# Multi-View Synthesis of Sparse Projection of Absorption Spectra Based on Joint GRU and U-Net
## 2 Mathematical Background  
### Tunable Dide Laser Absorption Tomography  
The TDLAS technique is a line-of-sight (LOS)-based path-integrated measurement technique characterized by fast response and high sensitivity. The technique is usually combined with computed tomography (CT) for the distribution measurements of gas temperatures, concentrations, and other parameters [26]. The basic theory of the TDLAS technique, the mathematical expression of Beer-Lambert's law, can be described as:  
\[A=\int_{-\infty}^{+\infty}-\ln\biggl{(}\frac{I_{t}(v)}{I_{0}(v)}\biggr{)}dv=PXS (T)\varphi(v)L=\alpha(v)L \tag{1}\]  
In this equation, \(I_{t}(v)\) and \(I_{0}(v)\) represent the intensities of the outgoing and incoming laser beams, respectively. \(P\) is the pressure in the region of interest, while \(X\) is the concentration of absorbing molecules in the region. The linear strength of the absorbing molecules, denoted by \(S(T)\), depends on the temperature \(T\). \(L\) is the length of the optical path through which the beam is absorbed. \(\varphi(v)\) is a linear function satisfying the normalization condition for \(\int_{-\infty}^{+\infty}\varphi(v)dv\equiv 1\). \(A\) is the path-integrated absorbance.  
Figure 1 illustrates the importance of the individual physics of the TDLAT measurement system as the laser passes through the combustion field to be measured. For the non-uniform characteristics of the two-dimensional temperature distribution of the high-energy combustion field, the area to be measured can be divided into \(M\times M\) grids, and assuming that the distribution within each grid is uniform, i.e., the combustion field area is discretized, Equation (1) can be transformed into Equation (2):  
\[A_{v_{n},i}=\sum_{j=1}^{I}\bigl{[}PS(v_{n},T_{j})X_{j}\bigr{]}_{i}L_{i,j}=\sum _{j=1}^{I}\alpha_{i}L_{i,j}(i=1,2,3,4\cdots I) \tag{2}\]  
Figure 1: Schematic of the discretization method for the temperature field under measurement.  
The discretization formula of Equation (2) can be written in the form of a matrix:  
\[\left\{\begin{array}{l}A_{v_{1},1}=\alpha_{v_{2},1}L_{1,1}+\alpha_{v_{1},2}L_{1,2 }+\cdots+\alpha_{v_{i},K}L_{1,K}\\ A_{v_{1},1}=\alpha_{v_{1},1}L_{1,1}+\alpha_{v_{1},2}L_{1,2}+\cdots+\alpha_{v_{i},K}L_{1,K}\\ A_{v_{2},2}=\alpha_{v_{2},1}L_{2,1}+\alpha_{v_{1},2}L_{2,2}+\cdots+\alpha_{v_{i},K}L_{2,K}\\ A_{v_{2},I}=\alpha_{v_{2},2}L_{1,1}+\alpha_{v_{2},2}L_{1,2}+\cdots+\alpha_{v_{i},K}L_{1,K}\end{array}\right. \tag{3}\]  
\[A_{v_{n},i}=L\cdot\alpha_{v_{n}} \tag{4}\]  
where \(A_{v_{n},i}\) denotes the integral absorption value of the detection path of the \(i\)th laser beam at frequency \(v_{n}\); \(j\) denotes the \(j\)th grid number, and \(K\) is the total number of grids in the discretized field; \(L_{i,j}\) denotes the path length of the \(i\)th laser beam through the \(j\)th grid and the weight of the grid.  
Under this discretized model, the problem of reconstructing the temperature distribution in the TDLAT system can be described as follows: solve the information asymmetry between the temperature distribution \(T_{j}\) and the concentration distribution \(X_{j}\) coupled according to the path integral absorption value \(A_{v_{n},i}\) and the lattice weights \(L_{i,j}\). In this paper, the two-dimensional reconstruction of the temperature field is realized by using the absorbance ratio of two absorption spectral lines according to the two-line ratio thermometry.