New classes inference, few‐shot learning and continual learning for radar signal recognition

IET Radar, Sonar & NavigationVolume 16, Issue 10 p. 1641-1655 ORIGINAL RESEARCHOpen Access New classes inference, few-shot learning and continual learning for radar signal recognition Jiaji Luo, Jiaji Luo orcid.org/0000-0002-3782-8333 College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, ChinaSearch for more papers by this authorWeijian Si, Weijian Si College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, ChinaSearch for more papers by this authorZhian Deng, Corresponding Author Zhian Deng [email protected] College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, China Correspondence Zhian Deng, College of Information and Communication Engineering, Harbin Engineering University, Harbin, 150001, China. Email: [email protected]Search for more papers by this author Jiaji Luo, Jiaji Luo orcid.org/0000-0002-3782-8333 College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, ChinaSearch for more papers by this authorWeijian Si, Weijian Si College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, ChinaSearch for more papers by this authorZhian Deng, Corresponding Author Zhian Deng [email protected] College of Information and Communication Engineering, Harbin Engineering University, Harbin, China Key Laboratory of Advanced Marine Communication and Information Technology, Ministry of Industry and Information Technology, Harbin, China Correspondence Zhian Deng, College of Information and Communication Engineering, Harbin Engineering University, Harbin, 150001, China. Email: [email protected]Search for more papers by this author First published: 09 June 2022 https://doi.org/10.1049/rsn2.12286AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Automatic radar modulation recognition plays a significant role in both civilian and military applications. With the rapid development of deep learning, convolutional neural networks have achieved demonstrated success in radar signal recognition. However, the convolutional neural networks usually only recognise trained classes, and when the dataset changes, the networks need to be retrained. However, in actual radar signal recognition applications, the model usually needs to predict new radar signals, and the size of the training set will continue to accumulate. Therefore, few-shot learning and rapid training on dynamic datasets become crucial. In this study, a lifelong learning system based on imprint few-shot learning and Net2Net knowledge transfer for radar signal recognition is proposed. The proposed algorithm adapts to the constant changes of the dataset, which can achieve new classes inference, few-shot learning, and knowledge transfer. The model is trained on the dataset containing 8 types of radar signals and achieves high recognition accuracy in the test dataset containing 12 types of radar signals. The recognition accuracy of the proposed algorithm achieves 91.8% at −2 dB. In addition, Net2Net knowledge transfer can improve the training efficiency on new datasets avoiding training from scratch. 1 INTRODUCTION The low probability of intercept (LPI) radar is widely used in electronic warfare and modern radar systems. How to quickly and accurately recognise the radar signals modulation has gradually become a research focus [1-5]. Traditional radar signal recognition requires manual feature extraction, and a classification algorithm to classify the manually extracted features [6, 7]. However, this method is complex and inefficient and cannot meet the needs of modern intelligent recognition systems. In recent years, with the success and continuous development of deep learning, researchers converted radar signals into time-frequency images (TFIs) through time-frequency analysis and used convolutional neural networks to automatically extract features and classify radar signals by supervised learning [3]. The radar signal recognition based on deep learning significantly improves the robustness and recognition accuracy [8-12]. Hoang et al. [13] presented an LPI radar waveform recognition technology based on a single-shot multi-box detector (SSD) and a supplementary classifier, which demonstrated excellent classification performance for LPI radar waveforms. Ni et al. [14] proposed an LPI radar waveform recognition strategy based on enhanced Fourier-based Synchrosqueezing Transform and multi-resolution deep feature fusion. Yu et al. [15] proposed an evaluation criterion to evaluate Convolutional Neural Networks (CNN) performance for radar intrapulse modulation recognition. Wei et al. [16] proposed a novel end-to-end sequence-based network including a CNN, a Bi-LSTM strengthening with self-attention mechanism, and a dense neural network to recognise eight types of representative intrapulse modulations of radar signals. In recent years, the performance of deep convolutional neural networks trained on millions of training data has outperformed humans in the field of large-scale image recognition. The current deep learning models are trained and tested in closed datasets where the classes and scale of the dataset are fixed [17]. However, the actual application scenario is far more complicated. In practice, humans can recognise new classes through one or a few samples, but deep neural networks trained on the large-scale dataset are unable to do so. The subject of few-shot learning is to learn to recognise previously unseen classes with very few annotated samples. Few-shot learning, or constantly expanding a deep learning model using limited new samples in an open environment, is still a difficult topic [18-22]. In the actual radar signal recognition, the dataset will continue to accumulate as the signals are detected. The radar signal recognition based on the traditional convolutional neural network can only recognise the known classes that have been trained with a substantial quantity of training data. However, in an open situation, the recognition accuracy will be compromised, because of unknown radar signals. Moreover, as the dataset continues to increase, standard training methods typically train numerous alternative models. Because the training and convergence process of each model takes a lengthy time, the data-driven model design will be slower [23]. In this paper, the imprint algorithm is utilised to achieve few-shot learning in the recognition of few-sample classes. The cosine similarity between the feature vectors of test samples and different classification weights is used to recognise samples of new classes. The weight of the new classes can be loaded into the model and updated online during the actual application process [20]. In addition, as the dataset accumulates, the Net2Net algorithm transfers the knowledge from the smaller teacher network to the larger student network without affection of recognition accuracy [24]. This strategy optimises training efficiency by keeping the student model in the optimal convergence state attained by the teacher model training. The proposed learning system achieves new classes inference, low-shot learning, model expansion, and knowledge transfer. It is trained uninterruptedly in an ever-increasing dataset, and the size of the model increased in lockstep with the size of the training dataset. As a result, the proposed learning framework can recognise new radar signals, and a larger student network is smoothly instantiated based on the teacher network. Combining the above methods, the deep learning model and training process is more in line with human cognitive logic. The proposed learning framework for radar signal recognition has three advantages: New classes inference. Smooth transition from smaller teacher models to larger student models. Continual learning under the dynamic dataset. 2 FEW-SHOT LEARNING AND CONTINUAL LEARNING FOR RADAR SIGNAL RECOGNITION The proposed learning framework applies to the actual application of radar signal recognition as shown in Figure 1. First, the teacher network is trained in the training dataset containing 8 classes. The test dataset comprises both 8 trained classes and 4 new classes with few training samples. The proposed algorithm utilises cosine similarity to recognise new radar signals and the imprint algorithm to embed the classification weight of the new classes into the model. As the amounts of datasets increases, the original teacher model is transferred to a wider or deeper student network through the Net2Net knowledge transfer for continual learning. This algorithm allows the model to transition smoothly, improving the convergence speed and training efficiency. FIGURE 1Open in figure viewerPowerPoint The few-shot learning and continual learning for radar signal recognition Radar signal and time frequency analysis In this paper, the radar signal interfered by additive white Gaussian noise can be expressed as y ( t ) = x ( t ) + N ( t ) = A e j θ ( t ) + N ( t ) $y(t)=x(t)+N(t)=A{e}^{\,j\theta (t)}+N(t)$ (1)where x(t) represents complex radar signal samples and N(t) stands for additive white Gaussian noise with zero mean value and variance σ 2 ${\sigma }^{2}$ . A represents the non-zero constant amplitude, and θ(t) denotes the instantaneous phase of the radar signal. The inherent difference between radar signals of different modulation type is the frequency variation over time. The radar signal is converted into TFIs through Time–Frequency Analysis. The pattern in the image corresponds to the frequency variation with time. The instantaneous phase θ(k) consists of instantaneous frequency f(k) and the phase function ϕ(k), which determine the modulation type of radar signals. The instantaneous phase θ(k) can be defined as θ ( k ) = 2 π f ( k ) k T s + ϕ ( k ) $\theta (k)=2\pi f(k)\left(k{T}_{s}\right)+\phi (k)$ (2) Twelve radar signals considered in this paper can be divided into two parts, Frequency Modulation (FM) and phase Modulation (PM). In the FM, the instantaneous frequency f(k) varies while the phase ϕ(k) is constant, and in the PM the phase ϕ(k) varies while the instantaneous frequency f(k) is constant as listed in Table 1. TABLE 1. Radar signals Modulation type f(k) ϕ(k) LFM f 0 + B τ p w k T s ${f}_{0}+\frac{B}{{\tau }_{pw}}\left(k{T}_{s}\right)$ Constant Costas fj Constant Barker Constant 0 or π Frank Constant 2 π M ( i − 1 ) ( j − 1 ) $\frac{2\pi }{M}(i-1)(j-1)$ P1 Constant − π M [ ( M − ( 2 j − 1 ) ) ] [ ( j − 1 ) M + ( i − 1 ) ] $-\frac{\pi }{M}[(M-(2j-1))][(j-1)M+(i-1)]$ P2 Constant − π 2 M [ 2 i − 1 − M ] [ 2 j − 1 − M ] $-\frac{\pi }{2M}[2i-1-M][2j-1-M]$ P3 Constant π ρ ( i − 1 ) 2 $\frac{\pi }{\rho }{(i-1)}^{2}$ P4 Constant π ρ ( i − 1 ) 2 − π ( i − 1 ) $\frac{\pi }{\rho }{(i-1)}^{2}-\pi (i-1)$ T1 Constant mod 2 π n ⌊ ( k t − j t ) j n T ⌋ , 2 π $\mathrm{mod}\left\{\frac{2\pi }{n}\lfloor (kt-jt)\frac{jn}{T}\rfloor ,2\pi \right\}$ T2 Constant mod 2 π n ⌊ ( k t − j t ) 2 j − k + 1 T n 2 ⌋ , 2 π $\mathrm{mod}\left\{\frac{2\pi }{n}\lfloor (kt-jt)\frac{2j-k+1}{T}\frac{n}{2}\rfloor ,2\pi \right\}$ T3 Constant mod 2 π n ⌊ n B t 2 2 T p ⌋ , 2 π $\mathrm{mod}\left\{\frac{2\pi }{n}\lfloor \frac{nB{t}^{2}}{2{T}_{p}}\rfloor ,2\pi \right\}$ T4 Constant mod 2 π n ⌊ n B t 2 2 T p − n B t 2 ⌋ , 2 π $\mathrm{mod}\left\{\frac{2\pi }{n}\lfloor \frac{nB{t}^{2}}{2{T}_{p}}-\frac{nBt}{2}\rfloor ,2\pi \right\}$ Abbreviation: LFM, linear frequency modulation. The Choi–Williams distribution based on the time-frequency distribution of Cohen's class has the advantages of high resolution and cross-term suppression. The resolution of the time-frequency analysis can be adjusted by changing the parameters of its exponential kernel function. The TFIs of radar signals based on Choi–Williams distribution is defined as C ( t , ω ) = 1 2 π ∭ ∞ f ( θ , τ ) x ( s + τ / 2 ) x ∗ ( s − τ / 2 ) e − j ( θ t + w τ − s θ ) d s d τ d θ $C(t,\omega )=\frac{1}{2\pi }\underset{\infty }{\iiint }f(\theta ,\tau )x(s+\tau /2){x}^{\ast }(s-\tau /2){e}^{-j(\theta t+w\tau -s\theta )}dsd\tau d\theta $ (3)where t and ω denote time and frequency axes, respectively. And f(θ,τ) is the exponential kernel function of the Choi–Williams distribution. The kernel function is regarded as a low-pass filter that can suppress cross-terms effectively. f ( θ , τ ) = exp − θ 2 τ 2 σ $f(\theta ,\tau )=\mathrm{exp}\left[\frac{-{\theta }^{2}{\tau }^{2}}{\sigma }\right]$ (4) The TFIs of different radar signals by Choi–Williams Distribution considered in this paper are shown in Figure 2. To effectively recognise radar signals, the TFI visually illustrates the relationship between the frequency variation with time. FIGURE 2Open in figure viewerPowerPoint The Choi–Williams distribution time-frequency images (TFIs) of different radar signals The Choi–Williams distribution provides better sensitivity with respect to signals buried in the noise. It provides better resolution both in time and frequency as compared to the short-time Fourier transform as shown in Figure 3. These properties are making the Choi–Williams distribution a common choice for contemporary time-frequency analysis. FIGURE 3Open in figure viewerPowerPoint The time-frequency images (TFIs) of different time-frequency analysis Before the feature extraction, TFIs are normalised to reduce the influence of the bandwidth of different radar signals. The TFIs are transformed to grey images as follows G ( x , y ) = I ( x , y ) − min ( I ( x , y ) ) max ( I ( x , y ) ) − min ( I ( x , y ) ) $G(x,y)=\frac{I(x,y)-\mathrm{min}(I(x,y))}{\mathrm{max}(I(x,y))-\mathrm{min}(I(x,y))}$ (5)where I(x,y) indicates the TFI by Choi–Williams Distribution, G(x,y) denotes the grey images, and (x,y) signifies each pixel in the TFIs. The grey TFIs contain significant components and information of radar signals. Imprint few-shot learning In order to enable the deep learning model to recognise the new classes in practical applications, the imprint algorithm is utilised for few-shot learning. The imprint algorithm establishes the symmetric relationship between the feature vectors and the classification weights, allowing the feature vector of the new classes to be used as the classification weight in the model [20]. 2.2.1 Architecture of the imprint layer The model consists of two parts. The first is the feature embedding extractor, ϕ : ℝ N → ℝ D $\phi :\mathbb{R}^{N}\to \mathbb{R}^{D}$ , which is mainly composed of convolutional layers. The extractor can map the input image x ∈ ℝ N $x\in \mathbb{R}^{N}$ to the D dimensional embedding vectorϕ(x). Unlike traditional convolutional neural networks, the end of the feature extractor uses the L2 normalisation layer to normalise the feature vector, so the output feature has a unit length, that is | | ϕ ( x ) | | 2 = 1 $\vert \vert \phi (x)\vert {\vert }_{2}=1$ . Second, the classifier f(ϕ(x)) maps the embedding vector to the logit score, and then calculates the category probability distribution of the input image through the softmax activation function. f i ( ϕ ( x ) ) = exp w i T ϕ ( x ) ∑ c exp w c T ϕ ( x ) ${f}_{i}(\phi (x))=\frac{\mathrm{exp}\left({w}_{i}^{T}\phi (x)\right)}{{\sum }_{c}\mathrm{exp}\left({w}_{c}^{T}\phi (x)\right)}$ (6)where wi represents the ith column of the weight matrix normalised by the classification layer, and the classification layer does not use the bias. Each column of the weight matrix of the classification layer can be regarded as a template for the corresponding category. During predicting, the last layer of the model calculates the inner product between the embedded features of the input image ϕ(x) and all category templates wi. Because the embedding features of the input image and the template weight are normalised to unit length, the prediction result is equivalent to finding the category corresponding to the nearest template in the embedding space based on cosine similarity. y ˆ = arg max c ∈ C w c T ϕ ( x ) $\widehat{y}=\mathrm{arg}\underset{c\in C}{\mathrm{max}}\,{w}_{c}^{T}\phi (x)$ (7) Compared with the nearest neighbour model, this model contains only one template for each class, and the classification weights can be directly embedded in the convolutional neural network to immediately calculate the prediction. Normalise the weight matrix of the final classification layer of the model to unit length by column to map the weight vector to the high-dimensional unit sphere. In the imprint algorithm, since the columns of embedded features and classification weights are normalised, the dot product w i T ϕ ( x ) ∈ [ − 1,1 ] ${w}_{i}^{T}\phi (x)\in [-1,1]$ can represent the cosine similarity of the two vectors. More importantly, normalisation can establish a symmetric relationship between the embedding vector and the classification weight, allowing them to be converted to one another. Cosine similarity w i T ϕ ( x ) ∈ [ − 1,1 ] ${w}_{i}^{T}\phi (x)\in [-1,1]$ cannot approximate the distribution of true one-hot encoding when using the softmax activation function. Therefore, a scale factor needs to be applied to scale the inner product. As a result, the softmax can be modified as f i ( ϕ ( x ) ) = exp s w i T ϕ ( x ) ∑ c exp s w c T ϕ ( x ) ${f}_{i}(\phi (x))=\frac{\mathrm{exp}\left(s{w}_{i}^{T}\phi (x)\right)}{{\sum }_{c}\mathrm{exp}\left(s{w}_{c}^{T}\phi (x)\right)}$ (8) There are two differences between the imprint model and the standard convolutional neural network. First of all, the normalised embedded feature vector and the weight have a symmetric connection and can be substituted with each other. Secondly, the last classification layer uses the inner product and cosine similarity for class prediction, without bias parameters. In this paper, cosine similarity is used to measure the similarity between the feature vectors of test samples and different classification weights. The new samples are distinguished by the maximum cosine similarity. If the maximum cosine similarity between the test samples and the templates of all known classes is less than the threshold, it indicates that the test sample belongs to a new class. The threshold of maximum cosine similarity for new classes inference is set to 0.85. If the maximum cosine similarity of the test sample is higher than the threshold, it indicates that the sample belongs to a known class. Otherwise, it means that the sample belongs to the new class. This method can effectively improve the recognition ability of the model in an open environment and improve the recognition performance. 2.2.2 Weight imprint The imprint algorithm [20] can directly set the embedded feature vector of the unknown sample as the classification weight of the new category. Consider a new sample x+, the imprint algorithm calculates the embedding vector ϕ(x+) and adds it to the weight matrix as a new column, Wi = ϕ(x+). As illustrated in Figure 4. Intuitively, the imprint algorithm can be regarded as the embedding feature of a few samples as a template for the new category and loaded into the classification layer of the model. FIGURE 4Open in figure viewerPowerPoint The architecture of the imprint algorithm If there are several samples for a new category x + ( i ) i = 1 n ${\left\{{x}_{+}^{(i)}\right\}}_{i=1}^{n}$ , the weight of the new category is derived by averaging the embedding vectors of the samples w ˜ + = 1 n ∑ 1 n ϕ ( x + ( i ) ) ${\widetilde{w}}_{+}=\frac{1}{n}{\sum }_{1}^{n}\phi ({x}_{+}^{(i)})$ . Normalise the weight vector w ˜ + = w ˜ + / ‖ w ˜ + ‖ ${\widetilde{w}}_{+}={\widetilde{w}}_{+}/{\Vert}{\widetilde{w}}_{+}{\Vert}$ and load it into the weight matrix of the classification layer to predict the new category. In addition, the model can be fine-tuned after loading new classification weights to improve its performance. It can significantly increase the convergence speed, and improve the training efficiency due to the symmetry between the feature vector and the classification weight. Net2Net knowledge transfer Knowledge transfer is applied to a variety of scenarios, including transfer learning [25] and knowledge distillation [23]. Transfer learning [25-27] applies the knowledge learnt in one problem to another. However, the transfer model needs to freeze a portion of the parameters of the pretraining model during the training stage, which prevents the model from being fully trained, decreasing the capacity of model feature extraction and generalisation. Knowledge distillation transfers the knowledge of a larger teacher model to a smaller student model [23, 28, 29]. However, in some practical application scenarios, with the accumulation of data, the size of the dataset gradually increases, so the scale of the corresponding model will also increase. Knowledge needs to be transferred from a smaller teacher model to a larger student model and knowledge distillation cannot meet this requirement. The Net2Net algorithm [24] quickly transfers the knowledge from the teacher network to the student network, resulting in a considerable increase in the training speed of the student network. In this paper, two Net2Net methods are adopted: Net2WiderNet replaces the original convolutional layer with the wider convolutional layer; Net2DeeperNet deepens the convolutional neural network by adding convolutional layers that match the identity requirements. Following the training of a smaller teacher network, knowledge can be transferred to a wider or deeper student network, assisting in the construction of a larger convolutional neural network. Two effective Net2Net strategies are utilised to initialise the student network to achieve the same recognition accuracy as the teacher network, after which it can be trained normally. Specifically, the teacher network is represented by y = f(x; θ), where x is the input of the network, y is the output of the network, θ denotes the parameter of the network model, and f represents the convolutional neural network. The Net2Net strategy is to select a set of parameters for the student network g x ; θ ′ $g\left(x;{\theta }^{\prime }\right)$ ∀ x , f ( x ; θ ) = g x ; θ ′ $\forall x,f(x;\theta )=g\left(x;{\theta }^{\prime }\right)$ (9) The Net2Net strategies have several advantages over other knowledge transfer algorithms: The larger student network maintains the same performance as the teacher network, and there is no need to devote excessive time to the training process during a period of low performance. During subsequent training, all of the parameters in the new student network can be fine-tuned to avoid the decline of generalisation ability caused by freezing the weight of the old model. 2.3.1 Net2WiderNet Net2WiderNet allows the convolutional layer in the model to be replaced with a wider convolutional layer, which contains additional convolutional channels. The weights W(i) and W(i+1) of the ith and i+1th layers need to be replaced to widen the ith layer. If the number of input channels of the ith layer is m, the number of output channels is n, and the number of output channels of the i+1th layer is p, then W ( i ) ∈ ℝ h × h × m × n ${W}^{(i)}\in \mathbb{R}^{h\times h\times m\times n}$ and W ( i + 1 ) ∈ ℝ h × h × n × p ${W}^{(i+1)}\in \mathbb{R}^{h\times h\times n\times p}$ . Net2WiderNet substitutes the ith convolutional layer with a convolutional layer whose output channel is q(q > n). Introduce the random mapping function g : { 1,2 , ... , q } → { 1,2 , ... , n } $g:\left\{1,2,...,q\right\}\to \left\{1,2,...,n\right\}$ , and satisfy g ( j ) = j j ≤ n random sample from { 1,2 , ... , n } j > n $g(j)=\left\{\begin{array}{l}j j\le n\hfill \\ \mathrm{random}\,\mathrm{sample}\,\mathrm{from}\left\{1,2,...,n\right\}\quad j > n\hfill \end{array}\right.$ (10) The weights of the ith and i+1th layers in the new widened student network are represented by U(i) and U(i+1), and the new weights satisfy U : , : , : , j ( i ) = W : , : , : , g ( j ) ( i ) U : , : , j , : ( i + 1 ) = 1 | { x | g ( x ) = g ( j ) } | W : , : , g ( i ) , : ( i + 1 ) $\begin{array}{c}{U}_{:,:,:,j}^{(i)}={W}_{:,:,:,g(j)}^{(i)}\\ {U}_{:,:,j,:}^{(i+1)}=\frac{1}{\vert \left\{x\vert g(x)=g(j)\right\}\vert }{W}_{:,:,g(i),:}^{(i+1)}\end{array}$ (11) The new convolution kernel weight in U(i) is created by selecting the random number defined in g, so every convolution kernel in W(i) may be copied. Each convolution kernel in U(i+1) is divided by the repetition factor 1 / | x | g ( x ) = g ( i ) | $1/\vert x\vert g(x)=g(i)\vert $ , to calculate the weight of the new convolution kernel. The parameter transformation generates a larger student network without compromising the recognition accuracy of the original teacher model. 2.3.2 Net2DeeperNet Net2DeeperNet can transform the teacher network model into a deeper network. Specifically, Net2DeeperNet replaces h(i) = ϕ1(h(i−1)) with two convolutional layers h(i) = ϕ2(ϕ1(h(i−1))). For any input features v, the new convolutional layer should satisfy ϕ2(v) = v. As a result, the additional convolutional layer maintains the input and output identity. Assuming that the weight of the added convolutional layer with the number of input and output channels n is U(i), the new weight satisfies U c , c , i , i ( i ) = 1 ${U}_{c,c,i,i}^{(i)}=1$ (12)where c denotes the coordinates of the centre point of the convolution kernel, and the size of the convolution kernel is odd. The channel index is represented by i. The value of the coordinate that satisfies Equation (12) is 1 and the bias and value of all other coordinates is 0 as shown in Figure 5. Net2DeeperNet increases the number of convolutional layers without sacrificing performance of the original teacher model. FIGURE 5Open in figure viewerPowerPoint The identity convolutional layer used in Net2DeeperNet Model design The specific model architecture is introduced in this section. The input data size of the model is 448 × 448. All convolutional layers use a 3 × 3 size convolution kernel and use a rectified linear unit as the activation layer. In the construction of the teacher model, the feature extractor consists of 7 stages, and each stage contains [1, 2, 2, 2, 2, 2, 1] convolutional layers, respectively. The number of channels in each stage is [32, 32, 64, 64, 128, 128, 256]. The average pooling layer with a step of two and a kernel size of two is employed at the end of each stage. The convolutional layers in the model generated by Net2WiderNet contain more convolution kernels and channels. In this paper, the wider student model increases the number of channels in each stage of the teacher model to [32, 64, 64, 128, 128, 256, 256]. Table 2 lists the specific model information of the wider student model. TABLE 2. The architecture of the Net2WiderNet model Teacher model Wider student model Stage1 [Conv3-32] × 1 [Conv3-32] × 1 Stage2 [Conv3-32] × 2 [Conv3-64] × 2 Stage3 [Conv3-64] × 2 [Conv3-64] × 2 Stage4 [Conv3-64] × 2 [Conv3-128] × 2 Stage5 [Conv3-128] × 2 [Conv3-128] × 2 Stage6 [Conv3-128] × 2 [Conv3-256] × 2 Stage7 [Conv3-256] × 1 [Conv3-256] × 1 Global average pooling Global average pooling Feature normalisation Feature normalisation Imprint layer Imprint layer Parameters 0.96 M 2.13 M FLOPs 1.92 M 4.25 M Each stage of the model generated by Net2DeeperNet contains additional convolutional layers. In this paper, the deeper student model increases the number of convolutional layers in each stage of the teacher network from [1, 2, 2, 2, 2, 2, 1] to [1, 4, 4, 6, 4, 4, 1]. The specific model information is shown in Table 3. TABLE 3. The architectu