基于约束优化的多光谱辐射真温反演算法

多光谱辐射测温是通过测量待测物某点的多个光谱辐射强度信息，通过普朗克公式反演获得真实温度。但是，通过普朗克公式获得的多光谱辐射测温方程组，是欠定方程组，即N个方程，N+1个未知数(N个未知的光谱发射率ε_λi和1个待求真温T)。目前，多采用事先假设一组发射率模型(发射率-波长或发射率-温度模型)，假设模型与实际情况如果相符，则反演结果能够满足要求，如果假设模型与实际情况不符，则反演结果误差很大。但是，发射率模型受温度、表面状态、波长等诸多因素影响，难以事先确定发射率模型。因此受未知光谱发射率的制约一直是多光谱辐射测温理论面临的主要障碍，能否在无需任何光谱发射率假设模型的情况下，实现真温和光谱发射率的直接反演一直是多光谱辐射测温理论研究的热点和难点。通过对参考温度模型的分析表明，多光谱辐射测温反演过程的实质是寻找一组光谱发射率，使得每个通道方程解得的真温都相同，如不相同则继续寻找合适的光谱发射率，直到每个通道解得的真温都相等。为此，提出将多光谱辐射测温参考温度模型的求解过程转换为约束优化问题，即在光谱发射率0≤ε_λi≤1的约束条件下，通过梯度投影算法不断寻找光谱发射率，带入多光谱辐射测温参考温度模型方程组后，计算温度反演值的方差，直到每个光谱通道方程获得的温度值应该近似相等，此时各个光谱通道的温度反演值方差最小，这样就把多光谱辐射真温和发射率的反演问题转换为约束优化问题。约束优化算法是解决这一类问题的主要方法，但为了满足Ax≥b的约束条件，将0≤ε_λi≤1分解为ε_λi≥0和-ε_λi≥-1的两个约束条件，从而满足了约束优化问题Ax≥b的约束条件。这样就可以通过约束优化算法在无需任何光谱发射率假设模型的条件下，直接求解真温和光谱发射率。实验采用六种不同光谱发射率分布模式(随波长递增、递减、凸波动、凹波动、“M”型波动、“W”型波动)的材料为研究对象，以验证新算法对不同材料光谱发射率分布反演的适应性，利用Matlab的minRosen函数，选择光谱发射率的初始值均为0.5(取中间值，提高计算效率)。针对六种不同光谱发射率模型的仿真结果表明，新算法无需任何有关发射率的先验知识，对不同发射率模型反演结果均表现较好，在真温1 800 K的情况下，绝对误差均小于20 K，相对误差均小于1.2%，新算法具有无需考虑任何光谱发射率先验知识、反演精度较高及适合于各种发射率模型等优点，进一步完善了多光谱辐射测温理论，在高温测量领域具有良好的应用前景。

Multi Spectral True Temperature Inversion Algorithm Based on Constrained Optimization Method

Multi-spectral radiation thermometry is a method for obtaining true temperature inversion by Planck formula by measuring a number of spectral radiant intensity information of a point to be measured. However, the multi-spectral radiation thermometry equations obtained by the Planck formula are the underdetermined equations, that is, N equations, and N+1 unknown (N unknown spectral emissivity ε_λi and a waiting true temperature T). At present, a set of emissivity models (emissivity-wavelength or emissivity-temperature model) are often assumed. If the assumed model is consistent with the actual situation, then the inversion results can meet the requirements, and if the assumed model does not match with the actual situation, the inversion result will have very large errors. The emissivity model is affected by many factors, including temperature, surface state, wavelength and so on. It is difficult to determine the emissivity model in advance. Therefore, the restriction of unknown spectral emissivity has been the major obstacle to the theory of multi-spectral radiation thermometry. The direct inversion of true temperature and spectral emissivity without any assumption of spectral emissivity has always been a hot and difficult issue in the theory of multi-spectral radiation t thermometry. Through the analysis of reference temperature model, the essence of multispectral radiation thermometry inversion is to find a set of spectral emissivity, so that the true temperature of each channel equation is the same, if different, we’ll continue to find the appropriate spectral emissivity, until the true temperature of each channel is equal. Therefore, it is proposed to convert the solving process of the multi-spectral radiation thermometry reference temperature model into a constrained optimization problem. That is, under the constraint of spectral emissivity 0≤ε_λi≤1, the spectral emissivity is constantly searched through the gradient projection algorithm. After taking the emissivity into the multi-spectral radiation thermometry reference temperature model equations, the variance of the temperature inversion values is calculated until the temperature values obtained for each of the spectral channel equations should be approximately equal. In this case, the variance of the temperature inversion value of each spectral channel is the smallest, so that the inversion problem of true temperature and emissivity of multi-spectral radiation is converted into a constrained optimization problem. The gradient projection method is the main method to solve this kind of problem. However, in order to satisfy the constraints of Ax≥b, we decompose 0≤ε_λi≤1 into two constraints of ε_λi≥0 and -ε_λi≥-1, so as to satisfy constraints optimization problem of Ax≥b. In this way, the true temperature and spectral emissivity can be directly solved by the gradient projection algorithm without any spectral emissivity assumption. Six kinds of materials with different spectral emissivity distribution modes (increasing, decreasing, convex wave, concave wave, “M” wave and “W” wave) were selected as the research objects, to verify the adaptability of the new algorithm to the inversion of the spectral emissivity distribution of different materials. Using the minRosen function of Matlab, the initial values of the spectral emissivity are all chosen to be 0.5 (taking the middle value to improve the computational efficiency). The simulation results of six different spectral emissivity models showed that the new algorithm does not require any prior knowledge of the emissivity and the inversion of different emissivity models by the new algorithm is better. The absolute error is less than 20 K and the relative error is less than 1.2% when the true temperature is 1 800 K. The new algorithm has the advantage that there is no need to consider any prior knowledge of spectral emissivity, high inversion precision and suitable for various emissivity models. It further improves the theory of multi-spectral radiation thermometry and has a good prospect in the field of high temperature measurement.