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Compressed Sensing Linear Array SAR Autofocusing Imaging via Semi-definite Programming
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摘要: 线阵合成孔径雷达(Linear Array Synthetic Aperture Radar, LASAR)3维成像技术是一种具有重要潜在应用价值的新体制成像雷达，压缩感知稀疏重构是近几年实现LASAR高分辨3维成像的热点研究之一。但相对于传统2维SAR，受线阵稀疏分布及阵列-平台2维联动，压缩感知LASAR成像面临回波数据欠采样、多维度高阶相位误差等问题，传统SAR自聚焦算法难以适用于压缩感知LASAR 3维稀疏自聚焦成像。为克服欠采样条件下多维度高阶相位误差对LASAR成像的影响，该文提出了一种基于半正定规划的压缩感知LASAR自聚焦成像算法。首先，结合压缩感知成像理论、图像最大锐度及最小均方误差准则，构造欠采样条件下稀疏目标的相位误差估计模型；其次，利用松弛半正定规划方法估计相位误差；最后，利用迭代逼近方法提高相位误差估计精度，实现压缩感知LASAR高精度稀疏自聚焦成像。另外，通过主散射目标区域提取，仅采用主散射区域进行相位误差估计，进一步提高自聚焦算法运算效率。仿真数据和实测数据验证了该文算法的有效性。Abstract: Linear Array Synthetic Aperture Radar (LASAR) is a novel and promising radar imaging technique. In recent years, Compressed Sensing (CS) sparse recovery has been a research focus for high-resolution three-Dimensional (3-D) LASAR imaging. Compared with the traditional two-Dimensional (2-D) SAR imaging, LASAR suffers from many problems, including under-sampling data and multi-dimensional and higher-order phase errors due to its sparse Linear Array Antenna (LAA) and the joint 2-D motions of the platform and LAA. The conventional autofocusing methods of 2-D SAR may be not suitable for CS-based LASAR 3-D sparse autofocusing. To address the multi-dimensional and higher-order phase errors in LASAR 3-D imaging with respect to under-sampling data, in this paper, we propose a sparse autofocusing algorithm based on semi-definite programming for CS-based LASAR imaging. First, by combining CS-based imaging theory, image maximum sharpness, and the minimum square error principle, we construct a LASAR phase-error estimation model based on under-sampled data. Next, we use semi-definite programming relaxation to estimate the phase errors. Lastly, we employ an iterated approximation method to improve the precision of the phase-error estimation and achieve the final CS-based LASAR autofocusing. To further improve the efficiency of the algorithm, we select only the dominant scattering areas for LASAR phase-error estimation. We present our simulation and experimental results to confirm the effectiveness of out proposed algorithm.
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图 3 沿航迹及切航迹随机分布相位误差下仿真点目标成像结果(上：全部回波数据；中：50%回波数据；下：25%回波数据)
Figure 3. The results of the point targets in the case of the along-track random phase errors and cross-track random phase errors(Top: all samples; Middle: 50% samples; Bottom: 25% samples)
图 2 沿航迹二次项及切航迹随机分布相位误差下仿真点目标成像结果(上：全部回波数据；中：50%回波数据；下：25%回波数据)
Figure 2. The results of the point targets in the case of the along-track quadratic phase errors and cross-track random phase errors(Top: all samples; Middle: 50% samples; Bottom: 25% samples)
图 6 地基LASAR合成阵列平面及回波数据
Figure 6. The virtual array antenna and the echo of the ground-based LSAR
图 8 路灯目标实测数据稀疏自聚焦成像结果(左：BP-PGA算法；中：IRLS-PGA算法；右：SDPSA算法)
Figure 8. The sparse autofocusing results of the experimental light data(Left: BP-PGA; Middle: IRLS-PGA; Right: SDPSA)
表 1 SDPSA算法
Table 1. SDPSA algorithm
输入：代价函数 ${J_{\rm{1}}}\left( {{{{f}}},{{{φ}}} } \right)$和 ${J_{\rm{2}}}\left( {{{{f}}},{{{φ}}} } \right)$ 输出：估计值 $\left( {{{{\stackrel \frown{f}}}} ,{{\stackrel \frown{{φ}}} } } \right)$ 初始化：迭代次数 $i = 0$，相位误差 ${{{{φ}}} ^{\left( {\rm{0}} \right)}} = 0$，门限 ${{δ}} $，迭代总次数 $K$ 若 ${\left\| {{{{{{\stackrel \frown{f}}}} }^{\left( {i + 1} \right)}} - {{{{{\stackrel \frown{f}}}} }^{\left( i \right)}}} \right\|_2}\Bigr/{\left\| {{{{{{\stackrel \frown{f}}}} }^{\left( i \right)}}} \right\|_2} > {{δ}} $和 $i \le K$，循环开始 步骤1 构造稀疏目标重构的代价函数： ${J_{\rm{1}}}\left( {{{{f}}},{{{{{\stackrel \frown{φ}}} } }^{\left( i \right)}}} \right) = {{λ}} {\left\| {{{f}}} \right\|_1}{\rm{ + }} \left\| {{{{y}}} - {{{R}}}\left( {{{{{{\stackrel \frown{φ}}} } }^{\left( i \right)}}} \right){{{Af}}}} \right\|_2^{}$ 步骤2 利用IRLS算法进行稀疏目标重构： ${{{{\stackrel \frown{f}}}} ^{\left( {i{\rm{ + 1}}} \right)}} = \arg \mathop {\min }\limits_{f} {J_{\rm{1}}}\left( {{{{f}}},{{{{{\stackrel \frown{φ}}} } }^{\left( i \right)}}} \right)$ 步骤3 构造相位误差估计的代价函数： ${J_{\rm{2}}}\left( {{{{{{\stackrel \frown{f}}}} }^{\left( {i + 1} \right)}},{{{φ}}} } \right) \approx {{{γ} }^{\rm H}}{{{{Q}}}^{\left( {i + 1} \right)}}{{γ} }$ 步骤4 利用SDP方法求解最优问题： ${{{{\stackrel \frown{X}}}} _{\rm{opt}}} = \arg \mathop {\min }\limits_{X} {\rm tr}\left({{{{Q}}}^{\left( {i + 1} \right)}}{{{X}}}\right),\ {\rm s.t.} \ \ {{{X}}} \succeq 0,{{{{X}}}_{ii}} = {g_n},n = 1,2, ·\!·\!· ,N$ 步骤5 利用 ${{{{\stackrel \frown{X}}}} _{\rm {opt}}}$估计相位误差 ${{{{\stackrel \frown{φ}}} } ^{\left( {i + 1} \right)}}$； $i \leftarrow i + 1$ 循环结束 返回结果： ${{{\stackrel \frown{f}}}} \leftarrow {{{{\stackrel \frown{f}}}} ^{\left( {i + 1} \right)}}$, $\stackrel \frown{{{{φ}}} } \leftarrow {\stackrel \frown{{{{φ}}} } ^{\left( {i + 1} \right)}}$ 
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