# Dual-Channel Mapping-Gas Column Concentration Inversion Method Based on Multispectral Imaging
## 2 Method
### Dual-Channel Mapping-Gas Column Concentration Inversion Algorithm Based on Multispectral Imaging  
In order to solve the problem whereby background radiation is difficult to obtain in gas detection, a dual-channel mapping-gas column concentration inversion algorithm is proposed in this paper. In this algorithm, the absorption spectrum band of the target gas is taken as the target spectral channel, the absorption spectrum band of the non-target gas is taken as the reference spectral channel, and the target gas is imaged through two spectral channels, respectively. According to Equation (2), the absorbance of the gas can be obtained from the grayscale image of the target gas radiation and the grayscale image of the background radiation. Part of the background radiation will be absorbed after passing through the target gas, so the background radiation cannot be obtained by direct measurement. Therefore, we use the infrared absorption characteristics of the gas and the background radiation calibration curve to convert the background gray image of the reference spectral channel into a background temperature image and, finally, into a background grayscale image of the target spectral channel. A schematic diagram and flow chart of this method are shown in Figures 1 and 2. The gray level information output by the detector can be expressed as [32; 33]  
\[DN=\int\limits_{\lambda_{2}}^{\lambda_{1}}\frac{\pi A_{d}}{4F^{2}}\frac{ \lambda}{hc}\eta(\lambda)T_{int}\frac{c_{1}}{\lambda^{5}}\frac{1}{exp\Big{(} \frac{c_{2}}{\lambda T_{d}}\Big{)}-1}d\lambda \tag{3}\]  
where \(DN\) is the output gray value of the detector; \(A_{d}\) is the pixel area of the detector; \(F\) is the F-number of the optical system; \(\lambda/hc\) is the number of photons per unit energy in the narrow band with a wavelength of \(\lambda\); \(\eta(\lambda)\) is the quantum efficiency of the detector; \(T_{int}\) is the integration time of the detector, \(c_{1}\) and \(c_{1}\) are the first and second radiation constants, respectively (according to research, it is known that \(c_{1}=3.7418\times 10^{-12}\) Wcm\({}^{2}\)  
Figure 1: Schematic diagram of gas column concentration inversion. and \(c_{2}=1.438786\) K \(\cdot\) cm [34]); and \(T_{i}\) is the temperature of the object. In the spectral range \(\lambda_{1}\sim\lambda_{2}\), the surface source blackbody under different temperatures is imaged several times, and the calibration function can be obtained by combining the temperature of the blackbody \(T_{i}\) and the output signal of the detector \(DN\). All the constants in Equation (3), except \(T_{i}\) and \(DN\), are constant. When \(\lambda_{1}\) and \(\lambda_{2}\) are relatively close, Equation (3) is simplified. When the background temperature is \(T_{B}\), the grayscale image of background radiation under the target spectral channel \(DN(T_{B})\) is  
\[DN(T_{B})=\frac{B_{1}}{\textit{exp}\Big{(}\frac{R_{1}}{T_{B}}\Big{)}-G_{1}} \tag{4}\]  
where \(B_{1}\), \(R_{1}\) and \(G_{1}\) are the three parameters obtained after fitting. If we switch the spectral channel to the reference spectral channel, similarly, the background radiation gray level image \(DN^{\prime}(T_{B})\) is  
\[DN^{\prime}(T_{B})=\frac{B_{2}}{\textit{exp}\Big{(}\frac{R_{2}}{T_{B}}\Big{)} -G_{2}} \tag{5}\]  
where \(B_{2}\), \(R_{2}\) and \(G_{2}\) are the calibration parameters under the reference spectral channel, respectively. \(DN(T_{B})\) can be expressed as follows according to Equations (4) and (5):  
\[DN(T_{B})=\frac{B_{1}}{\textit{exp}\Big{(}\frac{R_{1}}{R_{2}}\cdot\textit{ln }\Big{(}\frac{B_{2}}{DN^{\prime}(T_{B})}+G_{2}\Big{)}-G_{1}\Big{)}} \tag{6}\]  
Because the spectral band where \(DN^{\prime}(T_{B})\) is refers to the reference spectral channel, the acquisition of \(DN^{\prime}(T_{B})\) is not affected by the target gas. Equation (6) can obtain the background gray image under the target spectral channel which is not affected by the target  
Figure 2: Flow chart of gas column concentration inversion. gas.