Approach to multispectral thermometry with Planck formula and hybrid metaheuristic optimization algorithm

Baolin Zhao; Kaihua Zhang; Kaihua Zhang; Longfei Li; Yinxin He; Kun Yu; Yufang Liu; Yufang Liu; Yufang Liu

doi:10.1364/OE.503423

1. Introduction

Temperature is an important parameter of material state, and accurate temperature measurement has significant implications for product quality, industrial process control, and scientific research [1–4]. In recent years, multispectral thermometry as a non-contact temperature measurement technology has attracted much attention. Compared with the traditional contact temperature measurement method, multispectral thermometry has the characteristics of not damaging the temperature field of the target to be measured, high response rate, convenience and flexibility. It has been widely applied in various fields, including the assessment of working conditions in aerospace engine blades [5–7], temperature monitoring in steel metallurgy processes [8,9], and temperature monitoring of explosives or combustion materials [10].

In the process of radiation temperature measurement, the spectral emissivity of the object is an important factor that affects the accuracy and reliability of temperature measurement. During temperature inversion, the emissivity is usually unknown, posing a challenge to accurate temperature determination. The traditional method is to assume the emissivity model [5,11–13]. By summarizing and inducting the known material emissivity, the mathematical model of emissivity changing with wavelength is established. Then the object's temperature is obtained using methods such as the least squares method. However, in practical applications, the defined emissivity model by mathematical may not fully match the actual spectral emissivity characteristics of the object surface, leading to errors in the inversion temperature. To address this issue, some researchers have proposed the “secondary measurement method" [14–19]. This method does not rely on the relationship between emissivity and wavelength, but assumes that there is a linear relationship between emissivity and the true temperature of the object within a specific range. However, no matter what kind of mathematical model is established, there are always significant limitations in practical applications.

Considering the complexity and variability of emissivity, some researchers have introduced neural networks into multispectral thermometry. Using the learning ability and adaptive characteristics of neural network, a highly accurate emissivity models is obtained by nonlinear mapping, and good results have been got in multispectral thermometry [20]. However, the training of neural network needs a large number of samples and data labeling, and the untrained emissivity model has poor effect on temperature inversion. In order to eliminate the influence of emissivity on multispectral thermometry, researchers have introduced optimization theory and proposed a multispectral thermometry without emissivity model. They transform the problem of solving the target surface temperature into an optimization problem with the minimum deviation between the calculated temperature and the actual temperature at each wavelength, and use different optimization algorithms to solve it [20–25]. For example, Xingjian et al. proposed multiple optimization algorithms to solve multispectral thermometry [7,26–28], including the gradient projection (GP) algorithm, the interior penalty function (IPF) algorithm, the GIM-LSTM algorithm which combines the generalized inverse matrix (GIM) with the long short-term memory (LSTM) neural network algorithm, and the GIM-EPF algorithm which combines the generalized inverse matrix (GIM) with the exterior penalty function (EPF) algorithm. Zhang Yuchun proposed a ridge estimation-sequence quadratic programming constraint optimization algorithm for data processing in multi-spectral thermometry [29]. Wang Nian proposed a constrained optimization algorithm based on an exterior penalty function (EPF), achieving high-precision temperature measurement [30].

Although many researchers have used different optimization algorithms to obtained the true temperature in high-temperature, they all use Wien approximation to simplify the problem when constructing the optimization objective function. However, the Wien's approximation is suitable for short wavelengths or low-temperature application scenarios and may not meet all application scenarios requirements. Therefore, in this paper, we construct the objective function by deriving the Planck formula directly. Additionally, in order to reduce the amount of calculation in the optimization process, a feasible region constraint method is proposed to constrain the emissivity; A hybrid metaheuristic algorithm (3DE-MPG) combining differential evolution (DE) algorithm with three different variation operators and multi-population genetic (MPG) algorithm is designed to solve the true temperature. In this study, we demonstrate the effectiveness of 3DE-MPG algorithm by simulating six different emissivity models and experimenting with silicon carbide spectral data. Combined with the constructed objective function, the necessity of selecting hybrid meta heuristic algorithm is demonstrated, which provides a reliable temperature measurement method for the application of multispectral thermal measurement technology.

2. Principles of multispectral thermometry

2.1 Reference temperature model

According to Planck's blackbody radiation law, the spectral radiant emittance M(λ,T) of a blackbody with temperature T at a wavelength λ is given by

(1)$$\; M({\lambda ,T} )= \varepsilon ({\lambda ,T} )\frac{{{c_1}}}{{{\lambda ^5}}}\frac{1}{{{e^{{c_2}/({\lambda T} )}} - 1}}$$

where c₁= 374177185.22 (W·μm⁴·m²) is the first radiation constant, c₂ = 14387.77 (μm·K) is the second radiation constant and ε(λ,T) is the emissivity of the blackbody. Generally, the emissivity of blackbody is 1 by default.

For multispectral radiation thermometer with n wavelength channels, the object output signal V_i of the i-th channel can be given:

(2)$${V_i} = A({{\lambda_i}} )\varepsilon ({{\lambda_i},T} )\frac{{{c_1}}}{{\lambda _i^5}}\frac{1}{{{e^{{c_2}/({{\lambda_i}T} )}} - 1}}$$

where A(λ_i) is the response coefficient of the multispectral radiation thermometer, which is changed with wavelength λ and independent of temperature. ε(λ_i,T) is the emissivity of the object at the wavelength λ_i at temperature T, and λ_i is the wavelength of the i-th channel.

When the object is the standard radiation blackbody source, the output signal V_i’ of the i-th channel at reference temperature T’ for the same multispectral radiation thermometer can be given as:

(3)$${V^{\prime}_i} = A({{\lambda_i}} )\frac{{{c_1}}}{{{\lambda ^5}}}\frac{1}{{{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1}}$$

By dividing Eq. (2) by Eq. (3), the response coefficient of the multispectral radiation thermometer can be eliminated, and the functional relationship between the signal V_i and the true target temperature T can be obtained:

(4)$$\frac{{{V_i}}}{{{{V^{\prime}}_i}}} = \varepsilon ({{\lambda_i},T} )\frac{{{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1}}{{{e^{{c_2}/({{\lambda_i}T} )}} - 1}}$$

In Eq. (4), V_i and V_i’ can be obtained by multispectral radiation thermometer, and the temperature T’ of the reference blackbody is a known. Therefore, when the emissivity ε(λ_i,T) is determined, the actual temperature of the target can be determined accordingly. However, the emissivity of an object is affected by various factors, making it hardly to known the accurate emissivity of an object. For n-channel multispectral thermometers, the temperature of the same object at different wavelengths is the same. According to Eq. (4), n equations can be obtained, but there are n + 1 unknown parameters. This is an underdetermined system of equations, it can be transformed into a constrained optimization problem to solve this issue.

2.2 Establishment of objective function

According to Eq. (4), the temperature T_i for the i-th channel, which represents the temperature at wavelength λ_i, is shown below:

(5)$${T_i} = \frac{{{c_2}}}{{{\lambda _i}\ln \left( {\varepsilon ({{\lambda_i},T} )({{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1} )\frac{{{{V^{\prime}}_i}}}{{{V_i}}} + 1} \right)}}$$

In an ideal scenario, if the emissivity ε(λ_i,T) at the temperature of the material is known, the calculated temperature for any channel should be equal to the true temperature T of the object. Then the following relationship is existed:

(6)$$\sum\limits_{i = 1}^n {|{{T_i} - E({{T_i}} )} |} \textrm{ = }0$$

where E(Ti) is the average temperature of all channel calculated temperature and it can be shown as:

(7)$$E({{T_i}} )= \frac{1}{n}\sum\limits_{i = 1}^n {{T_i}}$$

Since the emissivity of the object at each wavelength is uncertain, the problem of solving for true temperature T can be transformed into an optimization problem, where the objective is to make the temperatures deviations of each channel as close to zero as possible. According to Eq. (6), the objective function F of the optimization problem can be established as follows:

(8)$$\min F = \frac{1}{n}\sum\limits_{i = 1}^n {|{{T_i} - E({{T_i}} )} |}$$

2.3 Constructing constraints of optimization problem

Firstly, according to the definition of emissivity, the emissivity ranges is between 0 and 1, so the following constraints exist for the emissivity of any object:

(9)$$0 < \varepsilon ({{\lambda_i},T} )< 1$$

A larger range of emissivity can increase the computational complexity of solving the optimization problem. Therefore, it is necessary to employ appropriate methods to narrow down the range of emissivity, allowing for more efficient resolution of the optimization problem. In section 2.4 of this paper, further constraints on the range of emissivity will be discussed.

Secondly, based on the relationship between the radiation signal V_i and emissivity ε(λ_i,T), it is possible to impose further constraints on the emissivity. By taking the logarithm of both sides of Eq. (4), the following expression can be obtained:

(10)$$\ln \left( {\frac{{{V_i}}}{{{{V^{\prime}}_i}}}} \right) = \ln \varepsilon ({{\lambda_i},T} )+ \ln ({{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_i}T} )}} - 1} )$$

According to Eq. (10), the relationship between the radiation signal V_i + 1 and the emissivity ε(λ_i + 1,T) of the (i + 1)-th channel can be obtained:

(11)$$\ln \left( {\frac{{{V_{i + 1}}}}{{{{V^{\prime}}_{i + 1}}}}} \right) = \ln \varepsilon ({{\lambda_{i + 1}},T} )+ \ln ({{e^{{c_2}/({{\lambda_{i + 1}}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_{i + 1}}T} )}} - 1} )$$

Subtracting Eq. (11) from Eq. (10), then:

(12)$$\begin{aligned} &\ln \left( {\frac{{{V_i}}}{{{{V^{\prime}}_i}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{{{V^{\prime}}_{i + 1}}}}} \right) - \ln \varepsilon ({{\lambda_i},T} )+ \ln \varepsilon ({{\lambda_{i + 1}},T} )\\ &=\ln ({{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_i}T} )}} - 1} )- ({\ln ({{e^{{c_2}/({{\lambda_{i + 1}}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_{i + 1}}T} )}} - 1} )} )\end{aligned}$$

In Eq. (12), set 0 < λ_i < λ_i + 1, since the temperature T’ of the reference blackbody radiation source is manually set and its value does not affect the inversion results, Therefore, we stipulate that T < T’. Order:

(13)$$f({\lambda ,T} )= \ln ({{e^{{c_2}/({\lambda T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({\lambda T} )}} - 1} )$$

It can be proven that the function f(λ,T) is monotonically increasing with the variable λ. Then, the detailed proof is as follows. First, calculating the partial derivative of f(λ,T) with respect to λ:

(14)$$\begin{aligned} \frac{{\partial f({\lambda ,T} )}}{{\partial \lambda }} &= \frac{1}{{{e^{{c_2}/({\lambda T^{\prime}} )}} - 1}}{e^{{c_2}/({\lambda T^{\prime}} )}}\frac{{ - {c_2}}}{{{\lambda ^2}T^{\prime}}} - \frac{1}{{{e^{{c_2}/({\lambda T} )}} - 1}}{e^{{c_2}/({\lambda T} )}}\frac{{ - {c_2}}}{{{\lambda ^2}T}}\\ &= \frac{1}{\lambda }\left( {\frac{{{c_2}}}{{\lambda T}}\frac{{{e^{{c_2}/({\lambda T} )}}}}{{{e^{{c_2}/({\lambda T} )}} - 1}} - \frac{{{c_2}}}{{\lambda T^{\prime}}}\frac{{{e^{{c_2}/({\lambda T^{\prime}} )}}}}{{{e^{{c_2}/({\lambda T^{\prime}} )}} - 1}}} \right) \end{aligned}$$

Defining function $g(x )= \frac{{x{e^x}}}{{{e^x} - 1}}$, where $x = \frac{{{c_2}}}{{\lambda T}}$, It is easy to determine that g(x) is monotonically increasing with respect to the variable x, When T < T’, being in ${{{c_2}} / {\lambda T}} > {{{c_2}} / {\lambda T^{\prime}}}$, $g({{{{c_2}} / {\lambda T}}} )> g({{{{c_2}} / {\lambda T^{\prime}}}} )> 0$, and $\lambda > 0$, then the left side of Eq. (17) is greater than zero, so the right side is also greater than zero, viz. Equation (14) is greater than 0. Accordingly, the function f(λ,T) is monotonically increasing with respect to the variable λ. Therefore, when 0 < λ_i < λ_i + 1, being in f(λ_i,T) < f(λ_i + 1,T), then:

(15)$$\ln ({{e^{{c_2}/({{\lambda_i}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_i}T} )}} - 1} )- ({\ln ({{e^{{c_2}/({{\lambda_{i + 1}}T^{\prime}} )}} - 1} )- \ln ({{e^{{c_2}/({{\lambda_{i + 1}}T} )}} - 1} )} )< 0$$

Thus, the following inequality can be obtained from Eq. (12):

(16)$$- \varepsilon ({{\lambda_i},T} )+ \varepsilon ({{\lambda_{i + 1}},T} )\frac{{{V_i}{{V^{\prime}}_{i + 1}}}}{{{V_{i + 1}}{{V^{\prime}}_i}}} < 0$$

Taking it as a constraint condition into the objective functional Eq. (8), the following constrained optimization problem can be constructed:

(17)$$\left\{ \begin{array}{l} \min F = \frac{1}{n}\sum\limits_{i = 1}^n {|{{T_i} - E({{T_i}} )} |} \\ - \varepsilon ({{\lambda_i},T} )+ \varepsilon ({{\lambda_{i + 1}},T} )\frac{{{V_i}{{V^{\prime}}_{i + 1}}}}{{{V_{i + 1}}{{V^{\prime}}_i}}} < 0 \end{array} \right.$$

2.4 Feasible region constraint for multispectral thermometry

According to Eq. (5), for each measured radiation signal V_i, the corresponding temperature can be calculated by using the given emissivity. Therefore, by subdividing the emissivity range into a discrete emissivity array $[{{\varepsilon_1},{\varepsilon_2},{\varepsilon_3},\ldots ,{\varepsilon_j}} ]$ with a certain spacing and substituting each emissivity value into Eq. (5), we can obtain an array of temperature $[T({{\lambda_i},{\varepsilon_1}} ),T({{\lambda_i},{\varepsilon_2}} ),\ldots ,T({{\lambda_i},{\varepsilon_j}} ),\ldots ,$ $T({{\lambda_i},{\varepsilon_q}} ) ]$. Among them, the temperature $T({{\lambda_i},{\varepsilon_j}} )$ calculated with the emissivity ${\varepsilon _j}$ that is closest to the true emissivity represents the temperature that is closest to the true temperature of the object. When the emissivity range is subdivided with higher precision, the obtained temperature $T({{\lambda_i},{\varepsilon_j}} )$ is infinitely close to the true temperature of the object. Applying the same processing method to each channel, a temperature matrix with dimensions of i × q can be obtained:

(18)$$\left[ {\begin{array}{c} {{T_{\varepsilon [ \cdots ]}}({{\lambda_1}} )}\\ {{T_{\varepsilon [ \cdots ]}}({{\lambda_2}} )}\\ \vdots \\ {{T_{\varepsilon [ \cdots ]}}({{\lambda_i}} )} \end{array}} \right] = \left[ {\begin{array}{cccccc} {T({{\lambda_1},{\varepsilon_1}} )}&{T({{\lambda_1},{\varepsilon_2}} )}& \cdots &{T({{\lambda_1},{\varepsilon_j}} )}& \cdots &{T({{\lambda_1},{\varepsilon_q}} )}\\ {T({{\lambda_2},{\varepsilon_1}} )}&{T({{\lambda_2},{\varepsilon_1}} )}& \cdots &{T({{\lambda_2},{\varepsilon_j}} )}& \cdots &{T({{\lambda_2},{\varepsilon_q}} )}\\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {T({{\lambda_i},{\varepsilon_1}} )}&{T({{\lambda_i},{\varepsilon_2}} )}& \cdots &{T({{\lambda_i},{\varepsilon_j}} )}& \cdots &{T({{\lambda_i},{\varepsilon_q}} )} \end{array}} \right]$$

Then, starting from the first row of the matrix, each row is sequentially selected as the reference row. For each element $T({{\lambda_i},{\varepsilon_j}} )$ in the reference row, the closest value among the elements in the other rows is searched and recorded in the corresponding column of the new matrix.

(19)$${T_{refer}}({referenc{e^{}}\;row = {T_{\varepsilon [ \cdots ]}}({{\lambda_1}} )} )= \left[ {\begin{array}{ccc} \cdots &{T({{\lambda_1},{\varepsilon_j}} )}& \cdots \\ \cdots &{closes{t^{}}\;t{o^{}}T{{({{\lambda_1},{\varepsilon_j}} )}^{}}i{n^{}}{T_{\varepsilon [ \cdots ]}}({{\lambda_2}} )}& \cdots \\ \cdots & \vdots & \cdots \\ \cdots &{closes{t^{}}\;t{o^{}}T{{({{\lambda_1},{\varepsilon_j}} )}^{}}i{n^{}}{T_{\varepsilon [ \cdots ]}}({{\lambda_i}} )}& \cdots \end{array}} \right]$$

Then calculate the standard deviation for each column of matrix T_refer. In theory, when the subdivision size is small enough, the mean value of the column with the smallest standard deviation can be approximate considered as the temperature of the object under that reference column. Apply the same processing method to different reference columns, and finally a total of j temperatures under different reference column can be obtained. Taking the average of these j temperatures can provide an approximate of the true temperature of the object. The corresponding ${\varepsilon _j}$ is the approximate emissivity of the object. In this paper, by selecting twice the minimum standard deviation to limit the emissivity range, the limited range of emissivity under each channel in constrained optimization can be got. The upper limit of emissivity is expressed as $({\varepsilon_{\max }^1,\varepsilon_{\max }^2,\ldots ,\varepsilon_{\max }^D} )$ and the lower limit is expressed as $({\varepsilon_{\min }^1,\varepsilon_{\min }^2,\ldots ,\varepsilon_{\min }^D} )$, Where D is the dimension of this problem. Through the constraint of feasible region, it helps to reduce the amount of calculation in the optimization process and improve the optimization efficiency.

This paper uses a set of randomly generated simulation data to illustrate the above process. The simulated data V_i has a true temperature of 1873K, while the reference temperature of the simulated data V_i’ is 1973K. When the emissivity is low, even a small deviation can result in significant inversion temperature deviations. Therefore, the emissivity range is set from 0.1 to 1, with a subdivision size of 0.01. The resulting contour map of the temperature matrix obtained by substituting the emissivity matrix into Eq. (5) is shown in Fig. 1(A). Different colors represent different temperatures, with blue indicating the lowest temperature and red indicating the highest temperature.

Fig. 1. The preprocessing steps for obtaining object temperature. A. The temperature matrix obtained by substituting the emissivity matrix into Eq. (5); B-I. The temperature matrix after reordering the columns according to Eq. (18) for different reference columns; J. The standard deviation for each wavelength.

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According to Formula 19, the temperature matrixes reordered based on different reference rows are obtained, and its contour maps are drawn in Figs. 1(B) to 1(I). The black solid line represents the true temperature isotherm, while the red solid line represents the isotherm of the mean temperature of the column in matrix T_refer corresponding to the minimum standard deviation. The temperature data in the red box is the reference row data. By calculating the standard deviation of each column, the discreteness of the column data can be reflected. A smaller standard deviation indicates a smaller deviation of the column temperature, allowing us to find an approximate isotherm at different wavelengths. Figures 1(B) to 1(I) demonstrate that the temperature obtained using the minimum standard deviation method is close to the true temperature of the object.

For the temperature matrix reordered based on different reference rows, the column standard deviations are calculated. And then find the minimum standard deviation at each wavelength, start from the maximum and minimum emissivity respectively to find the first point closest to the twice standard deviation. These two points corresponding to emissivity is the upper and lower limits of emissivity. The results are presented in Fig. 1(J), and the projection of the standard deviations on the plane is also shown in the same figure. Here, the black solid line represents the true emissivity of the object, while the red dotted line represents the emissivity corresponding to the minimum standard deviation. Additionally, the two purple solid lines represent the upper and lower limits of emissivity, respectively. Figure 1(J) reveals that the emissivity obtained using the minimum standard deviation method is similar to the true emissivity of the object. Moreover, the column standard deviations of the temperature matrix, obtained by reordering based on different reference rows, consistently converge near the true emissivity of the reference wavelength. Consequently, by choosing a reasonable range of standard deviation, we can establish a reliable emissivity range. This means that we can estimate the emissivity of the object using the minimum standard deviation method and obtain a reliable emissivity range.

3. Hybrid metaheuristic algorithms

Due to the objective function is to make the temperature deviation of each channel as close to 0 as possible, and the temperature deviations for each isotherm are indeed zero in Fig. 1(A). Consequently, the emissivity corresponding to each isotherm in Fig. 1(A) represents an optimal solution for the objective function. In other words, in the temperature solution process of multispectral temperature inversion, the emissivity corresponding to each isotherm near the true temperature of the object is the optimal solution of the objective function, which means that there are infinitely many optimal solutions for the objective function. For optimization problems with multiple optimal solutions, one commonly method is to run the algorithm multiple times with different initial operators to increase the chances of finding the true temperature of the object. Another common method is to integrate multiple algorithms. By combining multiple optimization algorithms into an integrated algorithm, using the advantages of different algorithms, comprehensively search the space to find multiple optimal solutions, and finally obtain the real temperature of the object according to the distribution characteristics of the solutions.

The hybrid metaheuristic algorithm is a method that combines the advantages of multiple metaheuristic algorithms. The basic idea is to combine multiple metaheuristic algorithms, switch or combine different metaheuristic algorithms according to certain rules or strategies in the search process, and use the advantages of each algorithm to improve the effect and efficiency of problem solving. differential evolution (DE) algorithm and multi-population genetic (MPG) algorithm are both types of metaheuristic algorithms. DE algorithm has strong global search capability and good convergence performance, while MPG algorithm has strong parallel search ability and can maintain the diversity of population in the process of searching the optimal solution. Therefore, the DE algorithm and MPG algorithm is chosen to construct a hybrid metaheuristic algorithm for solving multispectral temperature measurement problems.

3.1 Multi-population genetic algorithm

The multi-population genetic (MPG) algorithm is a variant of the conventional standard genetic algorithm (SGA) that optimizes the objective function by introducing multiple parallel populations. By setting different parameters and genetic operations for each population, the exploration ability and diversity of the MPG algorithm can be increased. For the optimization objective of multispectral temperature measurement, this will increase the possibility of finding multiple optimal solutions. The main steps of the MPG are as follows:

Step 1: Initialize the population ${P_i}$ by randomly generating individuals based on the variable ranges of the objective function. There are I populations in total, and each population has independent parameters and population size;
Step 2: Utilize the fitness evaluation function to calculate the fitness of all individuals in the population ${P_i}$.
Step 3: Select a certain number of individuals of high-fitness individuals from the population ${P_i}$ through the tournament selection scheme.
Step 4: Utilize the two-point crossover operation to generate a sufficient number of new individuals.
Step 5: Utilize the breeder GA mutation operator to generate new individuals;
Step 6: Evaluate the fitness of the newly created individuals in population ${P_i}$;
Step 7: Select individuals with high fitness based on the fitness of both new and old individuals in the population to form the next generation population. And just to note, the selection is performed within each population independently.
Step 8: Select the best individual in the population and put it into the elite population;
Step 9: Utilize the immigration operator to introduce the best individuals appearing in various populations during the evolutionary process into other populations after a certain number of evolutionary generations. The purpose of that the immigration operator introduces the best individuals from various populations into other populations is to enhance diversity of the population.
Step 10: Check termination conditions (e.g., maximum number of generations reached or satisfactory solution found). If the termination conditions are met, stop; otherwise, repeat from step 4.
Step 11: Return the best individual from the elite population.

3.2 Differential evolution algorithm

Differential evolution (DE) algorithm is proposed based on the basis of evolution ideas such as genetic algorithms [31]. Both of these algorithms are inspired by biological evolution, simulating processes such as reproduction, mutation, and selection to gradually improve individuals and approach the optimal solution. In comparison to genetic algorithms, DE algorithm uses differential vectors to guide the evolution direction of the current population. The mutation vector is generated by the parent differential vector and crossbred with the parent individual vector to generate a new individual. Differential evolution algorithms employ a direct selection strategy, where the new individuals directly compete with the parent individuals during selection. This selection strategy preserves excellent individuals and accelerates the convergence speed of the algorithm. Differential evolution algorithms have a more significant approximation effect compared to genetic algorithms, making it easier to find the optimal solution. The steps of the DE algorithm are similar to those of the MPG algorithm, but they are different in mutation operation, crossover operation and population selection. For the differential evolution algorithm, the following aspects are mainly explained:

3.2.1 Initialize population

For optimization variables $\varepsilon = ({{\varepsilon^1},{\varepsilon^2}, \cdots ,{\varepsilon^D}} )$, its upper and lower limits can be obtained from Section 2.4, where the upper limit is $({\varepsilon_{\max }^1,\varepsilon_{\max }^2,\ldots ,\varepsilon_{\max }^D} )$ and the lower limit is $({\varepsilon_{\min }^1,\varepsilon_{\min }^2,\ldots ,\varepsilon_{\min }^D} )$.

An individual is generated within a given variable range using a random method, and the generated population can be expressed as:

(20)$${\varepsilon _i} = (\varepsilon _i^1,\varepsilon _i^2, \ldots ,\varepsilon _i^D),\quad i = 1,2, \ldots ,NP$$

where NP is the size of population. The initializing value of the i-th individual $\varepsilon _{i,1}^j$ is taken as follows:

(21)$$\varepsilon _{i,1}^j = {\varepsilon _{{{\min }^j}}} + rand({0,1} )\cdot ({{\varepsilon_{{{\max }^j}}} - {\varepsilon_{{{\min }^j}}}} ),\quad j = 1,2, \ldots ,D$$

where rand(0,1) is a uniform random number between [0,1].

3.2.2 Mutation operation

The most significant difference between differential evolution algorithm and genetic algorithm is that the individual variation in differential evolution algorithm is achieved through differential mutation strategy. The differential mutation strategy selects three different individuals from the current population and generates the corresponding mutation vectors. Different variation strategies focus on different aspects. Combined with the characteristics of the optimization objectives of this work and the search requirements, the following three variation strategies are selected:

a. DE/best/1 $(22)$${V_{i,G}} = {\varepsilon _{best,G}} + F \cdot ({{\varepsilon_{{r_1},G}} - {\varepsilon_{{r_2},G}}} )$$$
b. DE/ target-to-best/1 $(23)$${V_{i,G}} = {\varepsilon _{i,G}} + F \cdot ({{\varepsilon_{best,G}} - {\varepsilon_{i,G}}} )+ F \cdot ({{\varepsilon_{{r_1},G}} - {\varepsilon_{{r_2},G}}} )$$$
c. DE/current-to-rand/1 $(24)$${V_{i,G}} = {\varepsilon _{i,G}} + F \cdot ({{\varepsilon_{{r_1},G}} - {\varepsilon_{i,G}}} )+ F \cdot ({{\varepsilon_{{r_2},G}} - {\varepsilon_{{r_3},G}}} )$$$

where ${V_{i,G}}$ represents the mutation vector of the i-th individual of the G-th generation population. ${\varepsilon _{best,G}}$ represents the best individual in the G-th generation population, F represents the scaling factor, and ${r_1},{r_2},{r_3}$ are random integers in $[{1,NP} ]$. DE/best/1 is a basic mutation strategy in DE algorithms. Due to the use of the best individual information in the current population, this strategy tends to search around the best solution and usually converges quickly. DE/target-to-best/1 is an improved strategy that balances both global and local search capabilities by considering both the current target vector and the best individual. DE/current-to-rand/1 is a more randomized mutation strategy that incorporates random individuals, granting it strong global search capabilities. Given that the objective function has multiple optimal solutions, a broad search for optimal solutions within the solution space is necessary. Therefore, multiple mutation strategies are employed to seek the optimal solution from different directions.

3.2.3 Crossover operation

The crossover operation is used to combine the mutated individuals with the current individuals to generate new individuals. The crossover operation is defined as follows:

(25)$$u_{i,G}^j = \left\{ \begin{array}{ll} v_{i,G}^j,& if({rand({0,1} )\le CR} ){\kern 1pt} {\kern 1pt} {\kern 1pt} or{\kern 1pt} {\kern 1pt} {\kern 1pt} ({j = {j_{rand}}} )\\ \varepsilon_{i,G}^j,&otherwise \end{array} \right.$$

where $u_{i,G}^j$ represents the new individual of the i-th individual of the G-th generation population. $v_{i,G}^j$ is the i-th mutation individual generated by the mutation vector. CR is the crossover probability, which is a constant value between 0 and 1. ${j_{rand}}$ is an integer randomly selected from $[{1,2, \cdots ,D} ]$.

3.2.4 Selection operation

In the DE algorithm, the individuals for the next generation are selected by comparing the fitness between each individual in the population and the corresponding offspring generated through crossover. The individual with a better fitness is chosen as a member of the next generation population.

(26)$${\varepsilon _{i,G + 1}} = \left\{ \begin{array}{ll} {u_{i,G}}&if ({f({{u_{i,G}}} )\le f({{\varepsilon_{i,G}}} )} )\\ {\varepsilon_{i,G}}& otherwise \end{array} \right.$$

3.3 Hybrid metaheuristic algorithm base on MPG and DE

Given the objective function characteristics of multispectral temperature measurement, which has multiple optimal solutions, the hybrid metaheuristic algorithm can be employed to optimize this objective function. Metaheuristic algorithm is an algorithm that solves optimization problems by imitating the evolution of nature, group behavior or other heuristic methods. Genetic algorithm, simulated annealing algorithm, differential evolution algorithm and ant colony optimization are all optimization algorithms based on natural phenomena, which belong to metaheuristic algorithm. The hybrid metaheuristic algorithm combines multiple metaheuristics and uses their respective advantages to improve the solving effect and efficiency of optimization problems.

The MPG algorithm performs parallel optimization by searching multiple populations simultaneously. Each population is assigned different optimization parameters, and information exchange among populations ensures their global search capability, thereby guaranteeing the diversity of solution space in the multispectral temperature optimization problem. DE algorithm also possesses strong global search capability and fast convergence speed. In this paper, three different mutation strategies are employed to search for the optimal solution within the solution space more comprehensively. In the designed hybrid metaheuristic algorithm, the combination strategy of independent operation is adopted to ensure the independence of each metaheuristic algorithm. By performing multiple optimizations, the algorithm searches for the optimal solution in the solution space and the temperature of the object can be determined based on the geometric distribution characteristics of the obtained optimal solutions. The flowchart of the designed multi-spectral radiation thermometry base on hybrid metaheuristic algorithm is illustrated in Fig. 2.

Fig. 2. The flowchart of the multi-spectral radiation thermometry algorithm.

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As shown in Fig. 2, the first step is to obtain the upper bounds $({\varepsilon_{\max }^1,\varepsilon_{\max }^2,\ldots ,\varepsilon_{\max }^D} )$ and lower bounds $({\varepsilon_{\min }^1,\varepsilon_{\min }^2,\ldots ,\varepsilon_{\min }^D} )$ of the object emissivity according to the method described in Section 2.4. Based on the obtained emissivity range, the populations of the MPG algorithm and the DE algorithm are initialized separately. The DE algorithm and MPG algorithm are respectively used to optimize the objective function, the optimization process was terminated either when the objective function value fell below 0.000001 or when the maximum number of iterations was reached. Specifically, for the DE algorithm, the maximum number of iterations was set to 400, while for the MPG algorithm, it was limited to 100 iterations. When using the DE algorithm to optimize the problem, three different mutation operators are used to perform the mutation operation, and ultimately three optimal solutions can be obtained. Since the established objective function has multiple optimal solutions, the temperature of the object can be obtained by measuring multiple times and taking the average. Therefore, the MPG algorithm and the DE algorithm are run repeatedly for several times, and the results obtained each time are stored in the set of optimal solutions. Finally, the object temperature is obtained by calculating the average value of the optimal solution set. In the experiment, the set number of repeated runs is 7. The designed hybrid metaheuristic algorithm based on the DE algorithm with three different variation operators and multi-population genetic (MPG) algorithm is abbreviated as 3DE-MPG.

4. Algorithms simulation

According to the designed 3DE-MPG algorithm, simulation experiments are conducted on six different materials labeled as A, B, C, D, E, and F. The emissivity trends of the six models are as follows: monotonically decreasing, monotonically increasing, increasing-decreasing-increasing, decreasing-increasing-decreasing, decreasing and then increasing, and increasing and then decreasing. The simulated temperatures are 873 K, 1273 K, 1573 K and 1873K, respectively, and the reference blackbody temperature is set to 1973K. All simulated data are obtained using the Planck blackbody radiation formula. In order to simplify the simulation experiment, it is assumed that the emissivity of the six different materials is the same at different temperatures. The emissivity of six materials is shown in Table 1. The wavelengths used for the simulation are 1.5 μm, 1.7 μm, 1.8 μm, 1.9 μm, 2.1 μm, 2.2 μm, 2.4 μm and 2.5 μm, respectively. In the emissivity preprocessing stage, the emissivity range is set to 0.1-1, with a subdivision interval of 0.005.

Table 1. The emissivity of six model

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According to the 3DE-MPG algorithm flow described in Fig. 2, simulations is conducted for the six models at different temperatures. The 3DE-MPG algorithm simulation at each temperature is run repeatedly for 100 times. The simulation results are listed in Table 2. Figures 3 and 4 show the temperature boxplot and data histogram obtained from 100 inversions for each of the six models at temperatures of 873 K and 1873K, respectively. In order to test the stability of the 3DE-MPG algorithm, 5% random noise is added to the simulated data. Subsequently, temperature inversions are performed 100 times for the data included 5% random noise under the same simulation environment. The inversion results of the six models at different temperatures are listed in Table 3. Similarly, the temperature boxplot and data histogram of the six models after 100 calculation times at 873 K and 1873K have been plotted in Fig. 5 and Fig. 6, respectively. In Figs. 3, 4, 5 and 6, DE-1 means that the mutation operator selected in de algorithm optimization is DE/ target-to-best/1, DE-2 means that the mutation operator selected is DE/current-to-rand/1, and DE-3 means that the mutation operator selected is DE/best/1.

Fig. 3. The temperature boxplots and data distribution diagram obtained from 100 inversions performed at 873 K with six different models.

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Fig. 4. The temperature boxplots and data distribution diagram obtained from 100 inversions performed at 1873K with six different models.

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Fig. 5. The temperature boxplots and data distribution diagram obtained from 100 inversions performed at 873 K with six different models (with 5% random noise).

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Fig. 6. The temperature boxplots and data distribution diagram obtained from 100 inversions performed at 1873K with six different models (with 5% random noise).

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Table 2. The average results of each temperature for six model

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Table 3. The average results of each temperature for six model (with 5% random noise)

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As can be seen from Table 2, the maximum average absolute error of the inversion temperature for the six models is 3.04 K at 873 K, 3.78 K at 1273 K, 6.66 K at 1573 K and 7.48 K at 1873K, respectively. It also can be seen that the average absolute error increases with the increase of temperature. The maximum relative error does not show a clear regular, and the maximum is 0.4%. After adding 5% random noise, it can be seen from Table 3 that the maximum average absolute error of the inversion temperature for the six models is 3.78 K at 873 K, 5.4 K at 1273 K, 7.12 K at 1573 K and 8.18 K at 1873K, respectively. The maximum average relative error is 0.45% when 5% random noise is added. Compared with that before adding noise, the maximum average absolute error of inversion temperature after adding noise is slightly increased, but the maximum average relative error is still within 0.5%.

From Figs. 3 and 4, when the true temperature is 873 K, it can be observed that the maximum absolute error of the inversion temperature obtained by the 3DE-MPG algorithm is 5.2 K (appearing in the inversion results of model A), with a corresponding relative error of 0.60%. While the true temperature is 1873K, the 3DE-MPG algorithm produces a maximum absolute error of 13 K (appearing in the inversion results of model B), with a corresponding relative error of 0.69%. When 5% random noise is added to signal, it can be seen from Figs. 5 and 6 that, when the true temperature is 873 K, the maximum absolute error of the inversion temperature obtained using the 3DE-MPG algorithm is 5.4 K (appearing in the inversion results of model C), with a corresponding relative error of 0.62%. While the true temperature is 1873K, the 3DE-MPG algorithm produces a maximum absolute error of 14.8 K (appearing in the inversion results of model C), with a corresponding relative error of 0.79%. Overall, the maximum relative error of the inversion temperature by 3DE-MPG algorithm is within 1%.

As can be seen from Figs. 3, 4, 5 and 6, the inversion temperature range obtained by each algorithm is different. In some cases, a specific algorithm may achieve smaller errors and smaller temperature fluctuations. For instance, when the true temperature is 873 K, the results obtained by the DE-2 model, which is using the DE/current-to-rand/1 mutation operator, has the smallest temperature fluctuation. When the true temperature is 1873K, the temperature fluctuation of the results obtained by DE-2 model is equivalent to that of other models, but the random error is larger than other models. As a whole, the inversion temperature obtained by 3DE-MPG algorithm has less temperature fluctuation and the higher stability. It also can be seen from Figs. 3–6 that different optimization algorithms (including different mutation operators) have different solving capabilities when optimizing the objective problems, and their solving capabilities for different models are also different. Relying on a single algorithm to solve the temperature may face more uncertainty. The hybrid metaheuristic algorithm, which combines MPG algorithm and DE algorithm with three different mutation operators, can balance the shortcomings of different algorithms in optimizing the temperature for different models, and make the inversion results more stable.

Figure 7 shows the emissivity comparison obtained by different optimization algorithms for the six models. It can be observed that the emissivity trend obtained by the different algorithm is consistent with the trend of true emissivity, with only slight numerical differences. In terms of computational efficiency, the DE algorithm with three different mutation operators takes an average of about 0.15 seconds for each computation, while the MPG algorithm takes an average of about 0.1 seconds per computation, and the feasible region constraint processing takes approximately 0.01 seconds. Therefore, the total time for getting one effective temperature inversion result is approximately 0.26 seconds. In conclusion, the 3DE-MPG hybrid metaheuristic algorithm is expected to be applied to real-time online multi wavelength temperature measurement. (Simulation environment: Python 3.10; Windows 10; Intel Core i7-10700 CPU @ 2.90 GHz; 16 G RAM).

Fig. 7. The emissivity comparison of different algorithms for six models.

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5. Experiment

In order to further verify the application of the proposed 3DE-MPG algorithm in practical temperature measurement, the emissivity of silicon carbide (SIC) has been measured by using the emissivity measurement apparatus, which has been reported in our others work [32]. And according to the spectral radiation data recorded during the measurement, the temperature of silicon carbide at different temperatures was calculated based on the proposed 3DE-MPG algorithm.

The emissivity measurement apparatus is shown in Fig. 8, consisting of a sample heating furnace, a standard radiation source, and measurement equipment (including a fiber optic spectrometer, reflecting mirror, etc.). The sample heating furnace employs silicon carbide as the heating element, with a maximum furnace temperature of 1573 K. The samples are heated through thermal conduction. To accurately and effectively monitor the surface temperature of the samples, we use high-precision K-type thermocouples. The gap between the samples and the thermocouples is filled with boron nitride, a material with high thermal conductivity. A blackbody with an effective emissivity greater than 0.995 is used as the standard radiation source, which can provide a standard blackbody radiation spectrum from 423 K to 1473 K, with a temperature uncertainty of ±2 K. Light emanating from either the samples or the standard blackbody first reflects off a concave mirror, followed by a right-angle prism mirror. Eventually, this light is coupled into an optical fiber, which leads to the fiber optic spectrometer. Using this setup, we measured the spectral radiation signals of silicon carbide at temperatures of 873 K, 973 K, 1073 K, and 1173 K, as well as the blackbody radiation signal at 1273 K. The wavelengths and radiation signals which is selected for temperature inversion are shown in Table 4. And Table 5 provides the emissivity value of silicon carbide at the aforementioned temperature, which is used to compare the emissivity obtained by inversion results with the true emissivity of SIC.

Fig. 8. The schematic diagram of emissivity measurement apparatus.

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Table 4. The signals of SIC and blackbody (unit: Counts)

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Table 5. The spectral emissivity of SIC

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From Table 6, it can be observed that the maximum absolute error in the inversion temperature of SIC is 8.46 K, with a corresponding relative error of 0.72%. While the maximum average absolute error is 4.94 K, with a corresponding relative error of 0.42%. This conclusion is approximately consistent with the precision of the previous simulation experiments. Similarly, the average absolute error of the inversion temperature increases with the increase of SIC surface temperature, but the average relative error remains stable within 0.5%. The average calculation time for obtaining one output of the inversion temperature is approximately 0.26 seconds. Therefore, the proposed 3DE-MPG algorithm can be applied to online multispectral temperature measurements. Furthermore, it can be observed from Fig. 9 that there is some deviation between the inversion emissivity and the true emissivity of silicon carbide, but the trend remains consistent. The inversion emissivity can be used to assess material emissivity characteristics but may not be suitable for high-precision applications.

Fig. 9. The comparison of inversion emissivity and true emissivity of silicon carbide.

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Table 6. The inversion results of SIC by the 3DE-MPG algorithm

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Under the same conditions, the proposed 3DE-MPG algorithm was used to invert the emissivity and temperature of silicon carbide. Each group data is calculated repeatedly for 100 times, and the final results are shown in Table 6. The inversion emissivity has been compared with the true emissivity and presented in Fig. 9.

6. Conclusion

A multispectral radiation temperature measurement method has been proposed in this paper, and the actual temperature of the object is solved by using uses intelligent optimization algorithm. Through the combination the method of simulation and experiment, the effectiveness of the proposed intelligent optimization algorithm is verified. The following are the main features of this paper:

1. The objective function for multispectral thermometry is directly derived from the Planck formula, making the presented multispectral thermometry applicable to various temperature conditions.
2. A feasible region constraint method for multispectral thermometry is proposed, which restricts the emissivity range to reduce computational complexity in the subsequent optimization process.
3. A hybrid metaheuristic algorithm (3DE-MPG) combining differential evolution (DE) algorithm with three different mutation operators and multi-population genetic (MPG) algorithm is proposed to solve the true temperature.
4. The simulation results show that the 3DE-MPG algorithm can invert effectively both the true temperature and emissivity of the object, with an average relative error not exceeding 0.42% and a maximum random relative error not exceeding 0.79%.
5. Each temperature inversion time is approximately 0.26 seconds.

Since the objective function is established by minimizing the deviation between the calculated temperature in different channels and the average temperature of each channel, there will be multiple optimal solutions near the true temperature of the object. When using the optimization algorithm without emissivity model to solve the true temperature of the object, it needs to repeat the calculation for several times to get the average value to achieve the desired effect. The 3DE-MPG algorithm combines two different evolutionary algorithms, where the DE algorithm employs three different mutation operators to perform mutation operations, enabling a comprehensive search for optimal solutions in the solution space. With the help of hybrid metaheuristic algorithm, the shortcomings of different algorithms in solving the true temperature of objects with different model emissivity can be balanced. As a result, the proposed 3DE-MPG algorithm shows high stability in multispectral thermometry, and can be applied to complex industrial online temperature measurement scenarios.

Funding

Science and Technology Department of Henan Province (222102220078); Key Scientific Research Project of Colleges and Universities in Henan Province (23A140002); Key Scientific Research Project of Colleges and Universities in Henan Province (22A140021); Natural Science Foundation of Henan Province (222300420011, 222300420209); Innovation Scientists and Technicians Troop Construction Projects of Henan Province (224000510007); National Natural Science Foundation of China (62075058, U1804261).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Material	Wavelength
Material	1.5 μm	1.7 μm	1.8 μm	1.9 μm	2.1 μm	2.3 μm	2.4 μm	2.5 μm
A	0.8563	0.7496	0.6506	0.3807	0.3312	0.2909	0.2535	0.2211
B	0.2346	0.2680	0.3088	0.5277	0.6065	0.6905	0.7925	0.9086
C	0.3646	0.5468	0.7405	0.7034	0.5081	0.3363	0.2163	0.1948
D	0.3466	0.2240	0.1930	0.7823	0.8925	0.9024	0.8117	0.6448
E	0.8638	0.9094	0.8325	0.1952	0.2224	0.3509	0.5520	0.7512
F	0.2144	0.1973	0.3018	0.9098	0.8533	0.7048	0.4973	0.3087

True Temperature		A	B	C	D	E	F
873 K	Average Inversion Temperature (K)	875.43	874.29	876.04	872.75	874.40	875.36
	Absolute Error (K)	2.43	1.29	3.04	0.25	1.40	2.36
	Relative Error (%)	0.28	0.15	0.35	0.03	0.16	0.27
	Standard Deviation	0.83	0.73	0.71	0.69	0.74	0.63
1273 K	Average Inversion Temperature (K)	1,275.10	1,273.48	1,276.78	1,270.36	1,274.88	1,275.63
	Absolute Error (K)	2.10	0.48	3.78	2.64	1.88	2.63
	Relative Error (%)	0.16	0.04	0.30	0.21	0.15	0.21
	Standard Deviation	1.54	1.49	1.50	1.26	1.32	1.43
1573 K	Average Inversion Temperature (K)	1,574.51	1,570.55	1,578.18	1,566.34	1,573.87	1,574.96
	Absolute Error (K)	1.51	2.45	5.18	6.66	0.87	1.96
	Relative Error (%)	0.10	0.16	0.33	0.42	0.06	0.12
	Standard Deviation	2.51	1.92	2.29	1.86	2.10	2.03
1873K	Average Inversion Temperature (K)	1,872.19	1,865.99	1,875.81	1,865.52	1,868.67	1,873.75
	Absolute Error (K)	0.81	7.01	2.81	7.48	4.33	0.75
	Relative Error (%)	0.04	0.37	0.15	0.4	0.23	0.04
	Standard Deviation	3.72	3.13	2.72	2.49	2.82	3.02

True Temperature		A	B	C	D	E	F
873 K	Average Inversion Temperature (K)	876.06	874.58	876.78	873.79	875.21	875.85
	Absolute Error (K)	3.06	1.58	3.78	0.79	2.21	2.85
	Relative Error (%)	0.35	0.18	0.43	0.09	0.25	0.33
	Standard Deviation	0.77	0.75	0.65	0.61	0.60	0.74
1273 K	Average Inversion Temperature (K)	1,275.51	1,275.25	1,278.4	1,271.36	1,276.48	1,277.72
	Absolute Error (K)	2.51	2.25	5.4	1.64	3.48	4.72
	Relative Error (%)	0.20	0.18	0.42	0.13	0.27	0.37
	Standard Deviation	1.61	1.43	1.32	1.21	1.44	1.36
1573 K	Average Inversion Temperature (K)	1,577.50	1,573.16	1,580.12	1,569.11	1,577.04	1,579.49
	Absolute Error (K)	4.50	0.16	7.12	3.89	4.04	6.49
	Relative Error (%)	0.29	0.01	0.45	0.25	0.26	0.41
	Standard Deviation	2.10	2.15	1.89	1.55	1.82	1.82
1873K	Average Inversion Temperature (K)	1,875.34	1,869.15	1,881.18	1,866.95	1,872.36	1,878.65
	Absolute Error (K)	2.34	3.85	8.18	6.05	0.64	5.65
	Relative Error (%)	0.12	0.21	0.44	0.32	0.03	0.30
	Standard Deviation	3.29	3.58	2.69	2.52	2.74	2.56

Temperature (K)		Wavelength
Temperature (K)		1.5 μm	1.6 μm	1.7 μm	2.1 μm	2.2 μm	2.3 μm	2.4 μm	2.5 μm
Blackbody	1273 K	16869	24342	33255	56956	56122	51652	38518	20424
SIC	873 K	617.2	1073.9	1758	4935.8	5426.7	5439.1	4435.2	2534.5
	973 K	1891.1	3076.3	4724.1	11040	11693	11350	8973.9	4994.8
	1073 K	4667.6	7199.5	10508	21153	21737	20575	15864	8637.7
	1173 K	10016	14761	20637	36632	36703	34047	25734	13719

Temperature (K)	Wavelength
Temperature (K)	1.5 μm	1.6 μm	1.7 μm	2.1 μm	2.2 μm	2.3 μm	2.4 μm	2.5 μm
873 K	0.806	0.807	0.806	0.816	0.822	0.824	0.827	0.829
973 K	0.816	0.811	0.812	0.812	0.819	0.816	0.818	0.822
1073 K	0.816	0.812	0.811	0.81	0.819	0.815	0.815	0.817
1173 K	0.819	0.818	0.817	0.809	0.816	0.814	0.813	0.813

Approach to multispectral thermometry with Planck formula and hybrid metaheuristic optimization algorithm

Abstract

1. Introduction

2. Principles of multispectral thermometry

2.1 Reference temperature model

2.2 Establishment of objective function

2.3 Constructing constraints of optimization problem

2.4 Feasible region constraint for multispectral thermometry

3. Hybrid metaheuristic algorithms

3.1 Multi-population genetic algorithm

3.2 Differential evolution algorithm

3.2.1 Initialize population

3.2.2 Mutation operation

3.2.3 Crossover operation

3.2.4 Selection operation

3.3 Hybrid metaheuristic algorithm base on MPG and DE

4. Algorithms simulation

5. Experiment

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (6)

Equations (26)

Optics Express

True Temperature (K)	Inversion Temperature (K)			Absolute Error (K)		Relative Error (%)		Calculation Time (s)
True Temperature (K)	Max	Min	Mean	Max	Mean	Max	Mean	Calculation Time (s)
873 K	874.46	871.10	872.78	1.90	0.22	0.22	0.03	0.26
973 K	976.90	971.24	973.93	3.90	0.93	0.40	0.10	0.26
1073 K	1075.20	1068.66	1071.34	4.34	1.66	0.40	0.15	0.27
1173 K	1170.82	1164.54	1168.06	8.46	4.94	0.72	0.42	0.26