SAR target recognition based on improved joint sparse representation
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 87 (2014)
In this paper, a SAR target recognition method is proposed based on the improved joint sparse representation (IJSR) model. The IJSR model can effectively combine multiple-view SAR images from the same physical target to improve the recognition performance. The classification process contains two stages. Convex relaxation is used to obtain support sample candidates with the ℓ1-norm minimization in the first stage. The low-rank matrix recovery strategy is introduced to explore the final support samples and its corresponding sparse representation coefficient matrix in the second stage. Finally, with the minimal reconstruction residual strategy, we can make the SAR target classification. The experimental results on the MSTAR database show the recognition performance outperforms state-of-the-art methods, such as the joint sparse representation classification (JSRC) method and the sparse representation classification (SRC) method.
Synthetic aperture radar (SAR) is a high-resolution imaging radar. It can work regardless of climatic circumstances and time constraint. Thus, it is widely applied in kinds of military and civilian areas such as disaster assessment, resource exploration, and battlefield reconnaissance. SAR target recognition plays an important role in the automatic analysis and interpretation of the SAR image data. Over the past several decades, although lots of algorithms are exploited in SAR target recognition [1–3], it is a challenging issue due to the complexity of the measured information such as speckle noises, variation of azimuth, and poor visibility. Therefore, there is still no commonly agreed-upon system that settles SAR target recognition so far.
SAR target recognition includes two important parts, feature extraction and classifier construction. For feature extraction, classic methods, such as principal component analysis (PCA) , independent component analysis (ICA) , linear discriminant analysis (LDA) , nonnegative matrix factorization (NMF) [7, 8], and their improved algorithms , have been successfully used in SAR target recognition. Beyond those, in consideration of most features in nature distributing as a manifold structure, the manifold-based feature extraction algorithms become a new trend [10, 11]. Though kinds of feature extraction methods have their own advantages, no method extensively can be accepted. As for the classifier, support vector machine (SVM) and K-nearest neighbor (KNN) are the most common choices. To improve the performance of SAR target recognition, the classifying results under different features are fused to make the final classifier . In addition, sparse representation which closely bonds the feature extraction with the classifier has gradually aroused researchers' attention. Some advantages of sparse representation for recognition are mentioned in  such as its insensitivity to feature extraction method under certain conditions and the natural discriminative information in sparse representation coefficients, i.e., feature extraction is implicit in recognition and the classifier can be designed according to the sparse representation coefficients. The results for face recognition show the great competitiveness compared with other methods . Due to these advantages of sparse representation, Thiagarajan et al.  and Estabridis  both introduced sparse representation in target recognition. Thiagarajan et al. explained sparse representation from the point of manifold, which indicates the strength of sparse representation for SAR target recognition. They selected random projections as the feature extraction method and solved sparse representation using the greedy algorithm. Knee et al.  used image partitioning and sparse representation based feature to handle SAR target recognition.
The preceding methods only take one SAR image as the input signal to decide which class the target in the image belongs to. In practice, we can obtain multiple-view SAR images of the same physical target. Thus, some tried to make use of multiple-view SAR images under the theory framework of sparse representation. Exploring the sparse representation for the multiple input signals at the same time is a joint sparse representation problem [17, 18]. Therefore, Zhang et al.  used the joint sparse representation (JSR) model to seek common sparse patterns between multiple-view SAR images. In the JSR model, multiple-view SAR images are integrated in a matrix form. Under this context, the JSR model finally becomes a mixed-norm problem. An efficient and accurate greedy algorithm, CoSaMP [20, 21], is utilized to solve the model, and the classification algorithm is named as joint sparse representation classification (JSRC) which is similar with sparse representation classification (SRC).
With the inspiration from the JSR model, we propose an improved joint sparse representation (IJSR) model for SAR target recognition with multiple-view images. Compared with the original JSR model, there are two improvements in the IJSR model. The first is that sparse representation for the single-view image is described by a ℓ1-norm minimization model. The second is that common patterns in sparse representation coefficients of multiple-view images are sought by low-rank matrix recovery. The ℓ1-norm minimization model has two benefits for SAR target recognition. One benefit is that the proper sparse level parameter which is hard to choose in the original JSR model is not needed anymore. Another benefit is that sparse representation coefficients of the ℓ1-norm minimization are more concentrated in one class, which enhances the discrimination of sparse representation coefficients. Different from the greedy algorithm in the original JSR model, the ℓ1-norm minimization usually produces more nonzero entries in sparse representation coefficients of SAR target images. With the excessive nonzero entries, it becomes difficult to seek support samples which refer to the samples in the dictionary that associate with the nonzero entries in sparse representation coefficients. To tackle this problem, we further make some hypotheses that the matrix of joint sparse representation coefficients associating with support samples is low rank, and the rest that excludes the joint sparse representation coefficients associating with support samples is a sparse matrix. These hypotheses are based on the following reasons. According to the common sparse pattern assumption in the JSR model, these images with close views share the same support sample set. The common sparse pattern means important sparse representation coefficients which correspond to the support sample set have the same indexes in the dictionary and occupy the most nonzero entries in sparse representation coefficients. The problem of seeking the support samples converts to a low-rank matrix recovery problem; meanwhile, the low-rank matrix recovery algorithm could directly obtain the proper sparse representation coefficients on support samples.
The paper is organized as follows: In Section 2, we review the joint sparse representation model and describe the classification strategy. Section 3 analyzes the disadvantages of joint sparse representation and proposes the improved joint sparse representation model along with the classification strategy. In Section 4, we verify the proposed method with experiments on publicly available MSTAR database and compare with the classical SRC method and the original JSRC method.
2 Joint sparse representation for SAR target recognition
In the real scenario, the multiple-view SAR images from one same target can be captured, and those images are highly correlated. When a uniform dictionary is used for these multiple-view images' sparse representation, an implicit correlation in the sparse representation coefficients can emerge. The correlation is defined as the common patterns which specifically mean the same positions of the nonzero entries in the sparse representation coefficients in the work of Zhang et al. . The JSR model, which can combine the sparse representation coefficients of multiple-view images to extract the common patterns, is introduced in SAR target recognition.
2.1 Joint sparse representation model
Supposing each image under different views has been translated to a vector y j , given J views of the same physical target, the J sparse representation problems can be defined together as
where D is the dictionary which usually consists of the training sample vectors, x j is the sparse representation coefficient vector associating with the j th inputting image vector y j , and K is a preset parameter that controls the sparse level. Using the matrix notations , , and , the above model can be rewritten as
where ║ · ║ℱ represents the Frobenius norm which calculates the sum of squares of every entry in the matrix, and ║ · ║0 is the ℓ0-norm of the matrix, which is defined as the number of nonzero entries in the matrix. Since is decomposed to compute the column one by one, this model cannot embody the correlation information between the multiple-view SAR images. To combine the sparse representation coefficients under the multiple views, an assumption that the multiple views of the same physical target share a common pattern in their sparse representation coefficient vector with respect to the same dictionary is made. The common pattern means the indexes of atoms in the dictionary that participate in the linear reconstruction of the inputting SAR images are the same for multiple-view SAR images, though the coefficients corresponding to the same atom may be different for each view. Specifically, this assumption allows all the J observations sparsely represented by a same small set of atoms selected from the dictionary while weighted with different coefficient values. This can be achieved by solving an optimization problem with the ℓ0/ℓ2 mixed-norm regularization as
where the is the mixed-norm of the matrix X which is defined by two computing processes. Firstly, the ℓ2-norm is applied on each row of the matrix, and then the ℓ0-norm of the resulting vector is computed as the result of the mixed-norm. The K training samples corresponding to the nonzero entries in the resulting vector are the support samples whose class labels reflect the label of the testing SAR target in some sense. The number of support samples is usually far less than the total number of samples.
2.2 Joint sparse representation classification
The classification strategy for the JSR model is similar with the SRC model, and the minimal reconstruction residual/error criterion is used. The classification model is defined as
where c and are the class labels, is the recovery for Y with only the c th training samples involve in the reconstruction, and the operation δc(∙) is redefined as preserving the rows corresponding to class c in the matrix X and setting all others to be zeroes. The Frobenius norm indicates that the decision is based on the total reconstruction error of multiple views. This whole classification algorithm is named as JSRC, and greedy algorithm can solve this problem in an approximate sense. Since greedy algorithm is one way to solve sparse representation without any transformation for the original sparse representation model, we call it ℓ0-norm model/minimization in this paper.
3 Improved joint sparse representation
In the JSR model, the common pattern is sought on the ℓ0-norm minimization model whose performance depends on a proper choice of parameter K. According to the ℓ0-norm minimization, the mixed-norm strategy is used to explore the common patterns in sparse representation coefficients of multiple SAR images. However, the proper K is hard to determine. Therefore, in this section, we propose an improved joint sparse representation model which firstly replaces ℓ0-norm minimization with ℓ1-norm minimization to avoid the parameter K selection problem and then adopts the low-rank matrix recovery strategy to seek the common patterns based on the characteristics of the ℓ1-norm minimization solutions.
3.1 Improved joint sparse representation model
As Section 2.2 says, greedy algorithm is one way to solve sparse representation in the approximate sense. Another way, which has strong theoretical foundations, is convex relaxation. Under the theoretical framework of convex relaxation, the ℓ0-norm in the original sparse representation model is replaced with the ℓ1-norm, and then the original model is converted as a convex quadratic programming problem. This solving strategy is called the ℓ1-norm minimization in this paper. Zhang et al. did not discuss the possibility to use convex relaxation in the JSR model . So, we firstly explore the potentiality of the ℓ1-norm minimization through an elaborate experiment.
There is a key parameter K in the JSR model. It represents the sparse level of the inputting signals and needs to be set manually. However, no algorithm can predict K accurately, and K may be a variable even with a fixed number of views. Figure 1 gives a pictorial illustration for the JSR coefficient matrix with different parameter K. Dimensionality of each sparse representation coefficient vector is 10, and every entry in sparse representation coefficient vector is represented with one block. Colored blocks indicate nonzero entries and white blocks indicate zero entries. Let us assume that the first five blocks in each sparse representation vector correspond to the samples from one same class, and the rest correspond to the sample set of another class. To demonstrate the performance of the different parameter K, we suppose the SAR images are from the first class target in Figure 1. If a proper parameter K is set, all support samples in the JSR coefficient matrix will concentrate in the first class which is shown as . However, the perfect choice of the parameter K is very difficult in real situation. If a too small K is selected, a JSR coefficient matrix with less support samples would be obtained. is the JSR coefficient matrix with K = 2. Though support samples in are still from the first class, the reconstruction error becomes bigger with less support samples. In worse case, if SAR images from different classes are similar, the support samples will distribute on different classes. Under this context, the recognition becomes more difficult. If a too big K is chosen, the JSR coefficient matrix would contain more support samples as whose parameter K is 5. As Figure 1 shows, the support samples scatter on different classes, and it results in two close residuals that may classify the target to the second class. To avoid seeking the perfect K, ℓ0-norm minimization is replaced with ℓ1-norm minimization without setting the parameter K.
A more important motivation behind this replacement is that a more discriminative ability is shown with the ℓ1-norm minimization in SAR target recognition according to our experiments. In our experiments, two kinds of sparse representation coefficients of three samples from BMP2, T72, and BTR70, which are the class labels in the public database MSTAR, are shown in Figures 2, 3, and 4. One kind of sparse representation coefficients is obtained via ℓ0-norm minimization and another one is solved through ℓ1-norm minimization. To be fair, we firstly solve ℓ1-norm minimization, and then, according to the number of nonzero entries in the ℓ1-norm solution, we specify the number of nonzero entries which is defined as the parameter K in the ℓ0-norm solution. The dictionary is composed of 698 training samples. Each dictionary atom index in Figures 2, 3, and 4 is associated with one training sample. The first 233 training samples are from BMP2. The coefficients associated with BMP2 are presented in blue lines which are ended up with a blue circle mark. The training samples that have index from 234 to 465 belong to T72, and the corresponding coefficients are indicated as red lines with a red circle mark in their ends. The rest of the training samples, whose coefficients are described by green lines with the green circle mark in their tails, come from BTR70. Though some big coefficients exist in the ℓ0-norm solution, sparse representation coefficients scatter on different classes. Meanwhile, the coefficients of the ℓ1-norm solution almost concentrate on one class as well as the right class. Obviously, more concentrated coefficients reveal more discriminative information.
With the experimental results and the analysis, we adopt the ℓ1-norm minimization based algorithm to solve sparse representation coefficients for SAR image under each view. The ℓ1-norm minimization model can be expressed as
This model can be solved by computing the sparse representation coefficient vectors one by one as well. However, there are another two problems for the ℓ1-norm minimization model. First, the solution using (5) usually contains more nonzero items. In ideal case, we expect a few nonzero items in because this can give us a clear position indication of support samples. Second, the sparse representation coefficients of each inputting image with close azimuth are obtained independently. Therefore, the combination between multiple-view images is lacking, which makes the solution lost the jointing meaning.
Although the sparse representation coefficients from different views may be different in the coefficient distribution, they share most support samples. The sparse representation coefficients with J views can be combined as the matrix . Nonzero items associated with support samples in sparse representation coefficients occupy the majority. With this characteristics, we can consider that the matrix is composed of a joint sparse representation coefficient matrix S that is named as the signal matrix and a noise matrix N. Since the number of nonzero entries of S is expected to be smaller to improve the discriminative ability of the support samples and the positions of the nonzero entries in each column are expected to be the same, we suppose S is low rank. With regard to the noise matrix N, since the inputting images are highly correlated, it should have only a few nonzero items. Therefore, it can be considered as a sparse matrix. The goal is to solve S which really helps the recognition. Under this context, the problem is converted into a low-rank matrix recovery problem. The low-rank matrix recovery can be defined as
where rank(∙) stands for the rank of matrix and λ balances the rank of the signal matrix S and the ℓ0-norm of the noise matrix N. Since it is hard to find the solution of (6), some relaxations are made to simplify it. The operation rank(∙) is replaced with the nuclear norm ║ · ║* which computes the sum of singular values of a matrix, and the operation ║ · ║0 is substituted with operation ║ · ║1 which is defined by adding every absolute value of entries in the matrix. Then, (6) can be rewritten as (7). This becomes a robust principal component analysis problem .
Apparently, the rank of the signal matrix rank(S) in the JSR matrix, the number of view J, and the proper sparse level K have close relations, which affects the recognition performance in some sense. With consideration of the computation cost, the number of views should be limited in a proper range. Generally, J is far less than the dimensionality of the inputting sparse representation coefficient vector. Therefore, the maximal rank of the signal matrix is definitely no more than the number of views. When rank(S) < K, the nonzero entries with the same indexes are not enough to reveal real support samples. The support samples in this case tend to be the linear combinations of K real support samples, which also can make the right recognition. When rank(S) = K, the low-rank matrix S is very likely to attain the K real support samples which contains explicit classification information. This is the best situation for recognition. When rank(S) > K, the low-rank matrix S fails to find the support samples. As a result, small coefficients tend to appear on nonsupport sample to meet the low-rank condition, while most sparse representation coefficients solved by (5) will remain in the signal matrix S. The reconstruction to the multiple-view SAR images may become worse than the reconstruction by ℓ1-norm solutions in (5) as small coefficients' influence. However, the recognition is still right for most cases due to the sparse representation coefficients via (5) almost concentrating on one class.
According to the above analysis, the IJSR model can be described as two stages. The first stage is seeking the ℓ1-norm solutions for multiple-view SAR images via (5). The second stage is combining the ℓ1-norm solutions from the first stage to recover a low-rank matrix which can indicate the common patterns through (7). Different from the JSR model, in the first stage, the ℓ1-norm minimization in the IJSR model avoids choosing a proper sparse level which is hard to predict. In addition, the solution for the ℓ1-norm minimization contains more discriminative information. In the second stage, discarding the mixed-norm strategy in JSR, the problem of finding the support samples is converted into a low-rank matrix recovery problem.
3.2 Improved joint sparse representation classification
Similar with the classification strategy in SRC and JSRC, we classify a testing sample based on how well the new low-rank matrix associated with each class reproduces the testing sample under J views. is an operator that has the same meaning with in Section 2.2. Here, represents a new matrix whose nonzero entries are the entries in the matrix S associated with class c. Let , C is the number of class, and the sub-matrix S c stands for a matrix composed of rows in S associated with the c th class. Then, can be defined as . The given testing sample matrix under J views can be approximated as
Based on the approximation residuals on each class, we can make the classification by the minimum approximation residual criteria, which can be described as
The improved JSR classification (IJSRC) algorithm is summarized in Algorithm 1.
In this section, our experiments are implemented on the public database MSTAR. All SAR images in the MSTAR database are X-band with 0.3 m × 0.3 m resolution. Three kinds of targets with depression angle 17° are chosen as the training samples and seven categories with depression angle 15° as the testing samples. The depression angle, class, serial number, and sample size are listed in Table 1.
The database is firstly preprocessed as follows: The logarithm transformation is made to turn the multiplicative speckle to the additive noise. To reduce the disturbance from the background of SAR image, a 50 × 50 sub-image which mainly contains the SAR target is extracted in the center of the original SAR image. Then, PCA is used as the feature extraction algorithm for its convenience and effectiveness.
4.1 One important precondition
The JSR model presumed that inputting samples that are from the same class share the same patterns which means samples from the same class should be a linear combination of the same support sample set . However, it is known that the SAR images of the same physical target change a lot along with the azimuth variation. So, it cannot be ensured that the images with a huge azimuth variation still share the same pattern. Thus, it is worth pointing out that the inputting samples sharing one same pattern has one important precondition that all multiple-view images involved in the joint decision should be similar, i.e., the multiple-view images should have close azimuths. One verification experiment is conducted. Two groups of five inputting images that belong to BMP2-c21 with depression angle 15° are sparsely represented. One group of images has greatly different azimuths. Another group of images has close azimuths. The dictionary atoms belong to BMP2-c21 with depression angle 17°. There are 233 training samples (i.e., 233 dictionary atoms). For convenience, we use the greedy algorithm to select the support samples. The number of support samples is set as 5 in this experiment. The testing sample and corresponding support samples indexes are shown in Tables 2 to 3.
The testing samples have greatly different azimuths in Table 2 and have close azimuths in Table 3. As shown in Table 3, five samples with close azimuths apparently have a more similar support sample set. For the samples with greatly different azimuths, the common support samples cannot be found as example in Table 2. It is obvious that the right recognition cannot be made if we adopt testing samples in Table 2. Therefore, we expect more samples with a closer azimuth interval in practice. Fortunately, in real case, one can capture more SAR images of one physical target in a much smaller azimuth interval. In this paper, all experiments are performed under the condition that multiple-view images have close azimuths.
4.2 Experimental results and discussions
To demonstrate the performance, our proposed IJSRC algorithm is compared with the state-of-the-art methods, such as SRC  and JSRC . Since the SRC algorithm is applied for the single image, the comparison experiment is implemented by concatenating the images under J views as a vector to form the final inputting vector in SRC. Then, the multiple-view sparse representation could be regarded as a single-view problem and solved by SRC. The SLEP toolkit  is applied for seeking the ℓ1-norm solutions in IJSRC. Considering the efficiency as well as the accuracy, we still use CoSaMP greedy algorithm in JSRC as  does.The first experiment is implemented to show the recognition performance of SRC, JSRC, and IJSRC with different feature dimensionalities. The results are shown in Figure 5. IJSRC outperforms SRC and JSRC when the dimension is less than 160. IJSRC achieves maximum recognition rate 98.535% with dimension 48. The maximum recognition rates for JSRC and SRC are 98.022% and 97.363%. That is to say, IJSRC performs better in low dimension. This result can well fit the practical requirement that SAR target recognition systems hope a better recognition result with a lower feature dimension. However, the recognition rate of IJSRC decreases with increasing of dimension, especially when the dimension exceeds 160. There is one reason in our opinion. Since the noises exist both in training samples and testing samples, the noises become strong with the increasing of the feature dimensionality, which can reduce the relevance of features for the multiple-view SAR images. Therefore, an improper low-rank matrix is generated by IJSRC, which leads to a bad recognition.
In the second experiment, we compare the recognition performance of SRC, JSRC, and IJSRC with different number of views. Figure 6 shows the recognition results with the dimensionality fixed as 64. Three approaches all present the ascending trend along with the increase of the number of view. Recognition rate of IJSRC grows faster than one of JSRC and SRC. As for the best recognition performance, the maximal recognition rate of JSRC is 98.75% and the maximal recognition rate for SRC is 99.927%, both with the number of view as high as 15. In comparison, IJSRC reaches 100% when the number of view reaches J ≥ 10.
Since IJSRC is improved on the foundation of JSRC, the third experiment is carried to exhibit the improvement of IJSRC through reconstructing the feature matrix of testing samples. Figures 7, 8 and 9 give the reconstruction errors of three examples from three classes by using the IJSR model and the JSR model, respectively. Three black bars in each subplot denote the reconstruction errors on three classes. As the reconstruction errors shown in Figure 7, JSRC gives a wrong prediction while IJSRC makes a right decision according to the minimum approximation residual criteria. For the class BMP2, the reconstruction error using the JSR model is maximal, which is the worst case in recognition. Figure 8 shows the situation that the JSR model infers a wrong result while the IJSR model obtains the right predication of the class label with a slightly smaller reconstruction error on T72 than on BMP2. In Figure 9, both the JSR model and the IJSR model can make the right prediction. However, the IJSR model slightly outperforms the JSR model with a smaller reconstruction error. Actually, the recognition rate of both the JSR model and the IJSR model can reach 100% on the class BTR70. Though we sometimes find that the reconstruction error on the right class in the JSR model is slightly smaller than the reconstruction error on the right class in the IJSR model, this situation tends to happen when the reconstruction errors on the right class are both remarkably smaller than the reconstruction errors on the wrong class. That is to say, though the IJSRC algorithm may have poor reconstruction to the inputting SAR images, the right recognition result is still guaranteed. This phenomenon fits the analysis with regard to rank(S) > K in Section 3.2. In most cases, the reconstruction via IJSRC outperforms the reconstruction with JSRC according to our experiment. Therefore, the IJSR model outperforms the JSR model generally.
An IJSR model for SAR target recognition under multiple views is proposed in this paper. In the IJSR model, the ℓ0-norm minimization is replaced by the ℓ1-norm minimization to solve the sparse representation of single-view SAR image, which can overcome the problem of choosing the proper sparse level and concentrates sparse representation coefficients in one class. Moreover, the low-rank matrix recovery strategy is proposed to seek the common support samples for SAR target recognition under multiple views. Experiments on the MSTAR database show that our algorithm outperforms JSRC and SRC in a low-dimensional feature space. With the increase of the number of view, the recognition rates of IJSRC increase faster and reach a higher point than those of JSRC and SRC. In conclusion, IJSRC generally outperforms JSRC and SRC.
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This research was supported by the National Natural Science Foundation of China (61201271, 61301269), the Fundamental Research Funds for the Central Universities (ZYGX2013J019, ZYGX2013J017), and the Sichuan Science and Technology Support Program (cooperated with the Chinese Academy of Sciences) (2012JZ001).
The authors declare that they have no competing interests.
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Cheng, J., Li, L., Li, H. et al. SAR target recognition based on improved joint sparse representation. EURASIP J. Adv. Signal Process. 2014, 87 (2014). https://doi.org/10.1186/1687-6180-2014-87