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Twodimensional bidirectional principal component collaborative projection feature for SAR vehicle target recognition
EURASIP Journal on Advances in Signal Processing volumeÂ 2022, ArticleÂ number:Â 91 (2022)
Abstract
With the continuous improvement in the resolution of synthetic aperture radar (SAR), there are many problems in the interpretation of highresolution SAR images, such as a large amount of data and low efficiency of target recognition. In this paper, a novel SAR target recognition method based on a twodimensional bidirectional principal component cooperative representation projection feature ((2D)^{2}PCACRP) is proposed. First, (2D)^{2}PCA is used to project the image into the lowdimensional feature space, and the redundant information in the highresolution SAR image is filtered while considering the spatial structure. Then, the spatial global separability feature and local structure feature of the target in the highresolution SAR image are extracted by CRP to form the (2D)^{2}PCACRP feature. Finally, based on this feature, the nearest neighbour classifier is used to complete the target recognition experiments on MSTAR data. The experiments of this study are divided into three parts using standard operation condition (SOC) samples, type change samples and radar incidence angle change data. The experimental results show that the proposed feature achieves better target recognition performance in highresolution SAR images.
1 Introduction
Target recognition in SAR images is one of the most important applications in SAR image interpretation. SAR target recognition extracts features from target slices and classifies them with a classifier to realize the recognition of target categories, models, attributes, etc. [1, 2]. SAR target recognition technology can primarily be divided into two stages: feature extraction and classification [3, 4]. First, effective target features are extracted from training samples, and then, an appropriate classification model is selected to achieve target recognition for many test samples. Classical SAR target recognition algorithms can be roughly divided into two categories: traditional classification methods and deep learning methods. Traditional classification methods include template matching recognition, Knearest neighbour, Bayesian [5] and support vector machine [6]. Most of these methods use prior knowledge, such as the probability distribution of SAR images, and their performances are strongly affected by imaging quality, such as SAR data noise and the settings of the classifier parameters. To apply traditional methods to SAR image target recognition more effectively, many scholars have improved feature extraction methods and classification models or introduced new methods in similar fields, such as the sparse representationbased classification (SRC) method. SRC assumes that there are sufficient training samples, which uses the training samples to form an overcomplete dictionary to linearly represent the test samples and then minimizes the reconstruction error of the test samples according to the sparsity of the coding coefficients to achieve target recognition. In recent years, SRC has shown good performance in object recognition [7], face recognition [8], and text detection. SRC was first used for SAR image target recognition by Thiagarajan et al. [7]. Yang et al. [9] proposed a kernel sparse coding classifier using pulse contour transformation to derive the compressed features of the target and clustering pairs in the kernel space and verified its recognition performance on MSTAR data. Hongliang et al. [10] separated the target region from the background and proposed a recognition method based on the sparse representation of the target local dictionary in SAR images. To take advantage of the correlation between SAR multiview targets, Zhang et al. [11] proposed an automatic target recognition method based on multiview joint sparse representation. However, it is typically difficult to obtain multiview images of objects in the same scene in practical situations. Lv et al. [12] achieved the multitask joint classification of multilevel deep features extracted by convolutional neural networks based on a multitask joint sparse representation classifier. Zhang et al. [13] used information decoupling to build a multiresolution dictionary and proposed a joint sparse representation model based on the multiresolution dictionary.
However, SRC must solve the norm minimization problem with constraints, and the computational complexity of this process is too high. Zhang et al. [14] used the L2 norm to replace the L1 norm in the SRC model to make the model have a closed solution and proposed a collaborative representationbased classification (CRC) method. The recognition accuracy of CRC and SRC is similar, but the number of computations is markedly lower. Concurrently, CRC can find the optimal training sample representation coefficient of the reconstructed test sample, which enhances the correlation between the test sample and similar training samples. Zhang et al. [14] confirmed that good classification can be achieved under experimental conditions with few samples. The robustness of its classification has been verified in the fields of face recognition [15, 16], hyperspectral image classification [17], etc. In the field of SAR target recognition, Geng et al. [18] proposed a joint collaborative representation method based on the Wishart distance for polarimetric SAR image classification. Zhang et al. [19] input the 3 types of features of SAR target slices into SRC and CRC to obtain 6 category labels and obtained the final recognition result according to the Yess decision fusion. Based on the model of Zhang et al., Wang et al. [20] added multiple adjacent multiview samples of the current sample as the input of the model and obtained the label of the current test sample under this feature. Finally, the tags are fused by the voting method to obtain the final recognition result.
The deep learning method [21] primarily uses the convolutional neural network (CNN) to complete the automatic extraction of target features in the image and then uses the fully connected layer or other classification models to achieve automatic target recognition. CNN models represented by AlexNet [22], VGGNet [23], GoogLeNet [24], ResNet [25], DenseNet [26] and SENet [27] have appeared in deep learning methods. CNNs have achieved excellent results in the field of optical image target recognition, and many scholars have also introduced them into SAR image target recognition. Chen S et al. introduced CNN into the SAR field and achieved recognition accuracy far exceeding traditional methods on the MSTAR SOC dataset [28]. Zhang et al. used a ResNet network to extract fixed spatial scatter features from each azimuth image and trained the network using joint supervision of softmax loss and centre loss [29]. Treating SAR images as sequences, Ruihang Xu et al. proposed a target classification method for SAR images based on a spatiotemporal holistic convolutional network (STECNet), which achieved better results than basic convolutional neural networks [30]. Currently, a large number of studies of CNNbased SAR image target recognition [31,32,33,34,35,36,37, 39] have reported better performances than traditional learning algorithms, but deep learning methods have stricter hardware requirements and are difficult to implement on edge devices. In addition, deep learning methods rely heavily on largescale and samplebalanced datasets and face problems such as fewer training samples, optimal design of deep models and longer training time.
To develop traditional learning processes into deep learning processes, the primary required change lies in the automatic extraction of effective features. Feature extraction determines the upper limit of the performance of the target recognition algorithm, and the classification model only makes the algorithm approach the upper limit continuously. Therefore, the study of effective feature extraction technology is important for target recognition.
In the field of image processing, feature extraction primarily reduces the dimensionality of image data and obtains a series of features representing the image. In addition to the sparse dictionary method mentioned in the traditional method, it also includes methods such as spatial transformation. Spatial transformation uses mathematical methods such as matrix decomposition and optimization to obtain global or local structural features in the corresponding lowdimensional space [5], which can effectively transform a sample of highdimensional data into lowdimensional features. This transformation has two advantages for target recognition: one is to remove the correlation between samples and increase the feature difference between samples of different categories; the other is to reduce the dimension of the input classifier by reducing the dimension of the sample features of the input classifier, which can reduce the computational load of the classification process.
The twodimensional bidirectional principal component analysis feature named (2D)^{2}PCA[6] and collaborative representationbased projections (CRP) feature [7] are two effective methods of data dimension reduction in spatial transformation. (2D)^{2}PCACRP is a twodimensional PCA that can analyse the principal component features of a sample set that is composed of twodimensional samples. Compared with PCA, (2D)^{2}PCA reduces the feature dimension and extracts more effective feature information. However, (2D)^{2}PCA only extracts global structural features of an image and ignores local structural information in the target that may improve recognition. CRP is an unsupervised spatial transformation feature. Similar to Fisherâ€™s criterion, CRP must find an optimal projection matrix to maximize the global separability of the sample based on the graph and minimize the local compactness. It is necessary to convert the SAR image containing redundant information into a onedimensional vector before extracting the CRP feature, which destroys the spatial structure of the SAR image. Therefore, the extracted CRP features may not be able to characterize the real target characteristics, resulting in a decrease in the performance of SAR target recognition.
To solve the problem of inefficient interpretation of massive highresolution SAR data, this study combines (2D)^{2}PCA and CRP to develop a spatial transformation feature called the (2D)^{2}PCACRP feature, which effectively reduces the dimension of target features and improves the recognition accuracy of targets using the intensity information of SAR images. First, (2D)^{2}PCA is used to project the image to the lowdimensional feature space and filter out the interference redundant information in the highresolution SAR image without destroying the spatial structure. Then, CRP is used to extract the global structural features and local structural features of the target in the highresolution SAR image.
2 Twodimensional bidirectional principal component collaborative projection feature
2.1 Twodimensional bidirectional principal component analysis
Principal component analysis (PCA) is a commonly used data analysis method that is often used for dimensionality reduction of highdimensional data and can be used to extract the primary feature components of the data. PCA is often used as a preprocessing method in computer vision and signal processing to reduce the dimension of big data. (2D)^{2}PCA is developed based on PCA. Compared with PCA, (2D)^{2}PCA does not need to vectorize the image matrix but directly processes it. Concurrently, (2D)^{2}PCA performs dimensionality reduction from the row and column directions to eliminate the correlation between rows and columns. Given a training sample set \({\mathbf{X}} \in {\mathbb{R}}^{m \times n \times N}\), where \({\mathbf{X}}\) is a threedimensional sample tensor composed of \(N\) images of size \(m \times n\), the specific steps of (2D)^{2}PCA feature extraction are as follows.
To perform row mapping of \({\mathbf{Y}}_{{\mathbf{1}}} {\mathbf{ = XA}}\), where \({\mathbf{A}}\) is a projection matrix composed of a set of orthogonal bases, \({\mathbf{Y}}_{{\mathbf{1}}}\) is the result of the original sample \({\mathbf{X}}\) mapped by \({\mathbf{A}}\), and the overall scatter matrix \({\mathbf{G}}_{t,r}\) corresponding to the row transformation is in (1):
To perform column mapping of \({\mathbf{Y}}_{{\mathbf{2}}} {\mathbf{ = BX}}\), where \({\mathbf{B}}\) is a projection matrix composed of a set of orthogonal bases, \({\mathbf{Y}}_{{\mathbf{2}}}\) is the result of the original sample \({\mathbf{X}}\) mapped by \({\mathbf{B}}\), and the overall scatter matrix \({\mathbf{G}}_{t,l}\) corresponding to the column transformation is shown in (2):
where \({\mathbf{X}}^{i}_{r}\) and \({\mathbf{X}}^{j}_{r}\) represent the ith row vector and the jth column vector of the rth sample, respectively; \(\overline{{{\mathbf{X}}^{i} }}\) and \(\overline{{{\mathbf{X}}^{j} }}\) represent the ith row vector and the jth column vector of the sample mean \({\overline{\mathbf{X}}}\), respectively. To obtain the optimal projection of the training sample \({\mathbf{X}} \in {\mathbb{R}}^{m \times n \times N}\), the eigenvalues and eigenvectors of the scatter matrices \({\mathbf{G}}_{t,r}\) and \({\mathbf{G}}_{t,l}\) are calculated, and the eigenvectors corresponding to the largest \(d_{t}\) and \(d_{r}\) eigenvalues of each are taken to form the projection matrices \({\mathbf{A}}\) and \({\mathbf{B}}\). Therefore, the original sample joint mapping form is shown in (3):
where \({\mathbf{Y}}_{{\mathbf{2}}}\) is the (2D)^{2}PCA feature of the original sample set after spatial projection dimensionality reduction; thus, the feature dimension of the sample project is \(d \in {\mathbb{R}}^{t \times r}\). This process retains the original sample data and spatial structure information as much as possible but also markedly reduces the dimension of the feature space.
2.2 Cooperative representationbased projection
CRP is an unsupervised discriminant projection method based on \({\mathbf{\ell }}_{2}\) norm regularized least squares. The core idea of CRP is to use all samples to represent the specified sample to calculate the edge weight of the specified sample and other samples through collaborative representation theory. In addition, when obtaining the projection matrix, the local information and global information of the sample are considered concurrently, and the local information is minimized. The optimal projection matrix is found by minimizing local compactness and maximizing total separability.
We assume that the training sample set is \({\mathbf{X}} = [{\mathbf{x}}_{{\mathbf{1}}} {\mathbf{,x}}_{{\mathbf{2}}} , \ldots ,{\mathbf{x}}_{{\mathbf{n}}} ] \in {\mathbb{R}}^{m \times n}\), where \({\mathbf{x}}_{{\mathbf{i}}}\) is a column vector composed of the mdimensional features of the ith sample. First, we construct an \({\mathbf{\ell }}_{2}\) graph according to collaborative representation theory, and the weight \(w_{i} \in {\mathbb{R}}^{n \times 1}\) of the sample \({\mathbf{x}}_{{\mathbf{i}}}\) reconstructed from the \({\mathbf{\ell }}_{2}\) graph is:
where \(\lambda\) is the regularization parameter, and \(w_{ij} (i \ne j)\) represents the contribution made by sample \({\mathbf{x}}_{j}\) when reconstructing sample \({\mathbf{x}}_{i}\). We let the derivative of Eq.Â (4) with respect to \(w_{ij}\) be 0, and it is easy to find its closed solution as follows:
According to the \({\mathbf{\ell }}_{2}\) diagram, CRP defines local compactness \({\mathbf{J}}_{L}\) and global separability \({\mathbf{J}}_{T}\) as:
where the local scatter matrix \(S_{L}\) and the global scatter matrix \(S_{T}\) S are, respectively, expressed as:
where \({\mathbf{I}}\) is identity matrix, \({\mathbf{W}}\) is the matrix of \(w_{ij}\).
To obtain the best projection matrix \({\mathbf{P}}\), it is necessary to solve two optimization problems of minimizing local compactness and maximizing total separability concurrently. Therefore, the final optimization function of CRP can be expressed as:
Thus, the projection matrix \({\mathbf{P}}\) is the first \(d\) largest nonzero eigenvalue corresponding to \(({\mathbf{S}}_{L} )^{  1} {\mathbf{S}}_{T} {\mathbf{P}} = \lambda {\mathbf{P}}\). The lowdimensional CRP feature of the final sample is:
2.3 (2D)^{2}PCACRP features
As mentioned in Sect.Â 2.1, (2D)^{2}PCA is a lowdimensional feature extraction method for twodimensional data. When it is used in highresolution SAR images, by compressing the twodimensional image row and column simultaneously, the redundant information in the SAR image can be removed without destroying the twodimensional matrix structure of the image, effectively retaining the image. The global scattering information of the target slice is typically removed from the information redundancy caused by noise or background, and the characteristic information related to the target area is retained. However, the disadvantage of (2D)^{2}PCA is that it only focuses on global features and ignores the local structure information in the target that is beneficial for recognition.
CRP is an unsupervised spatial transformation feature. The advantage of CRP is that it considers global and local features concurrently. Similar to Fisher's criterion, the process of CRP to obtain the best projection matrix is the process of simultaneously optimizing the global separability and local compactness based on the \({\mathbf{\ell }}_{2}\) graph. Therefore, the extracted CRP features can effectively represent the global and local features of the SAR image. However, when extracting CRP features, the twodimensional SAR image matrix containing redundant information must be stretched into a onedimensional vector, which destroys the spatial structure of the SAR image. After stretching, the redundant information is interlaced and mixed with the effective information, and the extracted CRP features can no longer characterize the effective features of the original SAR image target, which leads to the degradation of the SAR target recognition performance.
Therefore, we combine (2D)^{2}PCA and CRP to design (2D)^{2}PCACRP features to effectively reduce the target feature dimension and improve the SAR target recognition rate. First, we use (2D)^{2}PCA to project the highresolution SAR image into the lowdimensional feature space, remove the interference redundant information in the highresolution SAR image without destroying the spatial structure, and then use CRP to extract the redundant SAR image. With regard to the characteristic information of the target, the recognition of the target is finally achieved.
3 Methods
In this paper, the SAR vehicle target recognition algorithm flow based on the (2D)2PCACRP feature mentioned in Sect.Â 2.3 is shown in Fig.Â 1, which effectively reduces the dimension of target features and improves the recognition accuracy of targets in SAR images.
First, (2D)^{2}PCA is used to project the image to the lowdimensional feature space and filter out the interference redundant information in the highresolution SAR image without destroying the spatial structure. Then, CRP is used to extract the global structural features and local structural features of the target in the highresolution SAR image.
The steps of this method are as follows:
Step 1: (2D) ^{2}PCA feature matrix extraction. Considering that the original SAR image contains redundant information that interferes with target recognition, the original SAR image sample \({\mathbf{X}}\) is converted to a lower data dimension using the (2D)^{2}PCA (2D) 2PCA subspace before CRP feature extraction and using Eqs.Â (1)(3). We then denote the sample feature matrix after dimensionality reduction as \({\mathbf{X}}_{{(2D)^{2} PCA}}\).
Step 2: CRP feature projection matrix calculation. First, we establish the \({\mathbf{\ell }}_{2}\) graph about the sample space \({\mathbf{X}}_{{(2D)^{2} PCA}}\) according to Eqs.Â (4)(5). Then define the local compactness \({\mathbf{J}}_{L}\) and global separability \({\mathbf{J}}_{T}\) of the \({\mathbf{\ell }}_{2}\) graph according to Eqs.Â (6)(7). Finally the optimization problem of Eq.Â (10) obtains the characteristic projection matrix \({\mathbf{P}}\) of CRP by solving Eq.Â (10).
Step 3: (2D) 2PCACRP feature extraction. The CRP characteristics of training samples \({\mathbf{X}}_{{{\text{train}}}}\) and test samples \({\mathbf{X}}_{{{\text{test}}}}\) are expressed as \({\mathbf{X}}_{{{\text{train}}}} ^{\prime} = {\mathbf{P}}^{{\text{T}}} \cdot {\mathbf{X}}_{{{\text{train}},(2D)^{2} PCA}}\) and \({\mathbf{X}}_{{{\text{test}}}} ^{\prime} = {\mathbf{P}}^{{\text{T}}} \cdot {\mathbf{X}}_{{{\text{test}},(2D)^{2} PCA}}\).
Step 4: Nearest neighbour classification. The (2D) 2PCACRP features of samples \({\mathbf{X}}_{{{\text{train}}}} ^{\prime}\) and \({\mathbf{X}}_{{{\text{test}}}} ^{\prime}\) are sent to the nearest neighbour classifier, and the test sample is assigned a category label based on the Euclidean distance.
Finally, the complexity of (2D)^{2}PCACRP is analysed as following. Suppose there are \(M\) \(m\) by \(n\) training images, the number of projection vectors in PCA, 2DPCA and alternative 2DPCA is \(p\), \(d\) and \(q\). Then the compression ratios of PCA, 2DPCA, alternative 2DPCA and (2D)^{2}PCA are computed as \(Mmn/(Mp + mnp)\), \(Mmn/(Mmd + nd)\), \(Mmn/(Mmq + nq)\) and \(Mmn/(Mdq + nd + mq)\), respectively [38].
4 Results and discussion
Some experiments are performed to demonstrate the effectiveness of the proposed method in SAR vehicle target recognition under several condition and the results are discussed in this section.
The experimental data adopt the moving and stationary target acquisition and recognition named MSTAR published by Sandia Laboratory supported by DARPA. Whether at home or abroad, the research on SAR image target recognition is basically based on this dataset. All images in this dataset are highresolution spotlight sensor starlos imaging in the Xband that were created in the HH polarization working mode and have an image resolution of 0.3Â mâ€‰Ã—â€‰0.3Â m. The coverage of the imaging azimuth is 0Â°â€“360Â°. A target consists of one type of rocket launcher 2S1; four types of armoured vehicles, including BMP2, BRMD2, BTR60 and BTR70; one type of bulldozer D7; two types of tanks T72 and T62; and one type of air defence unit ZSU23/4. In addition, there are several types of targets with submodels and differences in specific model configurations. In addition, different types of target samples have different sizes, including 128â€‰Ã—â€‰128 pixels, 158â€‰Ã—â€‰158 pixels, 178â€‰Ã—â€‰178 pixels and 192â€‰Ã—â€‰192 pixels. Therefore, in the experiments, these samples must be preprocessed. For all types of SAR vehicle target samples, the centre interception or zero filling operation is performed, and the sample slice size is uniformly adjusted to 128â€‰Ã—â€‰128 pixels.
4.1 SOC samples
In this study, the training set and test set under the SOC conditions contain the MSTAR data of 10 types of targets with elevation angles of 17Â° and 15Â°. The comparison algorithms are CRP, PCA, (2D)^{2}PCA, (2D)^{2}PCA CRP and CAMCNN [39]. CAMCNN is a multiscale convolutional neural network (CNN) based on component analysis (CAMCNN) for synthetic aperture radar (SAR) automatic target recognition (ATR). The component information of a target is robust to the local variations of the target, which is not made the best of by traditional CNNbased methods. Specific experimental data settings are shown in Table 1. The training samples and test samples of each type include all submodels, which increases the difficulty of recognition compared to using only a single model. The confusion matrix obtained by the algorithm in this paper is shown in Table 2. The average recognition rate of the 10 types of targets is 95.63%. The average recognition rates obtained based on the other three spatial transformation features are shown in Table 3. The average recognition rates based on the PCA feature and CRP feature are 72.37% and 73.58%, respectively, which are similar. The (2D)^{2}PCA feature recognition rate is marginally higher but is far lower than the recognition accuracy based on (2D)^{2}PCACRP features because PCA and CRP both extract features based on the onedimensional form of the image and lose the spatial structure information that can characterize the detailed features of the target. However, with multitype target recognition, it is particularly important to accurately describe the detailed structural characteristics of the target. Therefore, the (2D)^{2}PCACRP feature has marked advantages when used for multiclass MSTAR vehicle target recognition under SOC conditions. It is worth noting that this method uses the multiscale CNN network structure and integrates the target scattering structure information, and its recognition accuracy is slightly higher than that of this method.
4.2 Type change samples
In specific classification tasks, similar targets typically appear in different models of variants or have different mounts installed. The primary structure of these variants is similar, but there are differences in specific details, such as whether a tank is equipped with fuel tanks, whether an antenna is deployed, whether an armoured vehicle target is equipped with artillery, etc. Therefore, the model recognition of the target brand of the vehicle is important. In this section, two sets of experiments are described. Experiment one considers four types of targets and a total of 8 specific models, and experiment two considers five submodels of T72 targets to complete two sets of SAR vehicle target model classification experiments.
4.3 Recognition of 4 types of targets
In this section, the training samples and test samples are 14 types of vehicles at elevation angles of 17Â° and 15Â°. As shown in Table 4, the BMP2 and T72 targets each contain three different variants. Pairwise methods have similar scattering performance on SAR images [4]. The average recognition rate based on the four spatial transformation features is shown in Table 6. The average recognition rate of the 8 submodels based on the four features from high to low is CRPâ€‰>â€‰(2D)^{2}PCACRPâ€‰>â€‰(2D)^{2}PCAâ€‰>â€‰PCA. The average recognition accuracy based on (2D)^{2}PCACRP features is 93.88%, and the performance is marginally lower than when SOC conditions exist. Specifically, from the confusion matrix obtained by the target recognition algorithm based on the (2D)^{2}PCACRP feature in Table 5, the primary reason for the decline in recognition rate is that the three submodels of the BMP2 target produce serious misclassifications. However, the BMP2 target will not be mistakenly classified into the other three types of targets and thus has almost no impact on the performance of the SOC's largetype target recognition experiment. In general, under the current experimental conditions, the (2D)^{2}PCACRP feature achieves good performance when recognizing the 14 types of SAR targets (Table 6).
4.4 T72 target type classification
To examine how the proposed method performs with specific model changes in the T72 target, the experimental conditions are set as shown in Table 7. The training samples and test samples are the A32, A62, A63, A64 and SN_S7 submodels of the T72 target at elevation angles of 17Â° and 15Â°. The confusion matrix of the algorithm in this paper for the recognition results of the five submodels of the T72 target is shown in Table 8. The recognition rates of each type of submodel are 99.27%, 97.81%, 94.89%, 95.26% and 98.95%, and the average recognition rate is 97.13%. These results are markedly higher than the recognition results of the other three features in Table 9, particularly the two features based on onedimensional vectors, which is consistent with the previous theoretical analysis.
4.5 Change in elevation angle
Affected by SAR imaging characteristics, there are marked differences in the imaging of samples at different elevation angles, and the greater the elevation angle difference is, the smaller the similarity between samples. To verify the robustness of the proposed algorithm under the change of the elevation angle, this section sets up the target recognition experiment under the large elevation angle. The training set still uses the samples at 17Â°, and the test set changes from 15Â° to the samples at the larger 30Â° pitch angle. Target types include three types, 2S1, BRDM2 and ZSU23/4, and the specific sample size of each category is shown in Table 10. The confusion matrix of (2D)^{2}PCACRP features at large pitch angles is shown in Table 11. The recognition rates of the three types of targets are 99.31%, 99.65% and 98.61%. All have achieved good recognition results. Compared with the recognition rate of the other three features shown in Table 12, the (2D)^{2}PCACRP feature achieves a recognition rate of 99.19%, which is higher than the two features of PCA and (2D)^{2}PCA. However, the recognition rate of the CRP feature for the 2S1 and BRDM2 targets is only 16.67% and 11.5%, respectively, which is strongly affected by the change in the pitch angle. The experimental results thus show that compared with the original CRP feature, the improved (2D)^{2}PCACRP feature is insensitive to changes in the pitch angle.
5 Conclusion
To solve the problem of a large amount of highresolution image data and low interpretation efficiency, this study uses three types of PCA, (2D)^{2}PCA and CRP to effectively reduce the spatial transformation characteristics of the SAR image feature space. In addition, we analysed the proposed methodâ€™s performance in a highresolution SAR target recognition task. Then, (2D)^{2}PCA and CRP were combined to develop a spatial transformation feature that effectively reduces the feature space dimension and improves the SAR target recognition rate. First, (2D)^{2}PCA is used to project the image into a lowdimensional feature space. While considering the spatial structure, the interference redundant information in the highresolution SAR image was filtered out. Then, CRP was used to extract the spatial global separability feature and local structure feature of the target in the highresolution SAR image. Finally, based on the (2D)^{2}PCACRP feature, the nearest neighbour classifier was used to complete the target recognition task. Experiments were performed under four experimental conditions in the MSTAR dataset, including SOC, 8 types of target changes, T72 model changes and pitch angle changes. (2D) 2PCACRP features achieved average recognition rates of 95.63%, 93.89%, 97.13%, and 99.19%, which were markedly higher than those of the three features of PCA, (2D)^{2}PCA and CRP.
When extracting spatial transformation features, this method only removes the correlation and reduces the feature dimension of the current sample, and does not use the category information of the sample. Similar targets usually have similar target characteristics, while different targets have obvious differences in target characteristics. Therefore, the next research will use the sample category information to extract more effective spatial transformation features.
Availability of data and materials
Please contact author for data requests.
Abbreviations
 SAR:

Synthetic aperture radar
 PCA:

Principal component analysis
 CRP:

Collaborative representation projection
 SOC:

Standard operating condition
 EOC:

Extended operating condition
 CRC:

Collaborative representationbased classification
 CNN:

Convolutional neural network
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Acknowledgements
The authors would like to acknowledge the anonymous reviewers and editors of this paper for their valuable comments and suggestions.
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This study was supported by the Natural Science Foundation of Hunan Province, China under Project 2021JJ30780.
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TT and CZ designed algorithms, TT provided project support, CZ carried out code design and performance simulation, TT, CZ and XZ completed the first draft of writing, and TT and ZX completed the revision of papers. All of the authors have read and approved the final manuscript.
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Tang, T., Zhang, C. & Zhou, X. Twodimensional bidirectional principal component collaborative projection feature for SAR vehicle target recognition. EURASIP J. Adv. Signal Process. 2022, 91 (2022). https://doi.org/10.1186/s13634022009259
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DOI: https://doi.org/10.1186/s13634022009259
Keywords
 Synthetic aperture radar
 Target recognition
 Feature extraction
 Principal component analysis
 Collaborative projection