and ISLR before and after compensation for each point
点 峰值旁瓣比(dB) 积分旁瓣比(dB) 补偿前 补偿后 补偿前 补偿后 P1 – 0.0981 – 11.7714 4.0537 – 10.2959 P2 – 0.2306 – 12.9285 0.3414 – 10.6454
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表 3 各点补偿前后距离向峰值旁瓣比和积分旁瓣比
Table 3. Range’s PSLR and ISLR before and after compensation for each point
点 峰值旁瓣比(dB) 积分旁瓣比(dB) 补偿前 补偿后 补偿前 补偿后 P1 – 13.1455 – 13.4884 – 11.2785 – 11.2984 P2 – 12.7365 – 13.2731 – 11.2785 – 11.4938
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表 4 本文方法与对比方法(BFGS方法)指标对比
Table 4. The method of this paper is compared with the method of comparison (BFGS)
指标 本文所提方法 对比方法 图像熵值 7.0949 7.0952 对比度 23.5014 23.4998 $ O{P_1} $和$O{P_2}$夹角 88.9365 °88.7987 °迭代次数 13 202 运行时间 0.737570 s5.835382 s
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表 5 不同SNR下两种方法成像指标对比
Table 5. Comparison of imaging indicators of the two methods under different SNRs
SNR值 图像熵值 对比度 $O{P_1}$和$O{P_2}$夹角 迭代次数 运行时间 本文所提方法 对比方法 本文所提方法 对比方法 本文所提方法 对比方法 本文所提方法 对比方法 本文所提方法 对比方法 10 dB 7.5515 7.5518 22.0521 22.0509 88.9554 °88.3489 °17 127 0.969786 s4.162207 s5 dB 8.5377 8.5378 18.4484 18.4475 88.9483 °88.6062 °10 128 0.568464 s4.012253 s0 dB 10.0005 10.0006 12.1812 12.1809 88.8443 °88.3880 °23 137 1.310209 s4.404005 s
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表 6 交叉极化下本文方法与对比方法指标对比
Table 6. The index of this method is compared with the comparison method under cross polarization
方法 图像熵值 对比度 迭代次数 运行时间 本文所提方法 6.7704 27.9896 13 0.805566 s对比方法(BFGS) 6.7754 28.0345 260 9.678188 s
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表 7 水平极化下本文方法与对比方法指标对比
Table 7. The index of this method is compared with the comparison method under horizontal polarization
方法 图像熵值 对比度 迭代次数 运行时间 本文所提方法 5.3443 96.2904 37 2.258587 s对比方法(BFGS) 5.3443 96.3270 122 5.285292 s
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表 8 实测数据雷达参数表
Table 8. Radar parameter table of measured data
参数 数值 载频 33 GHz 带宽 300 MHz 脉冲重复频率 150 Hz 采样率 500 MHz 脉冲数 1024
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表 9 本文方法与对比方法(BFGS)指标对比
Table 9. The method of this paper is compared with the method of comparison (BFGS)
方法 图像熵值 对比度 OP1和
OP2夹角(°)迭代
次数运行时间(s) 本文所提方法 6.9455 111.7340 50.4275 61 11.896714 对比方法 6.9457 111.8247 50.2595 269 29.319547
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