# Multi-View Synthesis of Sparse Projection of Absorption Spectra Based on Joint GRU and U-Net
## 2 Mathematical Background
### GRU Module for Absorbance Data  
The gate recurrent unit (GRU) network is a kind of recurrent neural network (RNN). It deals with time series data. It can solve some complexity and computational overhead problems existing in the long short-term memory (LSTM) network [27; 28]. It can also simplify the model structure to some extent and improve the training efficiency. For example, in the case of the stationary TDLAS temperature measurement system, the measurement of the combustion temperature field involves the measurement of the region of interest from multiple angles. The obtained projections represent the two-dimensional datasets obtained by the laminar scanning process. Each row of these data represents the projection of the laser absorption spectrum obtained by the detector at increasing angles. The features are sinusoidal throughout the image. Along a particular sinusoidal curve, the gray values of each detector element show a strong correlation. Therefore, we can treat each row of projected data as a sequence over the projected angle. The GRU network shows excellent performance in processing sequential data, so the inherent sequential nature and sinusoidal correlation of projection data make it ideal for synthesizing sparse projection data. The structure of the GRU network cell is shown in Figure 2; it controls the flow of information through the network by introducing two gating mechanisms, namely the reset gate and the update gate, and updating then. These gating mechanisms help the GRU network to better capture the intrinsic relationships in the sequence of lines and rows of the projected data, with a smaller number of parameters compared to the LSTM network. The flow of the projected data within the module is as follows:  
At the \(t\)th projection angle, Equation (5) defines the variable \(R_{t}\) as the gate of the control reset of the projection data:  
\[R_{t}=\sigma(X_{t}W_{xt}+H_{t-1}W_{hr}+b_{r}) \tag{5}\]  
where \(X_{t}\) is the absorbance data at the \(t\)th projection angle; \(H_{t-1}\) is the hidden state at the \((t-1)\)st projection angle, and this hidden state contains the relevant information about the previous projection angle; \(W_{xt}\) and \(W_{hr}\) denote the weight matrices corresponding to \(X_{t}\) and \(H_{t-1}\), respectively, at the input of the reset gate; \(b_{r}\) is the bias matrix of the reset gate; and \(\sigma\) is the activation function of the reset gate, which is essentially a nonlinear mapping sigmoid function that transforms the data into the range of 0-1 and acts as a gating signal. The variable \(Z_{t}\) is the gate for the control update of the absorbance data at the \(t\)th projection angle, which is defined by Equation (6):Similarly, \(\sigma\) is the activation function of the update gate; \(X_{t}\) and \(H_{t-1}\) are used as the inputs of the update gate; \(W_{xz}\) and \(W_{hz}\) denote the weight matrices corresponding to \(X_{t}\) and \(H_{t-1}\) at the input of the update gate, respectively; and \(b_{z}\) is the bias matrix of the update gate. In Equation (7), the variable \(H_{t}^{\prime}\) is the candidate hidden state at the projection angle \(t\) and is defined as follows:  
\[H_{t}^{\prime}=\tanh(X_{t}W_{xh}+(R_{t}\odot H_{t-1})W_{hh}+b_{h}) \tag{7}\]  
where the data after the reset are obtained by resetting the Hadamard product \(R_{t}\odot H_{t-1}\) of the gate \(R_{t}\) and the hidden state \(H_{t-1}\) of the projection angle. \(W_{xh}\) and \(W_{hh}\) denote the weights of the current input \(X_{t}\) and the reset data, respectively, and \(b_{h}\) is the bias for calculating the candidate hidden state at the projection angle \(t\). The data are deflated to the range of \(-1\)\(\sim\)\(1\) by the tanh activation function to obtain the candidate under the current projection angle \(t\) of the hidden state \(H_{t}^{\prime}\).