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高斯过程模型以贝叶斯神经网络为基础发展的一种新的核方法,可用于处理机器学习中的分类和回归等问题。由于其优越的鲁棒性,高斯过程模型已经在模式分类识别问题中得到应用和验证[21-24]。文中主要讨论以高斯过程为基础的多元分类器构造,从而满足SAR目标识别中多类别的分类需求。
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一般地,高斯过程模型包括似然函数定义、隐变量函数定义和后验概率计算三部分。基于高斯过程模型的分类算法利用高斯函数对分类过程的隐变量函数进行逼近,代表性的方法有Laplace法、Expectation Propagation法、Kullback-Leibler散度最小化法等。
对于一个包含N个观测数据的训练集D,
$D = $ $ \left\{ {\left( {{x_i},{y_i}} \right)} \right\}_{i = 1}^N$ ,其中${x_i} \in {R^d}$ 为第i个输入数据样本,$d$ 为样本维度,其对应的二值类别标签记为${y_i}$ ,${y_i}{\rm{ = }}1$ 表示正类,${y_i}{\rm{ = - }}1$ 表示负类。$X = \left[ {{x_i}, \cdots ,{x_i}, \cdots ,{x_N}} \right]$ 表示一个$N \times d$ 维矩阵。对于输入样本${x_i}$ ,相应地定义一个隐函数${f_i}$ 。$f = \left[ {{f_1}, \cdots ,{f_N}} \right]$ 包含所有输入样本的隐函数。同时,利用Sigmoid函数ρ(·),$\rho ({f_i}) = p({y_i} = 1/{f_i})$ ,将每一个隐函数输出约束在$[0,1]$ 之间。假设每个数据样本独立同分布,计算它们的联合概率分布如下:$$p(y/f) = \prod\limits_{i = 1}^N {p(\frac{{{y_i}}}{{{f_i}}})} = \prod\limits_{i = 1}^N {\rho ({y_i},{f_i})} $$ (1) 假设
${f_i}$ 为零均值高斯分布,则先验概率$p(f/X)$ 描述如下:$$p(f/X) = N(0,K) = \frac{1}{{{{(2\pi )}^{{{L}}/2}}{{\left| K \right|}^{1/2}}}}\exp \left\{ { - \frac{1}{2}{f^{\rm{T}}}{K^{ - 1}}f} \right\}$$ (2) 式中:
$K$ 指$f$ 的协方差矩阵。基于概率理论,隐函数后验概率计算如下:$$p(f/X,y,\theta ) = \dfrac{{p(y/f)p(f/X)}}{{p(y/X,\theta )}}$$ (3) 式中:
$p(y/f)$ 为似然函数;$p(y/X,\theta )$ 表示边缘概率。利用Laplace逼近法对后验概率$p(f/X,y,\theta )$ 进行求解,获得对应的估计值$q(f/X,y,\theta )$ 。采用二阶泰勒级数将$\log {\rm{ }}p(f/X,y,\theta )$ 在最大后验概率处$\hat f$ 处展开,得到结果如下:$$q(f/X,y,\theta ) = N(\hat f,{A^{ - 1}}) \propto \exp \left( - \frac{1}{2}{(f - \hat f)^{\rm{T}}}A(f - \hat f)\right)$$ (4) 式中:
$\hat f{\rm{ = }}\arg {\max _f}p(f/X,y,\theta )$ ;汉森矩阵$A = - \nabla \nabla p $ $ (f/X,y,\theta )\left| {f = \hat f} \right.$ 。$p(y/X,\theta )$ 与$f$ 之间相互独立,因此最大化$p(f/X,y,\theta )$ 与最大化公式(5)定义的$\varphi (f)$ 具有等效性,据此可求解$\hat f$ :$$ \begin{split} & \varphi (f){\rm{ = }}\log p(y/f) + \log p(f/X) =\\ & \log p(y/f) - \frac{1}{2}{f^{\rm{T}}}{K^{ - 1}}f - \frac{1}{2}\log \left| K \right| - \frac{{{L}}}{2}\log 2\pi \\ \end{split} $$ (5) 可得到后验概率为:
$$p(f/X,y,\theta ) \approx q(f/X,y,\theta ) = N(\hat f,{A^{ - 1}}) = N(\hat f,(W + {K^{ - 1}}))$$ (6) 另外,边缘概率分布可表示为:
$$\log p(y/X,\theta ) = - \frac{1}{2}{f^{\rm{T}}}{K^{ - 1}}f + \log p(y/\hat f) - \frac{1}{2}\log \left| B \right|$$ (7) 式中:
$\left| B \right| = \left| K \right|\left| {{K^{ - 1}} + W} \right| = \left| {{I_n} + {W^{\frac{1}{2}}}K{W^{\frac{1}{2}}}} \right|$ ;$\theta $ 代表超参数,可通过最大化公式(7)求解。对于给定的测试$x$ ,其对应的隐函数${f_ * }$ 的概率分布为:$${f_ * }/X,y,{x_ * } \sim N({K_ * }{K^{ - 1}}\hat f,{K_ * } - {K_ * }{\tilde K^{{\rm{ - }}1}}K_ * ^{\rm{T}})$$ (8) 式中:
$\tilde K{\rm{ = }}K + {W^{{\rm{ - }}1}}$ 。则测试样本对应的输出${y_ * } = 1$ 的概率为:$$\bar \rho ({f_ * }){\rm{ = }}\int {\rho ({f_ * })p({f_ * }/X,y,{x_ * })} {\rm{d}}{f_ * }$$ (9) 根据对应类别的概率大小,即可在二元分类的框架判定测试样本的目标类别。
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传统的高斯过程模型可直接用于二元分类,但实际问题往往涉及到多个类别的鉴别区分。为此,需要将二元分类推广到多元以实现多类别的直接分类。其中,将二元分类推广到多元分类的一种代表性方法就是一对一投票机制。文中采用该方法将传统基于高斯过程的二元分类器推广到多元分类,具体实施如下:
(1)在训练阶段,对1~k类的训练样本进行两两组合,得到
$C_k^2 = k(k - 1)/2$ 种组合方式,采用高斯过程模型对任一组合进行训练,得到相应的高斯过程二分类器${C_{i,j}}$ ,其中$i \in \left\{ {1, \cdots ,k} \right\},j \in \left\{ {1, \cdots ,k} \right\}$ (${\theta _{i,j}}$ 为超参数);(2)在分类阶段,通过投票机制对未知样本
${x_ * }$ 的所属类别进行判决。首先,将每个类别的初始票数均设置为0;然后,利用已经训练得到的$k(k - 1)/2$ 个二元分类器对测试样本进行分类,当分类器${C_{i,j}}$ 将${x_ * }$ 判别为第i类,则类别i的得票数加1,若分类器${C_{i,j}}$ 将测试样本判别为第j类,则类别j的得票数加1。最终,统计各类别的总得票数,得票数最高的类别即判决为测试样本${x_ * }$ 的所属类别。文中将高斯过程模型的二分类机制与一对一投票机制相结合获得多元分类器。结合后的多元分类器可直接采用多类别的训练样本进行训练,获得参数化的分类模型。此时,对于待识别的测试样本,可直接将其输入训练后的多类高斯过程分类器,通过计算总体得票数判定其所属目标类别。
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依托MSTAR数据集设置典型测试场景开展实验与验证。数据集中包含的目标类别如图2所示,相应的SAR图像由X波段机载雷达采集,方位及距离分辨率均为0.3 m。对于任一目标,其SAR图像覆盖全方位角以及典型俯仰角。基于MSTAR数据集可设置多种实验场景对所提方法进行测试,包括标准操作条件及典型扩展操作条件。
文中提出基于高斯过程模型的SAR目标识别方法,主要从分类决策融合提高整体识别性能。为此,在对比算法的选择现有SAR目标识别方法,包括NMF[9]、Mono[10]、BEMD[11]以及CNN[19]。这四类对比方法基本覆盖了现有SAR目标识别中最常用的特征和分类器类型。后续实验中,设置3个实验场景对所提方法进行考察,分别为场景1:标准操作条件,涉及10类目标;场景2:俯仰角差异,涉及3类目标;场景3:噪声干扰,涉及10类目标。
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表1给出了场景1标准操作条件描述,包含10类目标。训练集应用17°俯仰角SAR图像,获得分类器对于15°俯仰角样本进行测试。对比可见,两个集合之间目标型号保持一致,仅有2°俯仰角差异,因此整体相关性较强。图3显示了所提方法在当前场景下的结果(其中横纵坐标中“1~10”分别对应表1中首列自上而下的目标类别),单类识别率均高于98.5%(对角线元素所示),10类平均识别率达到99.28%,体现了提出方法的有效性。各类方法在标准操作条件下的平均识别率如表2所示,NMF、单演信号、BEMD以及CNN方法的识别率分别为98.04%、98.56%、98.82%和99.08%,均高于98%。由此可以看出,标准操作条件下的识别问题相对简单。通过比较各类方法的平均识别率,可以看出文中方法具有性能优势。CNN方法在当前条件下的识别率仅次于提出方法,得益于深度学习模型的分类能力。文中采用通过引入高斯过程模型获得统计意义上最优的分类模型,有效提升了标准条件下的目标识别性能。
表 1 场景1相关描述
Table 1. Description of the scenario 1
Type Training set (17°) Test set (15°) Configuration Scale Configuration Scale BMP2 9563 232 9563 194 BTR70 c71 232 c71 195 T72 132 231 132 195 812 194 s7 190 T62 A51 298 A51 272 BRDM2 E-71 297 E-71 273 BTR60 7532 255 7532 194 ZSU23/4 d08 298 d08 273 D7 13015 298 13015 273 ZIL131 E12 298 E12 273 2S1 B01 298 B01 273 表 2 场景1下结果对比
Table 2. Comparison of resutls under scenario 1
Method Average recognition rate Proposed 99.28% NMF 98.04% Mono 98.56% BEMD 98.82% CNN 99.08% -
扩展操作条件指的是由于SAR数据获取条件的变化导致测试样本与训练样本存在较大差异。典型地,SAR目标识别中的扩展操作条件包括目标型号差异、俯仰角差异、噪声干扰等。该实验在俯仰角差异条件下对提出方法进行测试,设置如表3所示的场景2。其中,17°俯仰角样本用于训练;30°和45°俯仰角样本均用于分类测试,可见训练和测试样本之间存在较大的俯仰角差异。在两个测试集上分别对各类方法进行性能测试,获得识别结果统计如图4所示。对比两个俯仰角下的结果,30°下的总体性能显著优于45°,说明大俯仰角差异会导致更大的图像差异。在两个角度下,文中方法均取得了最高的平均识别率,显示其更高的稳健性。高斯过程模型通过推导统计学上的最佳分类模型,能够更为有效地发掘真实类别之间的内在关联,因此识别方法对于俯仰角差异的稳健性得以提升。
表 3 场景2相关描述
Table 3. Desciption of scenario 2
Depression/(°) 2S1 BDRM2 ZSU23/4 Training set 17 299 298 299 Test set 30 288 287 288 45 303 303 303 -
噪声干扰是另一种典型扩展操作条件,主要是待识别SAR图像的信噪比(SNR)相对较低,导致与训练样本存在较大的差异。以表1的测试和训练样本为基础,向其中的训练样本添加不同程度的噪声,从而构造场景3下多个信噪比下的测试集。具体地,根据测试样本自身能力,按照预设的信噪比获得加性高斯噪声,将其混入原始测试样本,即可得到对应噪声水平的噪声干扰测试样本。然后,分别在各个噪声水平对各类方法进行测试,获得如图5所示的识别结果。可以看出,随着噪声水平的不断降低,各类方法的平均识别率呈现明显的下降趋势。对比可见,文中方法在各个噪声水平均取得最高的识别率,体现其更强的噪声稳健性。在高斯过程的推导过程中,充分考虑到可能的噪声影响。因此,其最终得到的分类模型对于噪声具有较强的稳健性。与俯仰角差异的情形类似,CNN方法在噪声干扰条件下的性能下降最为剧烈,主要是训练样本与测试样本存在较大的图像差异,导致最终的分类模型性能显著下降。
Application of Gaussian process model in SAR image target recognition
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摘要: 将高斯过程模型应用于合成孔径雷达(SAR)图像目标识别。高斯过程模型是基于贝叶斯框架的统计学习算法,通过结合核函数和和概率判别构建分类模型。与传统分类模型相比,高斯过程模型可以获得更高的分类效率和精度。方法实施过程中,采用SAR图像的特征矢量作为输入,以目标类别标签作为输出训练高斯过程模型。对于待识别样本,通过计算其在高斯过程模型下属于各个类别的后验概率判定其目标类别。实验中,依托MSTAR数据集在典型条件下开展测试。根据实验结果,所提方法在标准操作条件下对10类目标识别精度达到99.28%;在30°和45°俯仰角下的平均识别率分别为98.04%和73.13%;在噪声干扰各个信噪比条件下均保持最高性能。实验结果验证了所提方法的有效性和稳健性。Abstract: The Gaussian process model was applied to synthetic aperture radar (SAR) image target recognition. Gaussian process model was a statistical learning algorithm based on the Bayesian framework, which combines the kernel function and probability judgement to build the classification model. Compared with the traditional classification models, the Gaussian process model could achieve higher classification accuracy and precision. In the implementation of target recognition, the feature vectors from SAR images were used as the inputs while the target labels were employed as the outputs thus training the Gaussian process model. For the test sample to be classified, the posterior probabilities related to different classes were calculated thus determining its target label. In the experiments, typical situations were set up to test the proposed method using the MSTAR dataset. According to the experimental results, the proposed method could achieve 99.28% recognition accuracy for 10 types of targets under standard operating conditions. The average recognition rates at 30° and 45° depression angles were 98.04% and 73.13%, respectively. Under noise corruption, the best performance was achieved by the proposed method at each noise level. The results validated the effectiveness and robustness of the proposed method.
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表 1 场景1相关描述
Table 1. Description of the scenario 1
Type Training set (17°) Test set (15°) Configuration Scale Configuration Scale BMP2 9563 232 9563 194 BTR70 c71 232 c71 195 T72 132 231 132 195 812 194 s7 190 T62 A51 298 A51 272 BRDM2 E-71 297 E-71 273 BTR60 7532 255 7532 194 ZSU23/4 d08 298 d08 273 D7 13015 298 13015 273 ZIL131 E12 298 E12 273 2S1 B01 298 B01 273 表 2 场景1下结果对比
Table 2. Comparison of resutls under scenario 1
Method Average recognition rate Proposed 99.28% NMF 98.04% Mono 98.56% BEMD 98.82% CNN 99.08% 表 3 场景2相关描述
Table 3. Desciption of scenario 2
Depression/(°) 2S1 BDRM2 ZSU23/4 Training set 17 299 298 299 Test set 30 288 287 288 45 303 303 303 -
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