﻿ 用于线阵三维SAR成像的二维快速ESPRIT算法
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 雷达学报  2015, Vol. 4 Issue (5): 591-599  DOI: 10.12000/JR15065 0

### 引用本文 [复制中英文]

[复制中文]
Zhao Yi-chao, Zhu Yu-tao, Su Yi, Yang Meng, et al. Two-dimensional Fast ESPRIT Algorithm for Linear Array SAR Imaging[J]. Journal of Radars, 2015, 4(5): 591-599. DOI: 10.12000/JR15065.
[复制英文]

### 文章历史

, , ,
(国防科学技术大学电子科学与工程学院 长沙 410073)

Two-dimensional Fast ESPRIT Algorithm for Linear Array SAR Imaging
, , ,
(School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China)
Abstract: The linear array Synthetic Aperture Radar (SAR) system is a popular research tool, because it can realize three-dimensional imaging. However, owning to limitations of the aircraft platform and actual conditions, resolution improvement is difficult in cross-track and along-track directions. In this study, a twodimensional fast Estimation of Signal Parameters by Rotational Invariance Technique (ESPRIT) algorithm for linear array SAR imaging is proposed to overcome these limitations. This approach combines the Gerschgorin disks method and the ESPRIT algorithm to estimate the positions of scatterers in cross and along-rack directions. Moreover, the reflectivity of scatterers is obtained by a modified pairing method based on “region growing”, replacing the least-squares method. The simulation results demonstrate the applicability of the algorithm with high resolution, quick calculation, and good real-time response.
Key words: Linear Array SAR (LASAR)     Estimation of Signal Parameters by Rotational Invariance Technique(ESPRIT) algorithm     Superresulotion     Imaging     Region growing

1 引言

2 基于MIMO线阵的3维SAR成像模型

 图 1 基于MIMO线阵的3维SAR成像几何模型 Fig. 1 Linear array SAR imaging model based on MIMO

 $\begin{array}{*{20}{r}} {W\left( {t,u} \right) = \sum\limits_{q = 1}^Q {{\varsigma _q}\;{\rm{rect}}\left( {\frac{{t - {t_{nq}}\left( u \right)}}{{{T_{\rm{p}}}}}} \right)} }\\ { \cdot \exp \left\{ {{\rm{j}}\pi {K_{\rm{r}}}{{\left( {t - {t_{nq}}} \right)}^2}} \right\}}\\ { \cdot \exp \left\{ { - {\rm{j}}\frac{{4\pi {f_{0}}{r_{nq}}\left( u \right)}}{{\rm{c}}}} \right\}} \end{array}$ (1)

 \begin{aligned} {r_{{nq}}}\left( u \right) = & \Big( {{\left( {{x_n}\left( u \right) - {x_{\! q}}} \right)}^2} + {{\left( {{y_n}\left( u \right) - {y_q}} \right)}^2} \\ & + {{\left( {{z_n}\left( u \right) - {z_q}} \right)}^2} \Big) ^{1/2} \end{aligned} (2)

 ${r_{{nq}}}\left( u \right) \approx {r_0} + \frac{{{{\left( {uv - {x_{\! q}}} \right)}^2}}}{{2{r_0}}} + \frac{{{y_n^2} - 2{y_n}{y_q}}}{{2{r_0}}}$ (3)

PCA产生的误差相位为：
 $dr\left( i \right) \! = \! {r_{\text{s}}}\left( i \right) \! + \! {r_{\text{r}}}\left( i \right) \! - \! 2{r_{nq}}\left( i \right),i \! = \! 1,2,\cdots ,NM$ (4)

 $\begin{array}{*{20}{c}} {{S_{{\rm{ac}}}}\left( {t,{f_{u}}} \right) = {\varsigma _q}\;{{\mathop{\rm p}\nolimits} _{\rm{r}}}\left( {t - {t^\prime }} \right){{\mathop{\rm p}\nolimits} _{\rm{a}}}\left( {{f_{u}} - f_{u}^\prime } \right)}\\ { \cdot \exp \left( { - {\rm{j}}\frac{{4\pi {f_{0}}{r_0}}}{{\rm{c}}} + {\rm{j}}\frac{{4\pi {f_{0}}{y_n}{y_q}}}{{{\rm{c}}{r_0}}} - {\rm{j}}\frac{{2\pi f{_u}{x_{q}}}}{v}} \right)} \end{array}$ (5)

 $\begin{array}{*{20}{c}} {{S_{{\rm{cc}}}}\left( {t,u,\theta } \right) = {\varsigma _q}\;{{\rm{p}}_{\rm{r}}}\left( {t - {t^\prime }} \right){{\rm{p}}_{\rm{a}}}\left( {u - {u^\prime }} \right)}\\ { \cdot {{\rm{p}}_{\rm{x}}}\left( {\sin \theta - \sin {\theta _0}} \right)\exp \left( { - {\rm{j}}\frac{{4\pi {f_{0}}{r_0}}}{{\rm{c}}}} \right)} \end{array}$ (6)

 $\left\{ {\begin{array}{*{20}{c}} \!\! {{\rho _{\text{x}}} = \frac{{\lambda H}}{{2{L_{\text{s}}}}}} \\ \!\! {{\rho _{\text{y}}} = \frac{{\lambda H}}{{2{L_{\text{a}}}}}} \\ \!\! {{\rho _{\text{z}}} = \frac{\rm c}{{2B}}} \end{array}} \right.$ (7)

3 2维快速ESPRIT算法

(1)为了得到ESPRIT算法矩阵分解时奇异值的个数，结合线阵3维SAR成像模型，采用盖式圆方法估计点目标数量。

(2)为了降低运算量，对传统的2维ESPRIT算法进行改进，用L型搜索替代平面搜索，即在切航迹向和沿航迹向分别选择1行阵列估计2维位置。

(3)采用2维快速ESPRIT算法求出的2维位置不是一一对应的，会在平面内出现多种排列组合的情况，形成虚假目标。为了消除虚警，本文提出了基于“区域生长”的位置配对新方法，将配对的位置代入RD算法成像结果得到目标散射系数，相比于最小二乘估计，减少了运算量且降低了旁瓣的影响。

3.1 2维快速ESPRIT算法

 $\begin{array}{*{20}{c}} {X = \left[ {{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},{n_{\rm{a}}},1} \right),{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},{n_{\rm{a}}},2} \right), \cdots ,} \right.}\\ {{{\left. {{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},{n_{\rm{a}}},{N_{\rm{x}}}} \right)} \right]}^{\rm{T}}}} \end{array}$ (8)

 $R = {\left[ {\begin{array}{*{20}{c}} {{R_0}' }&r\\ {{r^H}}&{{r_0}} \end{array}} \right]_{{N_{\rm{x}}} \times {N_{\rm{x}}}}}$ (9)

 ${R_{\rm{U}}} = {\left[ {\begin{array}{*{20}{c}} {{\lambda ^\prime }}&0& \cdots &{{\rho _1}}\\ 0&{{\lambda ^\prime }}& \cdots &{{\rho _2}}\\ \vdots & \vdots & \ddots & \vdots \\ {\rho _1^*}&{\rho _2^*}& \ldots &{{r_{{N_{\rm{x}}}{N_{\rm{x}}}}}} \end{array}} \right]_{{N_{\rm{x}}} \times {N_{\rm{x}}}}}$ (10)

 ${\text{GDE}}\left( k \right) = {r_k} - \frac{{D\left( L \right)}}{{{N_{\text{x}}} - 1}}\sum\limits_{i = 1}^{{N_{\text{x}}} - 1} {{r_i}} > 0$ (11)

3.2 ESPRIT算法估计2维位置

 图 2 沿航迹向虚拟阵列示意图 Fig. 2 Illustration of the virtual array in along-track direction

ESPRIT算法利用子阵结构的旋转不变性估计信号参数，适用于两个完全相同的子阵，并且这两个子阵具有已知的间隔[8]。本文首先选取切航迹向的阵列作为子阵，切航迹向聚焦后的式(8)的值作为接收矢量X；将X循环移动单位距离d作为另一个接收矢量Y。因此，接收矢量表示为：
 $X = AS + {N_{\rm{1}}}$ (12)
 $Y = A\Phi S + {N_{\rm{2}}}$ (13)

 $A = {\left[{a\left( {{\theta _1}} \right),a\left( {{\theta _2}} \right),\cdots ,a\left( {{\theta _q}} \right)} \right]_{NM \times q}}$ (14)

 $a\left( {{\theta _i}} \right) = \left[{\exp \left( {{\rm{j}}\frac{{4\pi {f_{0}}{y_1}{y_i}}}{{{\rm{c}}{r_0}}}} \right),\cdots ,} \right.{\left. {\exp \left( {{\rm{j}}\frac{{4\pi {f_{0}}{y_{NM}}{y_i}}}{{{\rm{c}}{r_0}}}} \right)} \right]^{\rm{T}}}$

 ${\begin{array}{*{20}{l}} {{R_{XX}} = {\rm{E}}\left( {X{X^{\rm{H}}}} \right) = A{\rm{E}}\left( {S{S^{\rm{H}}}} \right){A^{\rm{H}}} + {\sigma ^2}I}\\ {{R_{XY}} = {\rm{E}}\left( {X{Y^{\rm{H}}}} \right) = A{\rm{E}}\left( {S{S^{\rm{H}}}} \right){\Phi ^{\rm{H}}}{A^{\rm{H}}} + {\sigma ^2}Z} \end{array}}$ (15)

 $Z = {\left[ {\begin{array}{*{20}{c}} 0&0& \cdots &0\\ 1&0& \cdots &0\\ 0&{}& \cdots &{}\\ 0&0& \cdots &0 \end{array}} \right]_{{N_{\rm{x}}} \times {N_{\rm{x}}}}}$ (16)

 $\left\{ {\begin{array}{*{20}{c}} {{C_{XX}} = {R_{XX}} - {\sigma ^2}I}\\ {{C_{XY}} = {R_{XY}} - {\sigma ^2}Z} \end{array}} \right.$ (17)

 $\begin{array}{*{20}{c}} {{C_{XX}} = U\Sigma {V^H}}\\ { = \left[ {{U_1}{U_2}} \right]\left[ {\begin{array}{*{20}{c}} {{\Sigma _1}}&0\\ 0&{{\Sigma _2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {V_1^H}\\ {V_2^H} \end{array}} \right]} \end{array}$ (18)

 $X = {[{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},1,{n_{\rm{x}}}} \right),{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},2,{n_{\rm{x}}}} \right), \cdots ,\;{S_{{\rm{ac}}}}\left( {{n_{\rm{r}}},{N_{\rm{a}}},{n_{\rm{x}}}} \right)]^{\rm{T}}}$ (19)

3.3基于区域生长的点目标2维位置配对方法

 图 3 2个点目标2维位置示意图 Fig. 3 Illustration of two-dimensional positions of three scatterers

${S_{{\rm{rx}}}}{\left( {i,j} \right)_{{N_{\rm{a}}} \times {N_{\rm{x}}}}}$相当于多个2维SAR成像结果的叠加，会产生更明显的旁瓣。当点目标的间隔距离小于分辨率时，相邻目标点的旁瓣叠加在一起，严重干扰对目标位置的判断，甚至会掩盖某些目标点的存在。由RD算法沿航迹向点扩展函数图 4可看出，两个相邻点目标之间一定存在一个凹点，这个凹点就是两个点目标聚焦值的平衡点。在平衡点的两侧，两个点目标的幅值各自占主导地位，即目标的散射中心一定分别位于这两个区域内。

 图 4 沿航迹向点扩展函数示意图 Fig. 4 Illustration of PSF in cross-track direction

(1)“生长区域”[18]：定义一个新矩阵$S{1_{{\rm{rx}}}}{\left( {i,j} \right)_{{N_{\rm{a}}} \times {N_{\rm{x}}}}} = {S_{{\rm{rx}}}}{\left( {i,j} \right)_{{N_{\rm{a}}} \times {N_{\rm{x}}}}}$，将RD算法成像结果的2维累加矩阵作为区域生长的“原图像”。设变量i表示点目标个数，即需要搜索的局部极值点个数，并置初值为零。

(2)“种子点”：遍历S1rx找出极大值，作为区域生长的“种子点”，将它在矩阵中的位置存于maq×q，设记录生长次数的变量k=0。

(3)“生长规则”：对种子点的四连通区域进行遍历，此时种子点取原矩阵S1rx中对应位置上的值。若遍历值比种子点的值大，此时种子点就是平衡点，停止此方向上的遍历，并将S1rx中种子点的值置零；若相反，则将此遍历点作为新的种子点重复进行此步骤，将原种子点和遍历点在S1rx中的值置零，并设k=k+1。

(4)“终止条件”：k<a是第3步的“终止条件”，a是个常数，作为遍历区域的半边长。第3步终止时，i=i+1，并跳到第2步继续进行，i<q作为整个算法的“终止条件”，q是点目标的个数。

 $\mathop S\limits^ \wedge = \left( {{A^H}{A^{ - 1}}{A^{\rm{H}}}X} \right)$ (20)

 图 5 用于线阵3维SAR成像的2维快速ESPRIT算法处理流程 Fig. 5 Flow of the proposed algorithm

4 实验结果与分析 4.1 多个点目标成像

 图 6 ESPRIT算法估计出的2维位置 Fig. 6 Estimation in cross and along-track directions by ESPRIT algorithm

 图 7 RD算法和本文算法2维俯视对比图 Fig. 7 Comparison of 2D image between RD algorithm and proposed algorithm

 图 8 RD算法和本文算法3维成像对比 Fig. 8 Comparison of 3D image between RD algorithm and proposed algorithm

 图 9 不同SNR下2维可分辨概率 Fig. 9 Probability of resolution versus the SNR in cross and along-track directions

4.2 不同散射系数的目标点成像

 图 10 RD算法和本文算法2维俯视对比图 Fig. 10 Comparison of 2D image between RD algorithm and proposed algorithm

5 结论

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