SAR Target Recognition via Monogenic Signal and Gaussian Process Model

e monogenic signal and Gaussian process model are applied to synthetic aperture radar (SAR) target recognition. e monogenic signal is used to extract the features of the SAR image. e Gaussian process model is a statistical learning algorithm based on the Bayesian theory, which constructs a classication model by combining the kernel function and the probability judgement. Compared with the traditional classication model, the Gaussian process model can obtain higher classication eciency and accuracy. During the implementation, the monogenic feature vector of the SAR image is used as the input, and the target label is used as the output to train the Gaussian process model. For the test sample to be classied, the target label is determined by calculating the posterior probability of each class using the Gaussian process model. In the experiments, the validations are carried out under typical conditions based on the MSTAR dataset. According to the experimental results, the proposed method maintains the highest performance under the standard operating condition, depression angle dierences, and noise corruption, which veries its eectiveness and robustness.


Introduction
Synthetic aperture radar (SAR) realizes the observation of the focused area through high-resolution imaging for intelligence interpretation. SAR target recognition classi es image chips of unknown targets through pattern recognition algorithms [1]. Generally, SAR target recognition methods rst perform feature extraction to obtain e ective feature vectors. In [2][3][4][5][6][7][8], the geometric shape features were used to design SAR target recognition methods, including target regions, shadows, and contours. In [9,10], e cient dimensionality reductions were achieved through projection algorithms such as principal component analysis (PCA) and non-negative matrix factorization (NMF). Image analysis algorithms including the wavelet, empirical mode decomposition, and monogenic signal were employed for SAR image processing with application to target recognition [11][12][13][14][15][16]. In [17][18][19], SAR target recognition methods based on the attribute scattering centers were developed. e application of the classi er needs to be combined with the typical characteristics of the extracted features. At this stage, most of the classi ers used in SAR target recognition came from the traditional pattern recognition eld. Representative ones are K nearest neighbors (KNN) [9,20], support vector machine (SVM) [20][21][22][23], sparse representation-based classi cation (SRC) [23][24][25][26][27][28][29] etc. In recent years, traditional neural networks have gradually moved to a deeper level, promoting the rapid development of deep learning theory.
In this paper, the monogenic signal and Gaussian process model are jointly applied to SAR target recognition. e monogenic signal can obtain an e ective feature description through multilevel time-frequency decompositions of the original SAR image, which can better re ect the target characteristics [12][13][14][15]. e Gaussian process model [42][43][44][45] is a statistical learning algorithm based on the Bayesian framework, and all its statistical characteristics are determined by the mean and covariance functions. erefore, the Gaussian process model is a nonparametric probability model. Compared with the traditional neural network, SVM, and other classification mechanisms, the Gaussian process model greatly reduces the parameter scale, simplifies the overall optimization process, and enhances the convergence of the model. In the application of Gaussian process model, prior knowledge can be represented as prior probability into the model, and the model's ability to describe actual problems can be improved by flexible selection of covariance functions. And the modeling and analysis of regression or classification problems can be realized. erefore, this paper chooses Gaussian process model as the basic classifier for SAR target recognition. First, a binary classifier is designed, and then a multiclass classifier is promoted through a oneto-one voting mechanism to meet the multiclass decisionmaking problem in SAR target recognition. In the specific implementation process, the feature vectors of the training and test samples are first extracted from the monogenic signal. e Gaussian process classification model is trained with the feature set of the training sample as input. For the feature vector of the sample to be identified, it is input into the trained Gaussian process model to obtain its target label. In the experiment, typical scenarios are set based on the MSTAR dataset to test the proposed method, whose performance is compared with some existing methods. e experimental results prove the effectiveness and robustness of the proposed method.

Feature Extraction Based on
Monogenic Signal e monogenic signal is an expansion of the analytic signal in a high-dimensional space. In particular, for image data, the two-dimensional monogenic signal can be used for characteristic analysis [12][13][14][15]. e monogenic signal is the product of the combination of the signal itself and its Riesz transform. e Riesz transformation of the signal f(z) is denoted as f R (z), where z � (x, y) T represents two-dimensional coordinates.
e monogenic signal f M (z) is calculated as follows: where both i and j are imaginary units. f(z) and its Riesz transform correspond to the real part and imaginary part of the monogenic signal, respectively. Accordingly, the decompositions of the monogenic signal are defined as follows: In the above equations, f x (z) and f y (z) correspond to the i-imaginary part and j-imaginary part of the monogenic signal, respectively; A(z) represent the amplitude information; and φ(z) and θ(z) correspond to the local phase and azimuth information, respectively. e three types of features obtained based on monogenic signal have different characteristics. Among them, A(z) mainly reflects the intensity distribution of the image. φ(z) and θ(z) reflect the local detail information and shape characteristics of the image. erefore, the joint use of the decompositions from the monogenic signal is conducive to construct more informative features.

Gaussian Process Model for Classification
Gaussian process model is a new kernel method developed on the basis of Bayesian neural network, which can be used to deal with classification and regression problems in machine learning. Due to its superior robustness, the Gaussian process model has been applied and verified in pattern classification and recognition problems [42][43][44][45].

Binary Classification.
Generally, the Gaussian process model includes three parts: likelihood function definition, hidden variable function definition, and posterior probability calculation. e classification algorithm based on Gaussian process model uses Gaussian function to approximate the hidden variable function of the classification process. Representative methods include Laplace method, Expectation Propagation method, and Kullback-Leibler divergence minimization method.
For a training set D containing N observations, , where x i ∈ R d is the ith input sample, d is the dimension, and its corresponding binary category label is denoted as y i , in which y i � 1 represents the positive class, and y i � − 1 represents the negative class.
, the output of each implicit function is constrained between [0, 1]. Assuming that each sample is independent and identically distributed, their joint probability distribution can be calculated as follows: Assuming f i a zero mean Gaussian distribution, the prior probability p(f/X) is described as follows: In (6), K refers to the covariance matrix of f. e posterior probability of implicit function is calculated as follows: 2 Mathematical Problems in Engineering where p(y/f) is the likelihood function and p(y/X, θ) represents the marginal probability. e Laplace approximation algorithm can be used to solve the posterior probability p(f/X, y, θ) and obtain the corresponding estimated value q(f/X, y, θ). e second-order Taylor series will be expanded at the maximum posterior probability logp(f/X, y, θ), and the results are as follows: where and f are independent of each other, so the problem can be reorganized as follows: e posterior probability can be obtained as In addition, the marginal probability distribution can be expressed as In equation (11), |B| � |K| · |K − 1 + W| � |I n + W 12 KW 12 |. θ represents hyperparameters, which can be solved by maximizing (11). For a given test sample x, the probability distribution of the corresponding implicit function f * is where K � K + W − 1 . e probability of the output corresponding to y * � 1 is According to the probability of the corresponding classes, the target label of the test sample can be determined using the binary classification.

Multiclass Classification.
e traditional Gaussian process model can be directly used for binary classification, but practical problems often involve the discrimination of multiple classes. erefore, it is necessary to extend the binary classification to multiple ones in order to realize the direct classification of multiple classes. Among them, a representative method of extending binary classification to multiple classifications is the one-to-one voting mechanism.
is paper uses this method to extend the traditional binary classifier based on Gaussian process mode for multiclass classification. e implementation process is as follows: (1) In the training stage, the training samples of class 1∼k are combined in pairs to obtain C 2 k � k(k − 1)/2 combinations. e Gaussian process model is used to train each combination to obtain the corresponding binary classifier (2) In the classification stage, the target label of unknown samples is judged through the voting mechanism. First, the initial number of votes is set for each class to 0; then, the k(k − 1)/2 binary classifiers which have been trained are used to classify the test samples. When the classifier C i,j determines the ith class, the number of votes for the ith class is increased by 1. If the classifier C i,j judges the test sample to be the jth class, the number of votes for the jth class is increased by 1. Finally, the total number of votes obtained for each class is counted, and the category with the highest number of votes is judged to be the target label of the test sample.
In this paper, the binary classification mechanism of the Gaussian process model is combined with the one-to-one voting mechanism to obtain a multiclass classifier.

Target Recognition Procedure.
In this paper, the monogenic signal and Gaussian process model are applied to SAR target recognition, which are used for feature extraction and classification, respectively. Figure 1 briefly shows the basic process of the proposed method. e main body is divided into two stages: training and testing. e main implementation steps are summarized as follows: Step 1: e monogenic signal is used to perform feature extraction on all training samples.
Step 2: e monogenic feature vectors of the training sample are used as inputs and the corresponding target labels as output to train a multiclass Gaussian classification model.
Step 3: e monogenic signal is used to perform feature extraction on the test sample to be classified.
Step 4: e monogenic feature vector of the test sample is input into the Gaussian process model after training, and the posterior probability of each class is calculated.
Step 5: e target label of the test sample is determined according to the principle of the largest number of votes.
Compared with the traditional SAR target recognition methods, the proposed one introduces Gaussian process Mathematical Problems in Engineering model, which provides a statistical way for decision. erefore, it can obtain the optimal decision in the sense of probability. By combining the decision criteria of the maximum posterior probability and the maximum number of votes, it is helpful to obtain more reliable target recognition results.

Description of the Dataset.
e MSTAR dataset is employed to set up typical test scenarios to carry out experiments and verification. e appearances of the targets included in the dataset are shown in Figure 2.
Some existing SAR target recognition methods are selected in the comparison algorithms, including NMF [10], Mono [12], BEMD [16], and ESENet [136]. ese four types of reference methods basically cover the most commonly used features and classifiers in existing SAR target recognition. In the subsequent experiments, three experimental scenarios are set to investigate the proposed method, namely: Scenario 1: standard operating condition, involving 10 types of targets, Scenario 2: depression angle differences, involving 3 types of targets, and Scenario 3: noise corruption, involving 10 types of targets. Table 1 gives a description of the standard operating condition in Scenario 1, including 10 types of targets. e training set uses SAR images at 17°depression angle, and the classifier is obtained to test the samples at 15°d epression angle. e comparison shows that the target configurations between the two sets remain the same, with only a 2°depression angle difference, so their overall correlation is strong. Figure 3 shows the results of the proposed method in the current scenario, the single-class recognition rate is higher than 98.5% (shown in diagonal elements), and the average recognition rate of 10 classes reaches 99.32%, which reflects the effectiveness of the proposed method. e average recognition rates of various methods under Scenario 1 are shown in Table 2.

Scenario 1.
e recognition rates of NMF, monogenic signal, BEMD, and ESENet methods are 98.26%, 98.69%, 99.02%, and 99.12%, respectively, which are all higher than 98%. It can be seen that the recognition problem under the standard operating condition is relatively simple.
By comparing the average recognition rates of various methods, it can be seen that the method in this paper has some performance advantages. e recognition rate of the ESENet method under the current condition is second only to the proposed method, which benefits from the classification ability of the deep learning model. In this paper, the discriminative features of the target in the SAR image are obtained through the monogenic signal, and the statistically optimal classification model is obtained by introducing the Gaussian process model, which effectively improves the target recognition performance under the standard operating condition.

Scenario 2.
e extended operating condition refers to the large differences between the test sample and the training sample due to the changes in the SAR data acquisition conditions. Typically, the extended operating conditions in SAR target recognition include target configuration differences, depression angle differences, and noise corruption. In this experiment, the proposed method is tested under the condition of different depression angles, and Scenario 2 shown in Table 3 is set. Among them, the samples at 17°d epression angle are used for training; the samples at 30°, and 45°depression angle are used for classification, which shows that there are large depression angle differences between the training and test samples. Different methods are examined on the two test sets, and the recognition results are obtained as shown in Figure 4. Comparing the results at the two depression angles, the overall performance at 30°is significantly better than 45°, indicating that large depression angle differences will lead to greater image differences. From the results at the two depression angles, the method in this paper has achieved the highest average recognition rate, showing its better robustness.
e monogenic signal can better analyze the relevant characteristics of the SAR images. e Gaussian process model can more effectively explore the internal correlation between the real target labels by deriving the best statistical model, so the robustness of the recognition method to the depression angle difference can be improved.

Scenario 3.
Noise corruption is another typical extended operating condition, mainly because the signal-tonoise ratio (SNR) of the SAR image to be identified is  Figure 5 are obtained. It can be seen that as the noise level continues to decrease, the average recognition rates of various methods show clear downward trends. e comparison shows that the method in this paper achieves the highest recognition rate at each noise     Mathematical Problems in Engineering level, reflecting its stronger noise robustness. In the timefrequency decomposition of the monogenic signal, the possible influence of noise is fully considered. erefore, the feature vector finally obtained has certain noise robustness.
In the derivation of the Gaussian process model, the possible noise influence is fully considered. erefore, the resulting feature vector has certain noise robustness. e classification model has strong robustness to noise. Similar to the case of the depression angle differences, the performance of the ESENet method under the condition of noise interference is the most severe mainly due to the large image difference between the training sample and the test sample, which leads to a significant decrease in the performance of the final classification model.

Conclusion
In this paper, the monogenic signal and Gaussian process model are applied to SAR image target recognition. e Gaussian process model constructs a Bayesian learning model by combining the kernel function and probability discrimination to obtain the best regression model in the statistical sense. e monogenic feature vector of the SAR image is used as the input, and the target label is the output. e regression mapping relationship between the two is constructed, so as to obtain a robust classification model. Based on the MSTAR dataset, three typical scenarios are set to test the proposed method and compared with the existing methods. e experimental results are as follows: In Scenario 1, i.e., the standard operating condition, the average recognition rate of the proposed method for 10 types of targets reaches 99.28%, which is higher than the four types of reference methods, verifying its effectiveness for multiclass target recognition problems under the standard operating condition. Under the condition of Scenario 2, i.e., depression angle differences, the average recognition rate of the proposed method for the test set at 30°and 45°depression angles is 98.04% and 73.13%, respectively, which is higher than the reference methods, verifying its robustness to the depression angle difference. In Scenario 3, i.e., noise corruption conditions, the proposed method maintains the highest average recognition rate under each noise level, showing its noise robustness.
Data Availability e dataset can be accessed upon request.

Conflicts of Interest
e authors declare no conflicts of interest.