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Research Article | Open Access
A Novel Semi-Supervised Learning Method Based on Fast Search and Density Peaks
Radar image recognition is a hotspot in the field of remote sensing. Under the condition of sufficiently labeled samples, recognition algorithms can achieve good classification results. However, labeled samples are scarce and costly to obtain. Our major interest in this paper is how to use these unlabeled samples to improve the performance of a recognition algorithm in the case of limited labeled samples. This is a semi-supervised learning problem. However, unlike the existing semi-supervised learning methods, we do not use unlabeled samples directly and, instead, look for safe and reliable unlabeled samples before using them. In this paper, two new semi-supervised learning methods are proposed: a semi-supervised learning method based on fast search and density peaks (S2DP) and an iterative S2DP method (IS2DP). When the labeled samples satisfy a certain requirement, S2DP uses fast search and a density peak clustering method to detect reliable unlabeled samples based on the weighted kernel Fisher discriminant analysis (WKFDA). Then, a labeling method based on clustering information (LCI) is designed to label the unlabeled samples. When the labeled samples are insufficient, IS2DP is used to iteratively search for reliable unlabeled samples for semi-supervision. Then, these samples are added to the labeled samples to improve the recognition performance of S2DP. In the experiments, real radar images are used to verify the performance of our proposed algorithm in dealing with the scarcity of the labeled samples. In addition, our algorithm is compared against several semi-supervised deep learning methods with similar structures. Experimental results demonstrate that the proposed algorithm has better stability than these methods.
Radar image recognition is a popular research area in the field of remote sensing [1–3]. With the development of imaging technologies and the expansion of radar image data, the requirement of real-time and accuracy of data processing becomes higher and higher. Under the condition where the number of the labeled samples is sufficient, a recognition algorithm can generally achieve satisfactory classification results with a strong sample representation ability [2, 4]. However, the labeled radar images are scarce compared to the case of optical images, and the cost of labeling is also very expensive. They can usually be interpreted by an experienced expert [5, 6]. Therefore, it is unrealistic to obtain a large number of labeled samples by manual annotation.
This paper focuses on how to use these unlabeled samples to improve the performance of a recognition algorithm in the case of limited labeled samples. This is a semi-supervised learning problem. Currently, semi-supervised deep learning achieves promising recognition performance, such as Ladder Network  and Temporal Ensembling . However, unlike those existing semi-supervised learning methods, we do not use unlabeled samples directly and, instead, look for safe and reliable unlabeled samples and then use these unlabeled samples to enhance the performance of the recognition algorithm. This is because the unlabeled radar images need to go through the detection stage in the process of acquisition [9, 10]. These samples may deteriorate the semi-supervised algorithms’ learning, especially when the number of the labeled samples and that of the unlabeled samples are somehow unbalanced. This will influence the performance of the semi-supervised algorithm. The negative effects of these unreliable and unlabeled samples on semi-supervised algorithms are analyzed comprehensively in [11, 12]. Therefore, it is very important for a semi-supervised algorithm to identify reliable unlabeled samples before we learn unlabeled samples’ features.
Effective use of unlabeled samples is a new and interesting topic for semi-supervised methods. These emerging semi-supervised methods are mainly divided into two categories: semi-supervision based on integrated resources and safe semi-supervision based on weights. Semi-supervised methods, based on integration resources, usually combine multiple semi-supervised models, comprehensively analyse the predictions of unlabeled samples, and choose reliable unlabeled samples to improve the recognition performance of the system. For example, Li et al.  proposed the S3VM-us method, which consists of a semi-supervised support vector machine (S3VM)  and a standard support vector machine (SVM) . The confidence of unlabeled samples is determined by both classifiers. If the evaluation results are consistent, the unlabeled samples are identified. Li et al.  also proposed a safe S3VM method (S4VM). We understand that the S3VM is based on the low-density hypothesis in order to detect a significant interval along the low-density boundary from the feature space to identify unlabeled samples. Unlike S3VM, S4VM was based on the fact that there may be more than one low-density boundary in the feature space. This approach considers all the possible situations, equivalently, integrating multiple S3VMs to pinpoint reliable unlabeled samples. Wang et al.  proposed a safety-aware semi-supervised method. It consists of a semi-supervised model and a supervised model, which minimized the square loss between the two models in order to detect reliable unlabeled samples. Similar to , Gan et al.  proposed a safe semi-supervised method which added a Laplace regularization term to the square loss function to enhance the reliability of unlabeled sample selection. Persello et al.  proposed a progressive S3VM with diversity (PS3VM-D) method. On the basis of multiple confidence measurements, reliable unlabeled samples were obtained by querying the samples nearby the margin band.
Weight-based semi-supervisory is based on the fact that the more unlabeled samples with similar weights to the labeled samples, the more reliable the system becomes. Therefore, the influence of unreliable unlabeled samples on the algorithmic performance is suppressed by reducing their weights. For example,  considered the unlabeled samples nearby the classification plane and suppressed their influence on the system performance by reducing their weights. In addition, [21–23] controlled the weights by density estimation, weighted likelihood maximization, and graph modelling.
The above semi-supervised methods use unlabeled samples to some extent, however, they also ignore the number of the labeled samples. If the labeled samples are too few, the performance of these algorithms is difficult to be guaranteed, which will inevitably affect the evaluation of the reliability of unlabeled samples. In addition, they lack investigating variability and similarity between unlabeled and labeled samples, which makes it difficult to understand the dynamics and interaction of unlabeled samples. Therefore, in this paper, two new semi-supervised learning methods are proposed: a semi-supervised learning method based on fast search and density peaks (S2DP) and an iterative S2DP method (IS2DP).
When the labeled samples satisfy a certain number, S2DP is used directly to identify reliable unlabeled samples. For one thing, it works with a new sample weighted kernel Fisher discriminant analysis (WKFDA) supervision method. Using the difference between the samples, the WKFDA method extracts the features of the labeled samples to help formulate the distribution of the unlabeled samples’ features, solving the problem of mismatch between them. And for another, it is combined with a clustering method: fast search and determination of density peaks (DP) proposed by Rodriguez and Laio in 2014 . Then, unlabeled sample features are further investigated so that the reliable unlabeled sample features are identified. Finally, an unlabeled sample labeling method based on clustering information (LCI) is designed to retrieve the labels of the unlabeled sample features.
When the labeled samples are insufficient, IS2DP is used to iteratively render reliable unlabeled samples. Since the labeled and the unlabeled samples may be uneven in numbers, the unreliable unlabeled samples tend to deteriorate the semi-supervised algorithm. The IS2DP first divides the unlabeled learning set into different subsets according to the size of the labeled sample set. This not only prevents the deterioration of the semi-supervised algorithm by a large number of unreliable samples but also speeds up the processing of the semi-supervised algorithm. Then, the S3VM is exploited to go through the semi-supervised samples which may be away from the hyperplane of the S3VM as the reliable semi-supervised samples are added to the labeled samples to improve the performance of the semi-supervised algorithm.
The rest of this paper is organized as follows. Section 2 gives a brief review of the approaches involved. Section 3 describes the proposed method in detail. Section 4 presents the experiments for the SAR images targets recognition. The conclusion is drawn in Section 5.
2.1. DP Algorithm
Clustering by fast search and detection of density peaks (DP) can quickly realize accurate detection and clustering of various shapes. Moreover, it is used to evaluate each cluster membership so as to determine reliable cluster members. The DP algorithm is mainly divided into the following three steps.
(1) Determination of Cluster Centers. In the DP, it is assumed that the cluster centers are surrounded by the neighbors with the lower local density and they are at a relatively large distance from any points with a higher local density. Based on the above cluster center assumption, for each sample , two quantities are calculated: the local density of the sample and the distance from a sample to the other with a high local density. In the decision map with and as the horizontal and vertical coordinates, respectively, their product iswhere the sample point with the larger is more likely to be the cluster center. Therefore, only is sorted in a descending order, and several corresponding samples are selected as the clustering center from the largest value.
(2) Clustering of Samples. After the clustering center has been determined, all the samples are assigned to be the nearest cluster centers. Compared with the other clustering algorithms, DP clustering process is simple and does not require iterative optimization of the loss function.
(3) Automated Evaluation of Cluster Members. In the clustering results, it is important to quantitatively evaluate the credibility of each sample cluster. The DP algorithm has this capability, compared to other clustering algorithms. It firstly defines a neighbourhood for each cluster. Then, the maximum value of the local density of the samples is found in each neighbourhood. Finally, in each cluster, all the samples with local density greater than are considered as the cluster core candidates, otherwise, they are considered as the cluster halo of the cluster. The samples in the cluster core are very similar to the central samples and belong to reliable samples. The samples in the cluster halo have a certain distance from the central sample, which is very likely to be noise and belongs to unreliable samples. In addition, there are some cross-clustering and isolated samples that are also unreliable.
In summary, after having clustered by the DP, the samples located at the cluster core are considered to be reliable cluster samples, whilst the others are unreliable samples. Compared to the conventional clustering algorithms, such as Clara  or Fanny , the DP has lower computational complexity and less computational time. It also well characterizes the distribution of the samples and achieves more accurate clustering results. Besides, the reliability of the clustering results can be provided, which makes the DP easy to be interactive with other algorithms. However, only considering the distance between the sample points can insufficiently characterize the data because it cannot accurately describe the samples with small difference between two categories. When the sample dimension is high, the distance matrix is large, which can reduce the efficiency of the algorithm. Therefore, choosing the appropriate feature extraction method is a key in the DP.
The S3VM is the extension of the support vector machine (SVM). A standard SVM is based on the structural risk minimization to classify the learning set by extracting the support vectors from the training set to find the optimal hyperplane. In case of the binary SVM, given the training set and the testing set , we have the following constriction optimization problem:where is the training sample and is the corresponding label, (,) ; () maps the data into the feature space; is the orthogonal vector between and the hyperplane; is the bias to measure the distance between L and the hyperplane; is the slack variable to represent the offset of ; is the cost factor to measure the weight between the optimal hyperplane and the minimum offset; is the number of the training samples.
For the S3VM, the iterative process is operated and the semi-labeled samples (selected from U in the previous step) are added to L. Their confidence is diverse in different iterative steps and they are given different cost factors, leading to the following function:where is the semi-labeled sample selected from U, with the slack variable (), cost factor () and semi-label ( ) and is the number of the semi-labeled samples.
The S3VM can deal with the nonlinear problem using the kernel methods and its semi-supervised samples with the bigger . But when the sample dimension is high, the computation speed would decrease. Therefore, the dimension reduction and effective semi-supervised samples are the critical aspects to the S3VM.
3. Proposed Methods
This paper presents two methods: S2DP and IS2DP. When the labeled samples exceeds a certain number, S2DP directly performs screening and classification of the reliable unlabeled samples. When the labeled sample is insufficient, IS2DP is used to continuously query reliable unlabeled samples and generate necessary samples to be added to the labeled samples in order to improve the recognition performance of S2DP. The S2DP and IS2DP are described below, respectively.
Figure 1 shows the flowchart of the proposed S2DP. First, we use WKFDA to extract the labeled sample features to build a new space. New features are obtained by projecting unlabeled samples into this new space. In this space, the new feature distributions are as close as possible between the intraclass features with a certain weight, and the interclass features are as far apart as possible to enhance the separability between the features. Secondly, the DP is used to cluster the generated features. Finally, the unlabeled samples are identified by the labeling method based on the DP clustering information (LCI). In Figure 1,L represents a set of the labeled samples, andU represents a set of the unlabeled samples, which respectively generate features with the labeled information (i.e., labeled features) and features without labeled information (i.e., unlabeled features) after going through WKFDA;C represents the clustering results of the DP. The WKFDA and LCI methods are described in the following section.
(1) WKFDA. Assume that represents all the samples of L and the th category is the subset of , where is the samples’ number of . is the sample weight vector of . It is used to control intraclass samples as close as possible with certain weights.
In case of the binary classification, it cannot simply multiply the weight by the corresponding sample. Firstly, the weight matrices and are generated:Secondly, the weight vector and the weight matrix are normalized using where represents the summation. The above weight matrix can be used to measure the information of the sample itself. Although and are made up of , their elements are different. The sum of each column’s elements of is equal to 1, and the trace of is equal to . Thirdly, the projection direction is calculated by Equation (6):where . is nonlinear mapping that maps the samples to a new feature space. In this new space, the sample’s mean, before and after the projection has been made, can be calculated by The interclass scatter matrix and intraclass scatter matrix , after the projection has been achieved, are calculated by where and . In order to satisfy the requirements of the maximum interclass interval and the minimum intraclass interval, this goal can be expressed as follows:which is called the generalized Rayleigh quotient. Then can be calculated according to the flowchart of the KFDA by solving the following optimization problem:where is the constant. By introducing the Lagrange multiplier, the function can be transformed to a Lagrange unconstrained extremum problem:Let , represent the partial derivative. This function solution is the eigenvector of . Once solving , for any sample , its projection iswhere is the kernel matrix of all the training samples and .
Adding weights to KFDA algorithm is a common way to improve the KFDA algorithm. The aim is to make the WKFDA algorithm better learn sample features. However, different ways of adding weights make the WKFDA algorithm focus on learning sample features differently. For example,  added weights to each kernel function. The purpose was to introduce the prior knowledge of samples to enhance the learning of sample features in the WKFDA algorithm. Reference  added weights to the within-class scatter matrix. The purpose was to make the WKFDA algorithm not only learn the features of different types of samples but also learn the features of same types of samples in the process of finding the best vector. Unlike these algorithms, the WKFDA algorithm in this paper adds weights to samples, and these weights can be calculated by using the similarity or iterative difference of the samples. The purpose is to make the intraclass samples close to a certain distance, so that the WKFDA algorithm can not only suppress overfitting due to the small number of labeled samples but also facilitate the absorption of spectral information of samples to improve the learning of sample features. Although the binary WKFDA is shown, the multi-WKFDA can be obtained in accordance with the promotion of the kernel Fisher discriminant analysis (KFDA) .
(2) LCI. After the labeled sample setL and the unlabeled sample setU have been extracted by the WKFDA method, the labeled and unlabeled features are obtained. Next, the labeled and unlabeled features go into the DP to produce a clustering resultC. The clustering resultC includes features such as cluster center, clustering core, clustering halo, and cross-clustering, but is insufficient to determine the labels of the unlabeled features. To solve this problem, we develop the LCI by using the clustering results and labeling information of the labeled features. LCI is able to label the clustering results of the unlabeled features. Because the unlabeled features are generated from the unlabeled samples, the unlabeled features and the unlabeled samples share the same labels. The basic flowchart of the LCI is shown in Figure 2.
We know that the features of clustering halo and cross-clustering are unreliable. Therefore, in Figure 2, the interference features inC need to be cleared to ensure that the subsequent unlabeled features are reliable. TheC clearing the interference is processed separately according to whether the labeled features are included in the cluster core. If there are labeled features in a certain cluster core, the unlabeled features of the cluster core are very similar to the labeled features. These unlabeled features are regarded as the best learning features, combined with the corresponding labeled features, for training the S3VM. At this time, in each iteration of the S3VM, the labeled features from the unlabeled features are added to the next iteration to improve the robustness of the S3VM algorithm. For the clustering cores which do not contain any labeled feature, the cluster centers are extracted and sent to the trained S3VM to obtain their labels. Once the unlabeled cluster centers are labeled, the unlabeled features of the corresponding clustering core will be assigned the label. In this way, all the clustering cores’ features are labeled, and the features that are not labeled are removed as noise. Finally, the unlabeled samples corresponding to the unlabeled features also have corresponding labels.
When the number of the labeled samples is small but reaches a certain amount, the S2DP uses the labeled features to investigate the distribution of the unlabeled features and also use the labeled features and the clustering result of the DP to obtain reliable unlabeled samples. However, when the number of the labeled samples is small, after the DP clustering has been achieved, the labeled features are not necessary in the cluster core, resulting in a low correlation between the labeled and unlabeled features. At this time, the labelled samples are difficult to represent the unlabeled samples, and S2DP is no longer applicable. In this case, the common solution is that the semi-labeled sample from the unlabeled set is queried in order to increase the number of the original labeled samples. In order to obtain the reliable semi-labeled samples, the S2DP needs to be modified iteratively.
The iterative semi-supervised method of the S2DP, namely, the IS2DP, is shown in Figure 3, whereU is the unlabeled learning set to query the semi-labeled samples,L is the labeled training set, is the semi-labeled samples set in each iteration, represents the final labeled training set, andT is the testing set.
The IS2DP specific process is described as follows. Firstly, in each iteration,U is randomly divided into several subsets (), which are combined withL to obtain (), (), …, () as the input of the S2DP. Secondly, the cluster cores are selected from () after the S2DP as the candidate semi-labeled samples, and their cluster centers are added to the training set as the labeled samples to train S3VM. For one thing, the number of the labeled samples sets is increased. And for another, it ensures that the labeled samples match the unlabeled samples since the cluster center represents the features of all the other samples in the cluster core. Therefore, the robustness of the S3VM is ensured. Thirdly, the semi-supervised sample of each iteration is obtained by S3VM. Finally, it needs to determine whether or not the iteration’s termination condition is met so that the number of the iteration is greater than the threshold. If not,L is updated,U is reduced, and the iteration process continues. Otherwise, the final labeled training is undertaken to classify the testing setT by the S2DP.
When the labeled sample is insufficient and necessary to query the semi-supervised samples, the IS2DP can query the reliable semi-supervised samples and classify the unlabeled samples. In fact, the IS2DP is equivalent to the S2DP when the labeled samples reach a certain number.
Our experiments use the SAR images from the Moving and Stationary Target Acquisition and Recognition (MSTAR) database, cofounded by National Defense Research Planning Bureau and the US Air Force Research Laboratory. The military targets contained in the database are collected at 15° and 17° depression angles, covering 360° azimuth angles. To display the intermediate experimental results in geometric space and highlight the significance and effectiveness of our method, the experiments in this paper use three types of military targets and one type of interference targets, which are T72, BMP2, BTR70, and SLICY. Of course, you can also choose other targets. Among these three types of military targets, BMP2 and T72 also contain different version variants. These variants have the same design blueprint, but from different manufacturers, they are slightly different in color and shape.
The optical images of the T72, BMP2, BTR70, and SLICY targets and the corresponding SAR images are shown in Figure 4. From optical images, the difference between these four types of targets is significant. However, the corresponding SAR images are difficult to distinguish by human vision due to speckle noise and similar spatial and spectral characteristics. The original resolution of these SAR image slices are 128128 and 4545. To facilitate the processing, we only take the 3232 resolution that contains the target and flatten these 2D images into one dimension. In order to show the separability of these data, we perform covariance operations on them in order to establish correlations between two-dimensional features. Figure 5 shows the correlation and box plot of the first 5-dimensional features. Figure 5(a) is the correlation of two dimension features, and Figure 5(b) is the corresponding box plot. In Figure 5(a), the lower left corner part is the scatter plot of two-dimensional features, and the upper right part is the correlation coefficient corresponding to the two-dimensional features. represents the total correlation coefficient of the relevant two-dimensional features. Positive numbers indicate positive correlations and negative numbers indicate negative correlations. The greater the absolute value of these numbers, the more relevant the features of the corresponding two dimensions. From the correlation coefficient, ’s absolute value is small which shows that the correlation is low, indicating that they are independent of each other. From the scatter plot, we observe that they are very similar, which increases the difficulty of the recognition algorithm. In addition, from the box plot, there are abnormal points in the upper and lower bounds of the data. If these points are not removed in the learning set features, the performance of the algorithm will be affected.
In order to evaluate the performance of the proposed method, we design three sets of experiments: the evaluation experiment of effectiveness, the evaluation experiment of generalization ability, and the experiment compared with the semi-supervised deep learning method. Among them, the first set of the experiments will be carried out under standard operating conditions (SOC), the latter two sets under different extended operating conditions (EOC). The SOC mean that the testing and the training conditions are very similar. For example, the target types of the training, the learning, and the test sets are the same. On the basis of SOC, the gap between the training and testing conditions is gradually extended to form different EOC. For example, the target types of training set, learning set, and test set are different variants. Even the learning set contains other interfering targets. Compared with SOC, EOC significantly increases the recognition difficulty of the algorithm. We will set up one SOC and two EOCs (EOC_1 and EOC_2) to carry out the above three sets of experiments. The specific configuration of these conditions is as follows.
(1) Data Configuration of the SOC. Table 1 shows the data configuration of SOC. It contains two sets of data: data and test sets. The data set is used for the algorithm training. According to the label of the samples, the data set is divided into the labeled and unlabeled samples. The labeled samples, also known as the training set, have a number ranging from 3 to 40 per class. The unlabeled samples, also known as the learning set, have 190 samples per class. Regardless of the training or learning set, their target depression angle is 17°. The testing set is used for algorithm testing. Its target depression angle is 15°. Regardless of data or test set, we use the same variants of the targets, that is, T72 series sn_132 tanks, BMP2 series sn_c21 armored vehicles, and BTR70 series sn_c71 armored vehicles. We will verify the effectiveness of the S2DP and IS2DP under these conditions in Section 4.1, including their core components (LCI and WKFDA).
(2) Data Configuration of the EOC_1. Table 2 shows the data configuration of EOC_1. In Table 2, the training set is the same as that of Table 1. And the testing set is not the same version variants as the training set and the learning set. For example, the T72 is the sn_s7 version in the test, but it is the sn_132 and sn_812 versions in the training and the learning sets, respectively. These conditions will help increase the recognition difficulty of the algorithm. Other conditions shown in Table 2, such as the number of data sets, the depression angle of data sets, and the depression angle of the test set, are the same as those shown in Table 1 and are not described here. We will verify the generalization ability of the S2DP and IS2DP under these conditions presented in Section 4.2.
(3) Data Configuration of the EOC_2. Table 3 shows the data configuration of EOC_2. It is formed by adding the interference target SLICY to the learning set of Table 2, further increasing the recognition difficulty of the algorithm. To highlight the advantages of the proposed algorithm, we will compare the S2DP based IS2DP algorithm with the semi-supervised depth learning method under EOC_2 in Section 4.3.
4.1. Effectiveness Evaluation Experiment
4.1.1. The Effectiveness of the WKFDA Feature Extraction
To verify the effectiveness of the WKFDA feature extraction, it is compared with the KFDA, kernel local linear discriminant analysis (KLFDA) , semi-supervised KLFDA (Semi-KLFDA)  and kernel principal component analysis (KPCA) . After these algorithms have extracted features, they all use the standard SVM as the final classifier. The experimental data configuration is shown in Table 1, and with the change of the number of the labeled samples, the overall accuracy rates (OA) of different methods are obtained, as shown in Figure 6. The horizontal axis represents the number of each type of target labeled samples corresponding to different experiments and the vertical axis represents the overall accuracy rate.
In Figure 6, the classification accuracy difference between the different algorithms is very clear. The WKFDA and KFDA both show higher accuracy, followed by the KLFDA and Semi-KLFDA, and finally KPCA. For the WKFDA and KFDA, when the number of the labeled samples is less than 24, the WKFDA’s classification results are better than the KFDA. When the number of the labeled samples is greater than 24, their classification results are almost the same. It shows that KFDA and WKFDA have good feature extraction capabilities, while the WKFDA is suitable for dealing with a small quantities of labeled samples. For the KFDA and Semi-KLFDA, when the number of the labeled samples is less than 20, the Semi-KLFDA’s classification results are better than the KFDA. When the number of labeled samples is greater than 20, their classification results are almost the same. For the KPCA, as the number of the labeled samples increases, its classification results are always poor.
In order to understand the above experimental results, we take a close look at the projections of the learning samples under the condition that the same number of the labeled samples is taken. Figures 7(a), 7(b), 7(c), 7(d), and 7(e) shows the projection of the learning set for KPCA, KLFDA, Semi-KLFDA, KFDA, and WKFDA algorithms respectively when the number of the labeled samples is 20. As can be seen from Figure 7, the projection result shown in (e) is the best, where we can classify the three targets, second best is (d) and then (c), (b), and (a). The quality of the projection results mainly depends on whether or not the feature extraction algorithm can effectively extract features from the SAR images.
For the KPCA algorithm, it only reduces the original features of the SAR images. As the number of the labeled samples increases, the classification accuracy of the KPCA features continues to increase. The original features of the SAR images are difficult to identify. Therefore, the classification accuracy of SVM based on the KPCA features is poor, shown in Figure 7(a).
For the KLFDA and Semi-KLFDA algorithms, they take advantage of the difference between the sample classes and extract features that are easily identifiable from the SAR images to certain extent. Therefore, their projection looks better than the KPCA algorithm. However, in the case where the overall features are not separable, the KLFDA and Semi-KLFDA algorithms overemphasize the local features, resulting in more confusing clutters in the projection space. This is observed from Figures 7(b) and 7(c).
For the KFDA and WKFDA algorithms, both of them well use the difference between different classes and the similarities in the same classes. Therefore, the projections shown in Figures 7(d) and 7(e) are better than those of the other methods. We know that the KFDA and WKFDA algorithms are supervised algorithms which guide the projection of the unlabeled sample features based on the features of the labeled samples. Therefore, whether or not these algorithms are good at learning the labeled sample features will affect the quality of the projection of the unlabeled sample features. In the process of the labeled sample feature learning, the KFDA algorithm forces the interclass samples to be as far apart as possible in addition to forcing the samples intraclasses to be as close as possible. At the same time, it may cause the algorithm to overfit and is difficult to guide the unlabeled sample features to be projected onto the optimal direction. The WKFDA algorithm is able to give the samples different weights so that the intraclass samples are close to each other with a certain weight. This can balance the concentration characteristics of the samples (the intraclass samples aggregate with each other and have a certain spatial structure) and can fully utilise the spectral information of the samples and reduces the algorithm’s overfitting. Therefore, the results shown in Figure 7(e) seem better than those of Figure 7(d).
To further explore the impact of weighting on the WKFDA algorithm, taking the same labeled samples, Figure 8 shows the results of the KFDA and WKFDA algorithms for learning the features of the labeled samples. Figure 8(a) shows the KFDA features. Although the interclass distance is significant, the intraclass samples are concentrated, almost grouping to a point, which is easy to cause overfitting of the algorithm. Figure 8(b) shows the WKFDA features. Under the condition that the interclasses is separable, the intraclass distance is relatively large, which is easy to learn the sample information and suppress the overfitting of the algorithm. Here, Figure 8(b) also shows that different weights can result in different intraclass distance. Compared with the weight of 100, the intraclass sample space is larger when the weight is 50, and the interclasses can be well spaced, which makes it easier to learn sample information.
4.1.2. The Effectiveness of the LCI for Labeling Unlabeled Samples
To verify the effectiveness of the LCI for labeling unlabeled samples, using the WKFDA features of the Section 4.1.1 experiments, LCI is compared with the SVM and S3VM classifiers under the DP clustering conditions. Using Table 1 as the experimental data, the same number of the labeled samples is selected from each type of targets in the training set. With the change of the number of the labeled samples, the OA trend chart of three methods is obtained, as shown in Figure 9. The horizontal axis represents the number of each type of the labeled target samples corresponding to different experiments, and the vertical axis represents the overall accuracy rate. As can be seen from Figure 9, the accuracy of LCI and S3VM is better than SVM. With more and more labeled samples, the accuracy of LCI and S3VM is almost the same.
We know that the SVM, as a supervised learning method, requires a large number of labeled samples. Because the DP algorithm cannot provide enough labeled samples for the SVM, the SVM classification results are poor. For S3VM and LCI, as a semi-supervised method, when the DP clustering outcome is reliable, they collect enough labeled samples to improve the recognition performance. Figure 10 shows the clustering results of the DP on the learning set when the number of the labeled samples is 21. In Figure 10(a), red circle, green circle, and blue circle are the cluster centers selected by the DP. The DP algorithm recommends that the learning set be divided into 3 categories, which is consistent with the actual situation. Figure 10(b) shows the clustering results of the DP. are the cluster cores, are clustered halos, and are clustering errors. From Figure 10(b), the DP clustering has only minor errors and the result is quite accurate, further demonstrating that the recognition accuracy of the S3VM and LCI is equivalent. In addition, these errors are located in the cluster halos. In the LCI algorithm, the cluster halos and the cross-clustering samples will be deleted to ensure that the final labeled samples are reliable. Therefore, compared to the S3VM, the LCI recognition results are more consistent.
4.1.3. Verifying the Recognition Performance of
In order to verify the recognition performance of S2DP, S2DP is compared with its similar semi-supervised methods. These similar semi-supervised methods are the semi-supervised algorithms that replace DP in S2DP with other classical clustering algorithms: Clara  and Fanny , namely, semi-supervised Clara and semi-supervised Fanny. The experimental data configuration is shown in Table 1. With the changing numbers of the labeled samples, the OA trend chart of three methods is obtained, as shown in Figure 11. The horizontal axis represents the number of each type of labeled target samples corresponding to different experiments, and the vertical axis represents the overall accuracy rate.
By comparing the S2DP with the semi-supervised Clara and semi-supervised Fanny, the classification results of the different methods are greatly influenced by the number of the labeled samples. When the number of the labeled samples is less than 24, the overall accuracy of the three methods is continuously improved with the increase of the labeled samples. For the curve smoothness, the curve of the S2DP looks consistent over the curves of the other two methods. When the number of the labeled samples reaches 15, the recognition accuracy of S2DP is higher than that of the other two methods. When the number of the labeled samples reaches 24, the three methods have the same recognition accuracy and the curve trend is stable, but the S2DP is still better than the other two methods. Therefore, the S2DP is superior to the other two algorithms in terms of stability and classification accuracy.
When the labeled samples are very few, the DP clustering results of S2DP are too divergent to represent the unlabeled samples. Only few labeled samples are generated from the cluster core samples. In the end, the classification accuracy of the S2DP will not be high. As the number of the labeled samples increases, more and more labeled samples are generated by the cluster cores, which are also quite reliable. The S2DP classification accuracy is greatly improved. The other two methods are similar. However, as the number of the labeled samples increases, it is difficult for semi-supervised Clara and semi-supervised Fanny to guarantee the reliability of the labeled samples from the unlabeled samples during the clustering process. Therefore, their stability is not as good as that of S2DP. Figure 12 shows the three algorithms generate labeled samples from the learning set when the number of the labeled samples is 21. are clustering errors. Obviously, the labeled samples generated by the S2DP algorithm are more reliable than the other two methods.
The Sections 4.1.1–4.1.3 experimental results show the relationship between the number of the labeled samples and the S2DP, verifying the validity of the WKFDA, LCI, and DP as the key step in the S2DP. It shows that the S2DP, compared with the other two methods, can achieve the best classification result when the initial labeled samples reach a certain number. But when the labeled samples are too few, its classification precision decreases. Therefore, the ability of the modified IS2DP to query semi-supervised samples needs to be verified.
4.1.4. Verifying the in Ability to Query the Semi-Labeled Samples
When the labeled samples are few, the IS2DP can select the semi-labeled samples from the unlabeled samples as the labeled samples. In the Section 1, we know that PS3VM-D is also a semi-supervised method, which considers reliable incremental samples as semi-supervised samples by sample similarity. Therefore, PS3VM-D is selected as a comparative semi-supervised algorithm. They are all based on the extracted features by WKFDA. The experimental data configuration is shown in Table 1. With the change of the number of the labeled samples, the OA trend chart of two methods is obtained, as shown in Figure 13. The horizontal axis represents the number of each type of the labeled target samples corresponding to different experiments and the vertical axis represents the overall accuracy rate.
When the number of the labeled samples is less than 25, the classification accuracy of PS3VM-D is obviously lower than that of IS2DP, indicating that IS2DP is more suitable for the case of too few labeled samples. When the number of the labeled samples is more than 25, the IS2DP and PS3VM-D have the same accuracy. It shows that the PS3VM-D also gets enough labeled sample information, and the classification accuracy is improved.
We know that the core of PS3VM-D is SVM. The optimal classification surface of PS3VM-D is mainly influenced by SVM. The PS3VM-D relies heavily on the labeled samples. It needs enough quantity to obtain a universal classification surface. Therefore, its classification performance varies significantly with the number of labeled samples and cannot remain stable until the labeled samples are sufficient. The classification performance of the IS2DP is largely determined by the DP and WKFDA, which makes the IS2DP more stable and accurate when the labeled samples are very few due to the sample description ability of the DP and the effective use of the labeled samples by the WKFDA.
4.2. Evaluation of Generalization Ability
The following will verify the generalization capabilities of S2DP and IS2DP under the EOC_1.
4.2.1. Verifying the Generalization Capabilities
In Section 4.1.2, the comparison between LCI and S3VM algorithm is actually the comparison of S2DP with S3VM based on WKFDA and DP (WKFDA+DP+S3VM). The recognition accuracy of S2DP and WKFDA+DP+S3VM is equivalent in the SOC experiments. Here, we continue to compare the S2DP and WKFDA+DP+S3VM. The experimental data configuration is shown in Table 2. With the change of the number of labeled samples, the OA trend chart of two methods is obtained, as shown in Figure 14. The horizontal axis represents the number of each type of the labeled target samples corresponding to different experiments, and the vertical axis represents the overall accuracy rate.
In Figure 14, the recognition accuracy of S2DP and WKFDA+DP+S3VM algorithms increases with the increasing number of the labeled samples, and their final accuracy is equivalent. However, the curve of the S2DP is relatively smooth. This shows that our method is stable and robust.
To verify this conclusion, we perform visual analysis of the key steps of the two methods, when the number of samples is 21. Figure 15(a) shows the features of the training and learning sets after the WKFDA processing. represents the learning samples and represents the initial labeled sample. Figure 15(b) is the actual classification map of the WKFDA features after the DP clustering has been achieved. represents the clustering core and represents the clustering halo. Figure 15(c) is the true classification map of Figure 15(b). represents the clustering core, represents the clustering error sample, and represents the sample of the next step of the algorithm to be identified. As can be seen from Figure 15(a), the three types of targets are more confused at the boundary, which means that, in the future, they will affect the performance of the recognition algorithm if these samples are not cleared. As can be seen from Figure 15(b), the DP algorithm divides the WKFDA features into 5 clusters. Among these 5 clusters, clusters 1 and 3 have cluster haloes, and clusters 2, 4, and 5 are all clustered cores. As can be seen from Figure 15(c), the initial labeled samples ( samples) are not included in clusters 4 and 5 and, therefore, the samples of clusters 4 and 5 need to wait for the next step of the algorithm to identify and label. Clusters 1, 2, and 3 contain initial labeled samples ( samples), so they get the same label as the initial labeled sample. In the clustering halos of clusters 1 and 3, there are many clustering error samples ( samples) caused by the confused samples shown in Figure 15(a). This means that, in the future, they will affect the performance of the recognition algorithm if these samples are not cleared.
For the WKFDA+DP+S3VM algorithm, in the S3VM training process, for one thing, the S3VM cannot clear the samples in Figure 15(c). And for another, for the samples in Figure 15(c), the S3VM can only identify them by traversing the samples. Therefore, the WKFDA+DP+S3VM algorithm is unstable and inefficient. For the S2DP algorithm, once the features of Figure 15(c) are input into the LCI, the LCI algorithm removes the unreliable features such as the clustering halos and cross-clustering features and makes full use of the cluster cores as reliable samples. Once the labeled samples are included in the cluster core, the other unlabeled samples are labeled with the labels of the labeled samples. For cluster cores that do not contain labeled samples, only the clustering center is identified, and the label of the whole cluster core can be obtained, which greatly improves the recognition efficiency. Figure 16 is a sequence diagram showing the recognition of the DP clustering result of Figure 15(b) by the LCI in the S2DP algorithm. Figure 16(a) is the visualization of the DP clustering results after removing the interference samples. Figure 16(b) is the result diagram of LCI’s final recognition of the DP clustering. As can be seen from Figure 16(a), both the confusing sample in Figure 15(a) and the sample in Figure 15(c) are removed, greatly improving the reliability of sample identification. As can be seen from Figure 16(b), clusters 4 and 5 are correctly identified, and at the same time, only 5 samples with incorrect identification are in the expanded labeled samples. Thus, S2DP is quite reliable. In this way, the above conclusions are verified.
4.2.2. Verifying the Generalization Capabilities
In Section 4.2.1, the S2DP is relatively stable, but its recognition accuracy is relatively low when the number of the labeled samples is less than 21. Therefore, the IS2DP is required to generate a large number of the labeled samples to improve the recognition accuracy of S2DP. Here, we compare the IS2DP+S2DP and S2DP. The experimental data configuration is shown in Table 2 with the change of the number of the labeled samples, and the OA trend chart of the two methods is obtained, as shown in Figure 17. The horizontal axis represents the number of each type of target labeled samples corresponding to different experiments; the vertical axis represents the overall accuracy rate.
From Figure 17, we can see that when the number of the labeled samples is less than 21, the recognition performance of IS2DP+S2DP is 10% higher than that of S2DP. With the number of labeled samples larger than 21, their classification accuracy is equivalent. To verify this conclusion, we apply 100 iterations onto IS2DP when the number of labeled samples is 15. The labeled samples generated by IS2DP are counted, as shown in Table 4. The accuracy rate of labeled samples generated from learning set is over 85%.
4.3. Comparison with Semi-Supervised Deep Learning
The semi-supervised deep learning algorithms, Ladder Network  and Temporal Ensembling , which contain a supervised and unsupervised learning process, similar to our algorithm. Therefore, we choose these two methods to compare with the IS2DP-based S2DP algorithm (IS2DP+S2DP). In addition to using SAR images as experimental data, we also use a set of publicly available optical image data to verify the effectiveness of our algorithm.
4.3.1. Testing with SAR Images
The experimental data configuration is shown in Table 3, and with the change of the number of the labeled samples, the OA trend chart of three methods is obtained, as shown in Figure 18. The horizontal axis represents the number of each type of labeled target samples corresponding to different experiments, and the vertical axis represents the overall accuracy rate.
In Figure 18, the recognition accuracy of the three methods is increasing with the increase of the labeled samples. From the curve smoothing, the accuracy curves of the Ladder Network and the Temporal Ensembling are fluctuating, especially the Temporal Ensembling. Comparing them, the accuracy curve of the IS2DP+S2DP is relatively consistent. From the classification accuracy, when the number of the labeled samples is less than 33, the results obtained by Ladder Network and Temporal Ensembling are not much different, but significantly lower than that of IS2DP+S2DP. When the number of the labeled samples reaches 33, the classification accuracy of IS2DP+S2DP is slightly better than that of Ladder Network. These results indicate that the learning set containing the interference samples has a great influence on the recognition performance of the Ladder Network and Temporal Ensembling. Because the Ladder Network and Temporal Ensembling were unable to remove these interference samples during the training process, their recognition accuracy was unstable and not high. Different from them, the IS2DP+S2DP can select reliable unlabeled samples and remove those interference samples, so its recognition performance is relatively stable and the accuracy is improved. When the number of the labeled samples is equal to 21, we will analyze the use of the learning set by the three methods below.
In the Temporal Ensembling algorithm, one neutral network conducts two different works, supervised learning and unsupervised learning. Figures 19(a)–19(c), respectively, show losses in these two processes and in the whole method. Observed from the curve fluctuation, supervised learning loss in Figure 19(a) is the most stable while unsupervised learning in Figure 19(b) fluctuates significantly. It demonstrates that neural network performs well in learning labeled samples, but is still unstable to handle the learning set, thus resulting in unstable overall loss as shown in Figure 19(c). Finally, the Temporal Ensembling algorithm utilizes the learning set by 56.45% only, which is calculated based on the trained neutral network’s recognition of the learning set. Recognition results of the learning set by the Temporal Ensembling algorithm is displayed in Table 5 (confusion matrix). Observed from the confusion matrix, the remaining 43.55% disturbs the learning process, for instance, by misrecognizing SLICY as BTR70 targets.
Similar to temporal ensembling algorithm, the neutral network in the Ladder Network algorithm consists of supervised learning and unsupervised learning as well. Figures 20(a)–20(c), respectively, show losses in these two processes and by the whole method. Observed from the curve fluctuation shown in Figure 20(a), supervised learning loss significantly fluctuates, probably because of inadequate labeled samples; in Figure 20(b), unsupervised learning performs stably, probably resulting from unsupervised learning (Autoencoder) embedded in the Ladder Network algorithm which could learn and recognize unlabeled samples and reduce certain interference. Thus, the overall loss shown in Figure 20(c) performs stably. Therefore, comparing with temporal ensembling, Ladder Network improves the utilization of the learning set to 68.42% (as shown in Table 6 confusion matrix), enhancing its recognition performance as well.
Differing from temporal ensembling and ladder network, the IS2DP+S2DP algorithm identifies reliable unlabeled samples by iterations before implementing feature learning, instead of directly learning features from the unlabeled samples. Here we employ 300 iterations on the IS2DP+S2DP algorithm for fair comparison. Figures 21(a)–21(c) show the screening of the reliable samples in the learning set during one iteration: (a) projection of the WKFDA algorithm on the learning set; (b) DP clustering result; (c) reliable samples labeled by LCI. In Figure 21, red circle, green circle, light blue circle, and blue circle represent BMP2, BTR70, T72, and SLICY target samples, respectively, and represents the labeled samples. Confused by SLICY interference targets, the WKFDA algorithm has some issue in projecting the learning set but performs well in dividing different samples during the DP clustering, and successfully identify SLICY during the LCI labeling process. Finally, IS2DP+S2DP improves the utilization of the learning set to 82.76% (as shown in Table 7). As 28.95% unreliable sample rejecting recognition will be deleted, only 10% false samples affects the performance; thus IS2DP+S2DP’s recognition performance can be improved.
4.3.2. Testing with Optical Images
To verify the effectiveness of the proposed method on other data sets, we use optical image data to test IS2DP+S2DP. These optical image data come from some publicly available databases, and the detailed data configuration is shown in Table 8. The images of cats and dogs are from the database of the Kaggle competition platform ; the images of panda are from the ImageNet database ; the images of airplanes, motorbike, and faces are from the caltech101 database .
In Table 8, we set more stringent conditions than EOC_2 for SAR images, which is closer to the reality. Specifically, our interested targets are cats, dogs and panda. However, our learning set contains not only unlabeled interested targets, but also other 3 types of interference targets (airplanes, motorbike, and faces) with the same number of unlabeled interested targets. Under such conditions, the Ladder Network, Temporal Ensembling, and IS2DP+S2DP are tested and compared. With the change of the number of the labeled samples, the OA trend chart of three methods is obtained, as shown in Figure 22. The horizontal axis represents the number of each type of target labeled samples corresponding to different experiments; the vertical axis represents the overall accuracy rate.
In Figure 22, the identification accuracy of our method IS2DP+S2DP is significantly better than Ladder Network and Temporal Ensembling. From the OA trend, the recognition accuracy of Ladder Network and Temporal Ensembling does not increase significantly with the increase of the number of the labeled samples, while IS2DP+S2DP is significantly improved. Compared with the results of the SAR image test (Figure 18), the results of the three algorithms in the optical image test are significantly lower. This may be because in the learning set, we both increase the numbers of the target types and the number of the confusion targets, which leads to the less satisfactory results in learning the target features. From Figure 22, the Ladder Network and Temporal Ensembling algorithms are subject to more serious interference, and their average recognition accuracy is about 45%, respectively. Our algorithm IS2DP+S2DP is also subject to certain interference, but when the number of samples per class reaches 21, its average recognition accuracy is about 70%, which is significantly higher than the Ladder Network and Temporal Ensembling algorithms. When the number of the labeled samples is equal to 21, we will analyze the use of the learning set by the three methods.
Figure 23 shows the use of the learning set by the Ladder Network in the last 270 iterations during 1000 iterations of training. Figure 23(a) is the recognition accuracy of the learning set by Ladder Network; Figure 23(b) is the Ladder Network’s loss value, where the blue line with square is the overall loss, the black line with circle is the supervised loss, and the green line with diamond is the unsupervised loss. From Figure 23(a), we know that the recognition accuracy is very low, about 33%. From Figure 23(b), we know that the supervised loss is low, while the unsupervised loss is high, which makes the overall loss difficult to reduce. Ladder Network is a complex network, which is intertwined by many components, but its core part mainly includes adding noise to samples, reconstructing samples and “skip connection" . It first augments the unlabeled samples by adding noise to obtain a wider range of generalization information, secondly retains the sample information as much as possible by reconstructing the unlabeled samples in a regularized manner, and finally combines unsupervised learning with supervised learning to form semi-supervised learning by skip connection. Compared with supervised learning, unsupervised learning is more important in Ladder Network. Therefore, although Ladder Network has been well learned in the labeled samples, it has not been well learned in using the unlabeled samples, resulting in the whole algorithm has not been well trained. Finally, the Ladder Network recognition accuracy is neither stable nor high.
Figure 24 shows the use of the learning set by the Temporal Ensembling in the last 270 iterations during 1000 iterations of training. Compared with Figure 23, the recognition accuracy of Temporal Ensembling for the learning set is increased, about 48%, but it is still relatively low. Different from the Ladder Network, Temporal Ensembling adds noise to all the samples, which makes the labeled samples augmented. At the same time, in the initial stage of the training, the Temporal Ensembling’s supervised learning plays an important role because of the small value of the unsupervised loss weighting function . Therefore, the Temporal Ensembling is well trained to some extent. As the value of the loss weighting function increases, the unsupervised learning gradually plays an important role in Temporal Ensembling. Although the unsupervised loss is very low, Temporal Ensembling has not been well trained in learning interested target features because of the large number of unreliable samples in the learning set. Finally, Temporal Ensembling still has low recognition accuracy for interested targets.
Unlike the Ladder Network and the Temporal Ensembling algorithms, the IS2DP+S2DP algorithm first removes the interference samples in the process of using the learning set and then learns the selected reliable samples. Table 9 shows the recognition results of IS2DP+S2DP algorithm for learning set after 300 iterations. The average accuracy is 80%, which is significantly higher than that of Ladder Network and Temporal Ensembling algorithms. Compared with Table 7, the average accuracy of Table 9 is lower. However, the correct rate of rejection of the 3 types interference target samples has not been reduced, and these correct rates have reached more than 80%. In addition, the rejection error rate of the IS2DP+S2DP algorithm for the target samples is quite low; for example, cats is 18/200 = 0.09; dogs is 15/200 = 0.075; panda is 20/200 = 0.1. These experimental results show that the proposed algorithm is effective in optical image testing.
In order to accurately identify remote sensing images when there are few labeled samples, two new semi-supervised learning algorithms have been proposed in this paper: S2DP and IS2DP. They use labeled sample information to filter out reliable unlabeled samples to improve the performance of the semi-supervised algorithms.
The novelty of this paper lies in the following: (a) the WKFDA has been derived to explore the features of the images; (b) based on the clustering information of the DP, the labeling method LCI has been designed to query reliable unlabeled samples and accurately classify the unlabeled samples; (c) in IS2DP, the unlabeled training set is divided into different subsets, which suppresses the deterioration of the algorithm by too many unreliable unlabeled samples in the learning process. Moreover, IS2DP uses S3VM twice to ensure reliable semi-supervised samples.
In the experiments for the actual SAR images recognition from the MSTAR database, the S2DP has made a significant improvement in terms of the classification accuracy and the stability in comparison with other existing methods. In addition, the IS2DP is effective and has applicable values to query the semi-labeled samples and is more suitable to deal with the situation where it lacks labeled samples.
How to make full use of remote sensing images to improve the performance of recognition algorithm has always been an open problem. Although the semi-supervised deep learning algorithm is susceptible to interfering samples, it has strong feature learning capabilities once the interfering samples have been removed. In the near future, we will try to further improve the feature learning ability of the S2DP and IS2DP algorithms by virtue of the semi-supervised deep learning.
AbbreviationsThe following abbreviations are used in this manuscript:
|DP:||Clustering by fast search and find of density peaks|
|EOC:||Extended operating conditions|
|KFDA:||Kernel Fisher discriminant analysis|
|KLFDA:||Kernel local Fisher discriminant analysis|
|KPCA:||Kernel principal component analysis|
|LCI:||Labeling method based on the DP clustering information|
|MSTAR:||Moving and Stationary Target Acquisition and Recognition database|
|OA:||Overall accuracy rate|
|PS3VM-D:||Progressive semi-supervised SVM with diversity|
|SAR:||Synthetic aperture radar|
|S2DP:||Semi-supervised learning method based on DP|
|SOC:||Standard operating conditions|
|SVM:||Support vector machine|
|WKFDA:||Weighted Kernel Fisher discriminant analysis.|
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (61771027; 61071139; 61471019; 61171122; 61501011; 61671035), the Scientific Research Foundation of Guangxi Education Department (KY2015LX444), the Scientific Research and Technology Development Project of Wuzhou, Guangxi, China (201402205), the Guangxi Science and Technology Project (Guike AB16380273), and the Research and Practice on Teaching Reform of Web Page Making and Design Based on the Platform of “E-Commerce Pioneer Park” (Guijiao Zhicheng 41). Professor A. Hussain was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant no. EP/M026981/1. E. Yang was supported in part under the RSE-NNSFC Joint Project (2017-2019), grant number 6161101383, with China University of Petroleum (Huadong). H. Zhou was supported by UK EPSRC under Grant EP/N011074/1, Royal Society-Newton Advanced Fellowship under Grant NA160342, and European Union’s Horizon 2020 research and innovation program under the Marie-Sklodowska-Curie Grant agreement no. 720325.
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