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稀疏表示[11-15]是基于压缩感知理论的一种新的信号处理方法,在图像处理领域得到了广泛运用。其基本思想是用各类训练样本构成的全局字典对测试样本
${\boldsymbol{y}}$ 进行线性表示,如公式(1)所示:$${\boldsymbol{\hat x}} = \arg \min {\left\| {\boldsymbol{x}} \right\|_0} \;{\rm{s}}{\rm{.t}}{\rm{. }}\left\| {{\boldsymbol{y}}{\rm{ - }}{\boldsymbol{Ax}}} \right\|_2^2 \leqslant \varepsilon $$ (1) 式中:
${\boldsymbol{A}} = [{{\boldsymbol{A}}_1},{{\boldsymbol{A}}_2}, \cdots ,{{\boldsymbol{A}}_C}] \in {{{R}}^{d \times N}}$ 代表各个类别组成的全局字典,${{\boldsymbol{A}}_i} \in {{{R}}^{d \times {N_i}}}$ 包含了第$i$ 个训练类的${N_i}$ 个样本;${\boldsymbol{\hat x}}$ 表示线性系数矢量;$\varepsilon $ 为重构误差大小。公式(1)中涉及${l_0}$ 范数,该优化为非凸优化问题,难以直接求解。因此,通常将其近似为${l_1}$ 范数优化进而转变为凸优化问题;也可以采用贪婪算法求解,如正交匹配追踪算法(Orthogonal Matching Pursuit,OMP)获得近似解。上述表示系数的大小体现了测试样本与不同类别中训练样本的相似性。具体应用到方位角估计中,根据SAR图像的方位角敏感性,那些与测试样本高度相关的样本实际上与测试样本保持接近的方位角。如图1所示,不同方位角下的三类MSTAR目标的SAR图像均具有较为明显的特性差异。参考文献[11-12]中提出的基于稀疏表示的方位角估计方法就是选用具有最小重构误差的训练样本得到估计方位角。实际上,SAR图像的相关性可以在一定的方位角区间内保持[13],因此综合一个区间内结果有利于得到更稳健的估计结果。
假定全局字典内各类别字典都是按照方位角由小到大排列,非零表示系数则集中在一定的方位角区间内,此时的线性表示系数具有块稀疏特性。为了得到更为准确的
${\boldsymbol{\hat x}}$ ,文中选用块稀疏贝叶斯学习[14-15]求解公式(1)。具有块稀疏特性的系数矢量如下式所示:$${\boldsymbol{x}} = {[\underbrace {{x_1},...,{x_{{d_1}}}}_{{\boldsymbol{x}}_1^{\rm{T}}},...,\underbrace {{x_{{d_{g - 1}} + 1}},...,{x_{{d_g}}}}_{{\boldsymbol{x}}_g^{\rm{T}}}]^{\rm{T}}}$$ (2) 其中,
${\boldsymbol{x}}$ 包含$g$ 个分块,但其中只有少量的分块包含非零元素。为了充分发掘这种块稀疏特性,块稀疏贝叶斯学习[13]采用高斯分布系数矢量建模如下:$$p({x_i};\gamma {}_i,\{ {\boldsymbol{B}}{}_i\} ) = N({x_i};0,{\boldsymbol{B}}{}_i)$$ (3) 式中:参数
$\gamma {}_i$ 和${\boldsymbol{B}}{}_i$ 分别描述置信度水平和块内的相关性。假定各个分块相互独立,则公式(3)可以重新描述为下式:$$p({x_i};\{ \gamma {}_i\} ,\{ {\boldsymbol{B}}{}_i\} ) = N({x_i};0,\varGamma )$$ (4) 式中:
$\varGamma$ 为块对角矩阵,其第$i$ 分量为${\gamma _i}{B_i}$ 。观测样本
${\boldsymbol{y}}$ 按照下式建模:$${\boldsymbol{y}} = {\boldsymbol{Ax}} + {\boldsymbol{n}}$$ (5) 式中:
${\boldsymbol{n}}$ 代表噪声项。采用均值为零方差为${\;\beta ^{ - 1}}$ 的高斯分布建模。由此,${\boldsymbol{x}}$ 的概率密度函数为:$$p({x_i}|\gamma {}_i;\beta ) = N(\varPhi {\boldsymbol{x}},{\beta ^{ - 1}})$$ (6) 块稀疏贝叶斯学习通过迭代的方式进行求解,根据估计得到的超参数
$\{ \gamma {}_i,{\boldsymbol{B}}{}_i\} $ 以及$\;\beta $ , 获得块稀疏系数的最大后验估计如公式(7)所示。具体的求解过程可参见参考文献[14-15]。$${\boldsymbol{\hat x}} = \displaystyle\sum {\varPhi ^{\rm{T}}}\beta {\boldsymbol{y}}$$ (7) -
根据稀疏表示的优化求解结果,分别在各个非零表示系数的分块内重构测试样本,如下式所示:
$${\boldsymbol{r}}(i){\rm{ = }}{\left\| {{\boldsymbol{y}} - {{\boldsymbol{A}}_i}{\delta _i}({\boldsymbol{\hat x}})} \right\|_2}{\rm{(}}i = 1, \cdots ,g)$$ (8) 式中:
${\delta _i}({\boldsymbol{\hat x}})$ 代表将${\boldsymbol{\hat x}}$ 中除了第$i$ 块的其他系数都置为0。按照最小误差的准则选取进行方位角估计的候选区间:$$K{\rm{ = }}\mathop {\arg \min }\limits_i ({\boldsymbol{r}}(i))$$ (9) 记第
$K$ 分块中的方位角序列为$[{\theta _1},{\theta _2}, \cdots ,{\theta _M}]$ ,对应的系数大小为$[{r_1},{r_2}, \cdots ,{r_M}]$ 。文中采用线性加权的方法得到方位角估计值。根据系数大小计算对应方位角的权值如下式所示:$${w_i} = {r_i}/\sum\limits_{j = 1}^M {{r_j}} (i = 1,2, \cdots ,M)$$ (10) 按照公式(11)中线性加权的方法得到估计的方位角。
$$\hat \theta = \sum\limits_{i = 1}^M {{w_i}} ·{\theta _i}$$ (11) 相比采用单个样本估计方位角的方法,这种联合整个方位角区间的方法既可以避免选择单个样本时可能出现的扰动误差,又可以通过加权综合的方式得到更为稳健的估计值。
根据以上分析,文中方法的基本流程如图2所示。具体可以描述为以下步骤:
Step 1:对训练样本按照方位角顺序从小到大排列,构成全局字典;
Step 2:对待估计的测试样本在字典上进行块稀疏重构,采用块稀疏贝叶斯算法求解稀疏表示系数;
Step 3:分别在各个分块区间内重构测试样本,根据最小重构误差选择候选区间;
Step 4:在候选区间内按照公式(10)、(11)的线性加权方法得到目标方位角的估计值。
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实验基于MSTAR数据集中三类目标的SAR图像开展,具体样本情况如表1所描述。其中,BMP2和T72均包括三个型号(不同型号采用Serial number (SN) 进行区分),BTR70仅有一个型号。测试样本采集于15°俯仰角,训练样本来自于17°俯仰角。训练样本的真实方位角可以从MSTAR图像的原始记录数据中读取,作为方位角估计精度计算的参考值。为了更好地体现提出方法的有效性,后续实验中将其与参考文献[4]中的最小外接矩形(Minimum enclosing rectangle,MER)法、参考文献[5]中的主导边界(Dominant boundary)法以及参考文献[11]中的基于稀疏表示(Sparse representation)的方法进行对比。
表 1 三类MSTAR目标的训练和测试样本
Table 1. Training and test samples of the three MSTAR targets
Type Depression angle/(°) Target BMP2 BTR70 T72 Training 17 233 (SN_1)232 (SN_2)233(SN_3) 233 (SN_1) 232 (SN_1)231 (SN_2)233 (SN_3) Test 15 195 (SN_1)196 (SN_2)19 6(SN_3) 196 (SN_1) 196 (SN_1)195 (SN_2)191 (SN_3) -
采用文中提出的算法对三类目标的测试样本进行方位角估计,通过对比方位角估计值与其真值得到各个测试样本的估计误差。表2统计了提出方法的方位角估计结果,其中认为估计角度与真值误差在±10°以内为正确估计,否则为错误估计。在此条件下,提出方法可以正确估计99%以上的测试样本中的目标方位角,这一结果充分证明了提出方法的高性能。表3进一步对提出方法的估计精度进行细化分析,按照误差区间统计分布情况,结果表明提出方法对测试样本的估计误差多数能够控制在5°以内,表明其具有很高的正确率和估计精度。更为细致的误差分布情况如图3所示,多数测试样本的方位角估计误差能够控制在2°范围内,进一步表明提出方法不仅能够取得高精度,还能保持很强的稳健性。
表 2 提出方法的对三类MSTAR目标测试样本的方位角估计结果
Table 2. Azimuth estimation results of the test samples of the three MSTAR targets by the proposed method
Target class Number of samples Number of errors Percentage of correct samples BMP2 (SN_1) 195 1 99.49% BMP2 (SN_2) 196 0 100% BMP2 (SN_3) 196 1 99.50% BTR70 (SN_1) 196 2 98.98% T72 (SN_1) 196 3 98.47% T72 (SN_2) 195 1 99.49% T72 (SN_3) 191 0 100% 表 3 提出方法在不同估计精度下的结果
Table 3. Results of the proposed method at different estimation precisions
Number of samples <5° <10° Mean Variance BMP2 587 573 585 2.01 1.85 BTR70 196 192 194 2.05 1.84 T72 582 569 578 2.16 1.77 Total 1365 1334 1357 2.07 1.81 表4对比了提出方法与其他方位角估计方法的总体性能。在要求的不同方位角估计误差约束下,提出方法的性能显著优于基于几何形状特征的最小外接矩形法和主导边界法。同时,提出方法还可以有效克服上述两类方法中存在的180°模糊问题。对比稀疏表示方法,提出方法性能具有优势,特别是在方位角估计误差要求很小的情况下,这种优势体现得更为明显。由于块稀疏贝叶斯学习考虑了SAR图像固有的方位角敏感性,可以更为准确地定位测试样本的方位角区间。通过文中的区间方位角加权方法,有利于得到更为准确的估计结果。表5对比了各类方法的效率,即估计单个测试样本所需的时间。提出方法的时间消耗最小,证明了其高效性。传统的基于目标二值区域的方法需要首先进行较为繁琐的图像预处理和目标分割,因此需要更多的时间消耗。相比基于稀疏表示的方法,文中采用的块稀疏贝叶斯学习效率更高,从而估计算法的效率更高。这些实验结果充分证明了采用块稀疏贝叶斯学习求解稀疏表示系数以及联合方位角区间估计方位角的有效性和稳健性。
图 3 提出算法的在不同估计精度下的正确估计样本数目
Figure 3. Numbers of correctly estimation samples by the proposed algorithm at different estimation precisions
表 4 各类方法在不同估计精度下的估计正确率
Table 4. Correct estimation percentages of different methods at different estimation precisions
Method type Threshold of error/(°) 2 4 6 8 10 Proposed 76% 88% 97% 99% 99% MER 13% 24% 39% 57% 68% Dominant boundary 55% 82% 93% 97% 99% Sparse representation 64% 80% 93% 98% 99% 表 5 不同方法的时间消耗
Table 5. Time consumption of different methods
Method type Average time consumption/ms Proposed 10.5 MER 45.2 Dominant boundary 40.2 Sparse representation 12.1
Target azimuth estimation of synthetic aperture radar image based on block sparse Bayesian learning
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摘要: 提出了一种基于块稀疏贝叶斯学习的合成孔径雷达(Synthetic aperture radar,SAR)图像目标方位角估计方法。SAR图像具有较强的方位角敏感性,因此对于具有某一方位角的SAR图像仅能与其具有相近方位角的样本具有较高的相关性。方法基于稀疏表示的基本思想,首先对所有训练样本按照方位角顺序排列为全局字典。在此条件下,待估计样本在该字典上的线性表示系数具有块稀疏特性,即非零表示系数主要聚集在字典上的某一局部区域。求解得到的块稀疏位置包含的训练样本可以有效地反映待估计样本的方位角信息。采用块稀疏贝叶斯学习(Block sparse Bayesian learning, BSBL)算法求解全局字典上的稀疏表示系数,并根据具有最小重构误差的原则获得最佳的局部分块。在获取最佳分块的基础上,方位角计算方法采用线性加权的方式综合了该分块区间内所有训练样本的方位角信息从而获得更为稳健的估计结果。所提出的方法在充分考察SAR图像方位角敏感性的基础上,综合运用局部区间内样本的有效信息,避免了基于单一样本估计的不确定性。为了验证所提出方法的有效性,基于Moving and stationary target acquisition and recognition (MSTAR)数据集进行了方位角估计实验并与几类经典方法进行对比分析。实验结果验证了所提出方法的性能优势。Abstract: A target azimuth estimation algorithm of Synthetic Aperture Radar (SAR) images based on block sparse Bayesian learning was proposed. SAR images were highly sensitive to target azimuth, the SAR image with a special azimuth only highly correlate with those samples with approaching azimuths. The proposed method was developed based on the idea of sparse representation. First, all the training samples were sorted according to the azimuths to construct the global dictionary. Then, the sparse coefficients of test sample to be estimated over the global dictionary should be block sparse ones, that was the non-zero coefficients mainly accumulate in a local part on the global dictionary. The solved positions of the blocks effectively reflect the azimuthal information of the test sample. The block sparse Bayesian learning (BSBL) algorithm was employed to solve the block sparse coefficients and then the candidate blocks were chosen based on the minimum of the reconstruction errors. With the optimal block, the estimated azimuth was calculated by linearly fusing the azimuths of all the training samples in the block thus a robust estimation result could be achieved. The proposed method considered the azimuthal sensitivity of SAR images and comprehensively utilized the valid information in a local discretionary, so the instability of using a signal reference training sample could be avoided. Experiments were conducted on moving and stationary target acquisition and recognition (MSTAR) dataset to validate effectiveness of the proposed method while compared with several classical algorithms. The experimental results validate the superior performance of the proposed method.
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表 1 三类MSTAR目标的训练和测试样本
Table 1. Training and test samples of the three MSTAR targets
Type Depression angle/(°) Target BMP2 BTR70 T72 Training 17 233 (SN_1)232 (SN_2)233(SN_3) 233 (SN_1) 232 (SN_1)231 (SN_2)233 (SN_3) Test 15 195 (SN_1)196 (SN_2)19 6(SN_3) 196 (SN_1) 196 (SN_1)195 (SN_2)191 (SN_3) 表 2 提出方法的对三类MSTAR目标测试样本的方位角估计结果
Table 2. Azimuth estimation results of the test samples of the three MSTAR targets by the proposed method
Target class Number of samples Number of errors Percentage of correct samples BMP2 (SN_1) 195 1 99.49% BMP2 (SN_2) 196 0 100% BMP2 (SN_3) 196 1 99.50% BTR70 (SN_1) 196 2 98.98% T72 (SN_1) 196 3 98.47% T72 (SN_2) 195 1 99.49% T72 (SN_3) 191 0 100% 表 3 提出方法在不同估计精度下的结果
Table 3. Results of the proposed method at different estimation precisions
Number of samples <5° <10° Mean Variance BMP2 587 573 585 2.01 1.85 BTR70 196 192 194 2.05 1.84 T72 582 569 578 2.16 1.77 Total 1365 1334 1357 2.07 1.81 表 4 各类方法在不同估计精度下的估计正确率
Table 4. Correct estimation percentages of different methods at different estimation precisions
Method type Threshold of error/(°) 2 4 6 8 10 Proposed 76% 88% 97% 99% 99% MER 13% 24% 39% 57% 68% Dominant boundary 55% 82% 93% 97% 99% Sparse representation 64% 80% 93% 98% 99% 表 5 不同方法的时间消耗
Table 5. Time consumption of different methods
Method type Average time consumption/ms Proposed 10.5 MER 45.2 Dominant boundary 40.2 Sparse representation 12.1 -
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