Random projection

In the compression process of traditional compressed sampling, a fixed measurement matrix is replaced by three sequential convolution layers owing to the substantial storage requirements of the fixed measurement matrix and the necessity to satisfy the restricted isometry property principle [25]. The convolution operation is a critical factor because it can be represented as a matrix multiplication [26]. This process is described as follows. $\begin{array}{ll}y\hfill & =\Sigma \left(w_3\Sigma \left(w_2\Sigma \left(w_1{\widehat{x}}_{\left(i\right)}+b_1\right)+b_2\right)+b_3\right)\hfill \\ \hfill & =w_3{w}_2{w}_1+w_3{w}_2{b}_1+w_3{b}_2+b_3\hfill \end{array}$ (1) Here, wi and bi (i=1,2,3) represent the weight and bias, respectively, of the first convolutional layer of the compression module, and y denotes the output. The convolution is expressed as a linear representation of the original neutron ToF signal (x). Hence, it serves as a measurement matrix in compressed sampling.

The kernel size of the three convolution layers are 1 × 8, 1 × 8, and 1 × 4, with the strides of 8, 8, and 4, and the number of filters is denoted as C, where C=256 × Sampling rate, in which sampling rate is the ratio of the number of measurement sampling points to the original signal. Therefore, after the convolution operation, the length of the signal will be reduced to a quarter of its original length. Following all three consecutive convolutional layers, a measurement of size 1 × 4 × C is obtained.

Rising dimension

The measured dimensions were significantly smaller than those of the original signals. Therefore, the measurement dimensions must be increased to match the original signal before initiating the reconstruction of the initial measurement. The most effective way to achieve this is by employing a convolution layer with a kernel size of 1 × 1, stride of 1, and 256 filters, and then a leaky rectified linear unit (LeakyReLU) activation function. This nonlinearity ensures that the measurements match the initial dimensions. After the initial signal passed through the first convolution layer, the measurement dimensions were 1 × 4 × 256 pixels. Subsequently, a reshaping layer was applied to convert the signal from 1 × 4× 256 to 1 × 1024 × 1.

Initial reconstruction

The inception module is a powerful tool for learning signal features at multiple scales, making it an essential component of neutron ToF signal analyses. With a design concept that combines multiscale feature extraction, dimension reduction, and expansion, the neutron ToF signal is time-sequential, with different stages exhibiting multiscale characteristics. This technology can simultaneously capture features at different scales by using convolutional kernels of various sizes. This multiscale feature extraction enables the network to simultaneously focus on local and global information, thereby capturing signal characteristics.

Therefore, we highlighted the challenges in determining the optimal convolutions to achieve the best convolutional effect for the convolutional layers. It is vital to incorporate convolutional layers of various sizes into an inception module. The network learns features of different dimensions using convolution kernels of various sizes (1 × 3, 1 × 5, 1 × 7, and 1 × 13), thereby gathering information from distinct perceptual domains. This approach can significantly improve the network performance. Skip connections were used in convolution to prevent a decline in network performance and enhance the reconstruction accuracy. This powerful technique is illustrated in Fig. 4.

Fig. 4Full-screen View

Schematic of skip connection

Moreover, the skip connections employed in signal processing do not increase the network parameters or computational complexity because they involve the addition of feature maps rather than multiplication. Therefore, skip connections are well suited for signal processing. The overall process is as follows: $x_{\text{output}}^{\left(n\right)}=\sigma \left(f_2^{\left(n\right)}\left(\sigma \left(f_1^{\left(n\right)}\left(x_{\text{input}}^{\left(n\right)}\right)\right)\right)+x_{\text{input}}^{\left(n\right)}\right)$ (2) Here, x_input and x_output represent the input and output, respectively, in the nth line (n=1,2,3,4); $f_1^{\left(n\right)}$ and $f_2^{\left(n\right)}$ represent the first and second convolution, respectively, in the nth line; and σ represents the LeakyReLU function.

When the inception module outputs four branches, feature maps are stacked together through a concatenation operation, and the integrated feature maps are mapped to be consistent with the original neutron ToF signal through a 1 × 1 convolution kernel. This process can be expressed as follows: $\begin{array}{c}\text{Con}\_x_{\text{inception}}=F(\text{Concat}(x_{\text{output}}^{\left(1\right)}\mathrm{,}{x}_{\text{output}}^{\left(2\right)}\mathrm{,}\\ x_{\text{output}}^{\left(3\right)}\mathrm{,}{x}_{\text{output}}^{\left(4\right)}))\end{array}$ (3) Here, Concat(·) represents an operation that integrates four branches to output feature graphs, F(·) represents the 1 × 1 convolution operation, and Con_x_inception represents the feature graphs extracted by the inception module.

Recurrent neural network (RNN) is a deep neural network extensively employed for modeling time-series data. LSTM, a variant of the RNN, effectively addresses the gradient explosion or vanishing problems encountered in RNNs. To the best of the our knowledge, LSTM [27] has not yet been applied to the compression sampling of neutron ToF signals. In this regard, LSTM was used to extract local temporal features from the initial reconstruction signal output of the inception module, thereby improving the overall performance of signal reconstruction.

The LSTM unit comprised various components including a storage unit, an input gate, an output gate, and a forgetting gate. The activation and current-status updates for each gate were calculated as follows: $i_t=\sigma \left(W_{\text{f}}\left[h_{t-1}\mathrm{,}{x}_t\right]+b_i\right)$ (4) $f_t=\sigma \left(W_{\text{f}}\left[h_{t-1}\mathrm{,}{x}_t\right]+b_f\right)$ (5) ${\tilde{c}}_t=\sigma \left(W_{\text{f}}\left[h_{t-1}\mathrm{,}{x}_t\right]+b_c\right)$ (6) $c_t=f_t\odot c_t+i_t\odot {\tilde{c}}_t$ (7) $i_t=\sigma \left(W_{\text{i}}\left[h_{t-1}\mathrm{,}{x}_t\right]+b_{\text{i}}\right)$ (8) $h_t=o_t\tan h\left(c_t\right)$ (9) Here, it represents the input gate calculated using the sigmoid activation function, xt represents the input data, and h_t-1, which is a hidden node, is the output. Furthermore, ft represents the forgotten gate that determines the c_t-1 features used in the ct calculation. The neural network layer determines the updated value of the cell state, ${\tilde{c}}_t$. Then, the activation function tan h is applied to xt and h_t-1; it selects the value ${\tilde{c}}_t$ from the feature to update ct. The output gate is calculated in the same manner as the input gate. Additionally, ht is calculated using ot and ct, where σ represents the sigmoid activation function, and W_i and b_i are the weight and bias of the corresponding layer, respectively.

The LSTM network had a hidden layer of 250 cells, and the activation function was tan h, which is linear. The dense-layer output was 1024, and the final output dimensions were 1024 × 1. Finally, through the initial reconstruction module, the different captured features were integrated and mapped to keep the original neutron ToF signal dimensions consistent. This process can be expressed as follows: $x_{\text{output1}}=F\left(\text{Con}\_x_{\text{output}}\right)$ (10) Here, F(·) represents the convolution operation, and x_output1 represents the initial reconstruction signal.

Secondary reconstruction

Attention mechanisms have revolutionized computer vision, allowing targeted focus on critical regions of an image, while disregarding irrelevant parts. The application of attention mechanisms has led to significant progress in various aspects of computer vision, such as image classification, object detection, and semantic segmentation. The human brain has a similar cognitive process: quickly identifying crucial areas within complex visual scenes and processing them in greater detail. This cognitive process within the visual system is represented by the following equation: $\text{Attention}=f\left(g\left(x\right)\mathrm{,}x\right)\mathrm{,}$ (11) where x denotes the input data; $g\left(x\right)$ denotes the features extracted from the input data and the attention obtained; and $f\left(g\left(x\right)\mathrm{,}x\right)$ denotes the attention generated to process the input data. With respect to the self-attention [28] mechanism, Eq. (11) evolves as follows: $Q\mathrm{,}K\mathrm{,}V=\text{Linear}\left(x\right)$ (12) $g\left(x\right)=\text{Soft}\mathrm{ max}\left(QK\right)$ (13) $f\left(g\left(x\right)\mathrm{,}x\right)=g\left(x\right)V$ (14) Designing different $g\left(x\right)$ and $f\left(x\right)$ produces different attention mechanisms.

Therefore, self-attention is a valuable tool for assisting models in understanding the connections between different positions within a sequence. It complements convolution by facilitating the modeling of long-range and multilevel dependencies among signal regions. The approach used by the model involves encoding each element of the sequence and analyzing the similarity between them to determine the most critical and contextually relevant elements.

Ultimately, the reconstruction network primarily comprises a self-attention module that plays a pivotal role in signal reconstruction by effectively capturing local temporal information and positional relationships. This technique enhances the quality of the reconstruction. This process is expressed as follows: $\widehat{x}=\text{SA}\left(x_{\text{output1}}\right)\mathrm{,}$ (15) where SA(·) represents the self-attention operation, and $\widehat{x}$ represents the final reconstructed neutron ToF signal.

Evaluation of reconstruction performance

This section compares ISACSNet with different types of traditional reconstruction algorithms to determine the reconstruction accuracy. Using the Bernoulli random matrix as the measurement matrix, we employed PRD and correlation coefficient to assess the reconstruction error of the neutron ToF signal.

Figure 6 illustrates the average PRD of the neutron ToF signal test sets under different sampling rates for various methods. The x-axis represents the sampling rate, and the y-axis represents the average percentage of residuals. Our approach consistently achieved the lowest PRD values among the six methods, indicating that it consistently exhibited the lowest reconstruction errors and maintained a relative stability. Under the same sampling ratio conditions, the reconstruction performance of the proposed method surpassed that of the other methods. For instance, at a sampling ratio as low as 10%, our method outperformed (PRD value: 47.08%) other popular methods, such as IHT, BP, SAMP, T-MSBL, and FOCUSS, which exhibited PRD values of 31.64%, 27.31%, 28.93%, 37.43%, and 3.81%, respectively. As shown in Fig. 6, the performance of the DL-CS method is significantly better than those of the traditional compressed sampling methods at low sampling ratios. For example, at a sampling ratio of 1%, the PRD of the proposed method is only 9.17%.

Fig. 6Full-screen View

PRDs of different reconstruction algorithms

The correlation coefficients for various reconstruction algorithms when reconstructing a test signal of size 1024 with different sampling ratios are provided in Fig. 7 and Table 2. The proposed method achieved a correlation coefficient of 0.9715 at a sampling ratio of 1%, whereas the correlation coefficients for the other methods were all below 0.2. Additionally, as the sampling ratio increased, our approach tended to stabilize, maintaining correlation coefficient values above 0.9715, outperforming the other methods. This approach highlights its advantages at low sampling ratios, demonstrating its effectiveness in achieving good reconstruction results.

Fig. 7Full-screen View

Correlation coefficient of different reconstruction algorithms

Figure 8 illustrates the original and reconstructed signals obtained using the different reconstruction methods at a sampling ratio of 10%. Our proposed reconstruction method outperformed traditional compressed sensing methods at a sampling rate of 10%, with a PRD of 5% and correlation coefficient of 0.998 (Fig. 8) and showing a better reconstruction performance with the reconstructed signal peaks closer to the original signal.

Fig. 8Full-screen View

Reconstructed neutron ToF signals of various reconstruction algorithms at a sampling ratio of 10%

Investigation of reconstruction time

The reconstruction time of a signal plays a critical role in neutron ToF compressed sampling. We reconstructed 100 signals from the test set by using different algorithms at various sampling rates. We also evaluated the reconstruction performances of the different algorithms by comparing their average reconstruction times.

Table 3 and Fig. 9 show the reconstruction times of the different test algorithms at different sampling rates. As shown in Fig. 9, the proposed ISACSNet method had the shortest and most stable reconstruction times. As shown in Table 3, for sampling ratios in the range of 1–50%, the proposed method required a reconstruction time of 0.0108–0.0136 s.

Fig. 9Full-screen View

Reconstruction time of different reconstruction algorithms at different sampling rates

At the same sampling rate, our proposed algorithm could reconstruct 1024 signals, which is one to four orders of magnitude faster than the other methods. Our proposed method fully exploits the computational power of the graphics card, resulting in shorter processing times compared to traditional methods that rely solely on CPU computation. This advantage is evident because traditional iterative methods struggle with parallel transformation when solving optimization problems. Each layer of deep learning is designed for parallel computation. By leveraging the parallel computing power of the GPU, our method enables high-speed inference. Even if traditional generation selection methods can be parallelly modified and executed on a GPU, they cannot match the efficiency of deep learning owing to the numerous assumptions and limitations in their theoretical design. Therefore, the deep learning reconstruction network proposed in this study capitalizes on hardware parallelization, resulting in significantly shorter reconstruction times compared to traditional compressed sampling methods.