## 2 Methods  
### Multispectral Radiometric Thermometry  
Conventional high-speed cameras and thermal imagers often experience low temperature measurement accuracy due to their strong dependence on surface emissivity, which is typically challenging to determine with precision [19]. In contrast, multispectral radiation thermometry, which retrieves temperature information using more than three spectral bands, can significantly mitigate the impact of emissivity uncertainty. This method enhances the robustness and accuracy of temperature estimation, particularly in complex radiative environments such as forest fires [20]. [PERSON]'s blackbody radiation law describes the relationship between the radiant exitance of a blackbody and its wavelength at a certain temperature (\(T\)) and can be expressed by Equation (1):  
\[M_{b}(\lambda,T_{b})=C_{1}\lambda^{-5}(e^{\frac{C_{2}}{T_{b}}}-1)^{-1} \tag{1}\]where \(M_{b}(\lambda,T_{b})\) is the blackbody radiantitance, \(\lambda\) is the wavelength, \(T_{b}\) is the blackbody temperature in K, and \(C_{1}\) and \(C_{2}\) are the first and second radiation constants, which are usually taken as \(C_{1}=3.7418\times 10^{-16}\) W \(\cdot\) m\({}^{2}\) and \(C_{2}=1.4388\times 10^{-2}\)m \(\cdot\) K.  
The radiant brightness \(L_{b}(\lambda,T_{b})\) of a blackbody can be obtained as follows:  
\[L_{b}(\lambda,T_{b})=\frac{C_{1}}{\pi}\cdot\lambda^{-5}\cdot(e^{\frac{C_{2}}{ \lambda T_{b}}}-1)^{-1} \tag{2}\]  
When \(e^{\frac{C_{2}}{\lambda T}}\gg 1\), the Wien formula can be utilized instead of [PERSON]'s blackbody law:  
\[L_{b}(\lambda,T_{b})=\frac{C_{1}}{\pi}\cdot\lambda^{-5}\cdot\mathbf{e}^{- \frac{C_{2}}{\lambda T_{b}}} \tag{3}\]  
The emissivity of a blackbody is about 1, and the emissivity \(\varepsilon(\lambda_{i},T)\) of different objects is usually less than 1. Therefore, when the multispectral camera tests an object with a temperature of \(T\), the gray value \(G_{i}\) of the radiance image output of the first channel is:  
\[G_{i}=A_{\lambda_{i}}\varepsilon(\lambda_{i},T)\frac{C_{1}}{\pi}\cdot\lambda_{ i}^{-5}\cdot\mathbf{e}^{-\frac{C_{2}}{\lambda T}} \tag{4}\]  
where \(A_{\lambda_{i}}\) is a calibration constant related to wavelength \(\lambda_{i}\) and the optical response of the multispectral camera, and \(\varepsilon(\lambda_{i},T)\) is the spectral emissivity at temperature \(T\). Joining Equations (3) and (4) provides:  
\[G_{i}=A_{\lambda_{i}}\varepsilon(\lambda_{i},T)L_{b}(\lambda,T) \tag{5}\]  
Taking logarithms on both sides of Equation (4) gives:  
\[\frac{C_{2}}{\lambda_{i}T}=\ln\frac{A_{\lambda_{i}}C_{1}\lambda_{i}^{-5}}{G_{ i}}+\ln\varepsilon(\lambda_{i},T) \tag{6}\]  
When \(x=\varepsilon(\lambda_{i},T_{i})\), \(a_{i}=-\frac{C_{2}}{\lambda_{i}}\), \(b_{i}=\ln\frac{A_{\lambda_{i}}C_{1}\lambda_{i}^{-5}}{G_{i}}\), the temperature of each channel \(T_{i}\) and the average temperature of each channel \(E(T_{i})\) can be expressed as:  
\[T_{i}=\frac{a_{i}}{b_{i}+\ln x} \tag{7}\]  
\[E(T_{i})=\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\frac{a_{i}}{b_{i}+\ln x}\biggr{)} \tag{8}\]  
According to Equation (4), the temperature of each channel should be consistent and equal to the true temperature when emissivity \(\varepsilon(\lambda_{i},T)\) is known, so the theoretical deviation of the calculated temperature for each channel should converge to 0:  
\[\sum_{i=1}^{n}[T_{i}-E(T_{i})]^{2}=0 \tag{9}\]  
\(E(T_{i})\) is the average temperatures of all channels, obtained via the standard optimization function \(min\,f(x)\):  
\[min\,f(x)=\sum_{i=1}^{n}[T_{i}-E(T_{i})]^{2}\to 0 \tag{10}\]  
Since the spectral emissivity of most objects ranges between 0.1 and 1, it can be constrained:  
\[ \begin{cases}\varepsilon(\lambda_{i},T)\geq 0.1\\ -\varepsilon(\lambda_{i},T)\geq-1\end{cases}  \tag{11}\]By constraining the range of emissivity, the multispectral thermometry problem can be transformed into a constrained optimization problem of the following form:  
\[\left\{