Data processing method for simultaneous estimation of temperature and emissivity in multispectral thermometry

Zhuangtao Tian; Kaihua Zhang; Kaihua Zhang; Yanfen Xu; Kun Yu; Yufang Liu; Yufang Liu

doi:10.1364/OE.470056

1. Introduction

Multispectral thermometry has the advantages of fast response, contactless measurement and no upper temperature limit [1–4]. It has been largely applied in many fields, such as evaluation of the working state of rocket engine, performance test of thermal shielding materials for gas turbine, and temperature monitoring during metal smelting [5–8]. The simultaneous estimation of true temperature and spectral emissivity of the materials is possible by using multispectral thermometry. Based on the data of n spectral channels of multispectral thermometer, n radiation equations including n+1 unknown parameters are constructed [9]. Consequently, the radiation equations are ill-posed and tough to solve because of the unknown emissivity.

Conventional methods are based on the fixed assumption models for emissivity versus wavelength or temperature. By the linear or nonlinear assumption models between emissivity and wavelength, the ill-posed radiation equations can be converted into well-posed equations. Then, the converted equations are solved through least square, solving equations, etc. [4,10–14]. Based on the assumption that the emissivity has a linear relationship with temperature at different wavelengths, the “secondary measurement method” was developed [9,15–18]. This method can calculate the temperature and emissivity of two consecutive temperature measuring points simultaneously. However, in the actual process of temperature measurement, the change of spectral emissivity of the object is often inconsistent at different measurement positions. Therefore, the fixed hypothesis models are difficult to obtain.

With the development of machine learning technique, the neural network algorithm without the emissivity assumption model has been applied to multispectral thermometry [19–21]. Yang et al. constructed a combined neural network emissivity model, which can obtain the spectral emissivity and real temperature of object according to the measured brightness temperature data [19]. Song et al. designed a twice recognition method based on BP neural network, which can convert the measured brightness temperature to real temperature [20]. Chen et al. established a multispectral thermometry based on the adaptive emissivity model, which combined neural network with genetic algorithm to obtain high accurate temperature inversion results [21]. However, it is necessary to obtain a mass of correct and reliable data samples ahead of time to train the neural network model.

In recent years, some methods of multispectral thermometry that require neither the assuming of the emissivity models nor the obtaining of the data samples, have been presented [22–24]. The core idea of these methods is to transform the data processing problem as the constrained or unconstrained optimization problem, i.e., make the deviation between the calculated temperature and the actual temperature of each spectral channel infinitely close to zero. Then, the optimization algorithm is utilized to address the problem. Xing et al. processed multispectral data by the gradient projection and internal penalty function algorithm [22]. Liang et al. utilized a set of emissivity acquired with the generalized inverse matrix method as the initial value of the iteration algorithm to inverse temperature of the rocket nozzle [23]. Yu et al. used the temperature deviation function and its gradient to calculate the real temperature and spectral emissivity of the zirconia by the improved Newtown algorithm [24]. They all can calculate the true temperature and spectral emissivity of the material simultaneously. However, in the above study, the objection function for multispectral thermometry has multiple local minimal values [25]. When the optimization results of the function are close to the optimal value of 0, the errors of calculated temperature may be still large. In other words, even if the optimal results of the function are effective, the calculation results of temperature also may be ineffective.

In order to achieve effective inversion simultaneously of true temperature and spectral emissivity. Based on the deviation between the calculated temperature and the actual temperature, a novel data processing model of multispectral thermometer is established by adding the relevant constraints of emissivity under three various relationships among the detection signal and the reference signal. Then, Sequential Randomized Coordinate Shrinking (SRCS) algorithm and Multiple-Population Genetic (MPG) algorithm, which are more suitable for multi-local extremum function, are used to optimize the data processing model. Six hypothetical materials with different emissivity distributions and the measured data of silicon carbide and tungsten are simulated to verify the effectiveness of the proposed data processing method.

2. Principles of multispectral thermometry

2.1 Reference temperature model

For multispectral radiation thermometer with n wavelength channels, the output signal V_i of the ith channel can be given:

(1)$${V_i} = {A_{{\lambda _i}}}\varepsilon ({\lambda _i},T)\frac{1}{{\lambda _i^5({e^{{C_2}/{\lambda _i}T}} - 1)}}\textrm{ }(i = 1,2, \cdots n)$$

Where A_λi is a calibration constant of the thermometer, which is temperature-independent but wavelength-dependent. At a given wavelength, A_λi is related to the detector’s spectral responsivity, optical element transmittance, geometrical size and the first radiation constant. ɛ(λ_i, T) is the emissivity of material at true temperature T, λ_i is the effective wavelength of the ith spectral channel, and C₂ is the second radiation constant.

When C₂ / (λ_iT) << 1, Eq. (1) can be replaced by Wien approximation:

(2)$${V_i} = {A_{{\lambda _i}}}\varepsilon ({\lambda _i},T)\lambda _i^{ - 5}{e^{ - \frac{{{C_2}}}{{{\lambda _i}T}}}}\textrm{ }(i = 1,2, \cdots n)$$

At the reference temperature T´ of blackbody, the output signal V_i´ for the ith spectral channel is:

(3)$$V_i^{\prime} = {A_{{\lambda _i}}}\varepsilon ({\lambda _i},{T^{\prime}})\lambda _i^{ - 5}{e^{ - \frac{{{C_2}}}{{{\lambda _i}{T^{\prime}}}}}}\textrm{ }(i = 1,2, \cdots n)$$

Where ɛ(λ_i, T´) is the emissivity of the blackbody, and its value can be treated as 1.

According to the ratio of Eq. (2) and Eq. (3), the reference temperature model can be established:

(4)$$\frac{{{V_i}}}{{V_i^{\prime}}} = \varepsilon ({\lambda _i},T){e^{\frac{{{C_2}}}{{{\lambda _i}}}\left( {\frac{1}{{{T^{\prime}}}} - \frac{1}{T}} \right)}}$$

This model only needs to measure the output signal V_i´ of each spectral channel at a given reference temperature T´ to complete the calibration of the thermometer, but presents one serious drawback. In multispectral thermometry there is invariably one unknown parameter more than there are radiation equations. To overcome the drawback, the promble of solving underdetermined equations will be converted into a constrained optimization problem.

2.2 Model of data processing for multispectral thermometry

2.2.1 Establishment of objective function

Constraint optimization problem can be described as follows:

(5)$$\left\{ \begin{array}{c} \min f(x)\\ Ax \ge b \end{array} \right.$$

where x is the decision variable, i.e. the variable to be optimized, Ax ≥ b is the linear inequality constraint condition of objective function f(x), A is the restraint coefficient matrix, and b is the restraint vector.

The Eq. (4) shows that if emissivity ɛ(λ_i, T) is known, the calculated temperature by each spectral channel is the selfsame and equal to true temperature. Therefore, the ideally deviation of calculated temperature for every channel is zero, the following formula can be obtained:

(6)$$\frac{1}{n}\sum\limits_{i = 1}^n {\frac{{T_i^2}}{{{E^2}({T_i})}}} = 1$$

Where T_i is the calculated temperature in each spectral channel, and its expression is:

(7)$${T_i} = \frac{1}{{\frac{1}{{{T^{\prime}}}} + \frac{{{\lambda _i}}}{{{C_2}}}[\ln \varepsilon ({\lambda _i},T) - \ln (\frac{{{V_i}}}{{V_i^{\prime}}})]}}$$

E(T_i) is the average calculated temperature for all spectral channels, it can be expressed as:

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

Since the emissivity is unknown, Eq. (6) can be transformed into an optimization problem according to the principle of minimizing deviation, and make the temperature deviation infinitely close to zero:

(9)$$\min F = \left|{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {\frac{{T_i^2}}{{{E^2}({T_i})}}} } \right) - 1} \right|\to 0$$

Thus, the objective function F is established as follows:

(10)$$\min F = \left|{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {\frac{{T_i^2}}{{{E^2}({T_i})}}} } \right) - 1} \right|$$

2.2.2 Constructing constraints of optimization problem

We set constraints on optimization problem from the following two aspects.

Firstly, the inherent range of object emissivity should follow 0 < ɛ(λ_i, T) < 1, it can be expressed as:

(11)$$\begin{array}{l} \varepsilon ({{\lambda_i},T} )> 0\\ - \varepsilon ({{\lambda_i},T} )> - 1 \end{array}$$

Secondly, according to the relationship between radiation signals and emissivity at different wavelengths, the following constraints of emissivity will be established:

Taking the logarithms on both sides of Eq. (4), it can be expressed:

(12)$$\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) = \frac{{{C_2}}}{{{\lambda _i}}}\left( {\frac{1}{{{T^{\prime}}}} - \frac{1}{T}} \right) + \ln \varepsilon ({\lambda _i},T)$$

According to Eq. (12), the (i+1)th spectral channel is shown:

(13)$$\ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) = \frac{{{C_2}}}{{{\lambda _{i + 1}}}}\left( {\frac{1}{{{T^{\prime}}}} - \frac{1}{T}} \right) + \ln \varepsilon ({\lambda _{i + 1}},T)$$

Since 0 < ɛ(λ_i, T) < 1 and 0 < ɛ(λ_i₊₁, T) < 1, set 0 < λ_i < λ_i₊₁ then $\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) > 0$. Equation (12) minus Eq. (13) is constructed:

(14)$$\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) + \frac{{{C_2}}}{{{T^{\prime}}}}\left( {\frac{1}{{{\lambda_{i + 1}}}} - \frac{1}{{{\lambda_i}}}} \right) = \ln \varepsilon ({\lambda _i},T) - \ln \varepsilon ({\lambda _{i + 1}},T) + \frac{{{C_2}}}{T}\left( {\frac{1}{{{\lambda_{i + 1}}}} - \frac{1}{{{\lambda_i}}}} \right)$$

By transforming Eq. (14) can be given:

(15)$$\ln \left( {\frac{{\varepsilon ({\lambda_i},T)}}{{\varepsilon ({\lambda_{i + 1}},T)}}} \right) - \ln \left( {\frac{{{V_i}V_{i + 1}^{\prime}}}{{V_i^{\prime}{V_{i + 1}}}}} \right) = {C_2}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right)\left( {\frac{1}{T} - \frac{1}{{{T^{\prime}}}}} \right)$$

Where ${C_2}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) > 0$. Being the blackbody’s reference temperature T´ is set manually, and the inversion result is not affected by value of reference temperature, we stipulate that $\left( {\frac{1}{T} - \frac{1}{{{T^{\prime}}}}} \right) < 0$. Thus, the right side of Eq. (15) is less than zero, the following inequality can be obtained:

(16)$$\varepsilon ({\lambda _i},T) < \varepsilon ({\lambda _{i + 1}},T)\frac{{{V_i}V_{i + 1}^{\prime}}}{{V_i^{\prime}{V_{i + 1}}}}$$

Adding ${C_2}\left( {\frac{1}{T} + \frac{1}{{{T^{\prime}}}}} \right)\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right)$ to both sides of Eq. (14):

(17)$$\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) + \frac{{{C_2}}}{T}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) = \ln \varepsilon ({\lambda _i},T) - \ln \varepsilon ({\lambda _{i + 1}},T) + \frac{{{C_2}}}{{{T^{\prime}}}}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right)$$

Next, according to three various relationships between detection signal V and reference signal V´, the constraints of emissivity are discussed.

When $\frac{{{V_i}}}{{V_i^{\prime}}} > \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}$, being $\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) > 0$ and $\frac{{{C_2}}}{T}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) > 0$, then, the left side of Eq. (17) is greater than zero, so the right side is also greater than zero, it can be transformed into the following inequality:

(18)$$\varepsilon ({\lambda _{i + 1}},T){e^{\frac{{{C_2}}}{{{T^{\prime}}}}\left( {\frac{1}{{{\lambda_{i + 1}}}} - \frac{1}{{{\lambda_i}}}} \right)}} < \varepsilon ({\lambda _i},T)$$

When $\frac{{{V_i}}}{{V_i^{\prime}}} = \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}$, known that $\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) = 0$, and $\frac{{{C_2}}}{T}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) > 0$, according to Eq. (17), the same inequality as Eq. (18) can be obtained.

When $\frac{{{V_i}}}{{V_i^{\prime}}} < \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}$, considering that $\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \ln \left( {\frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}} \right) < 0$ and $\frac{{{C_2}}}{T}\left( {\frac{1}{{{\lambda_i}}} - \frac{1}{{{\lambda_{i + 1}}}}} \right) > 0$, no obvious conclusions were reached.

From what has been discussed above, the following conclusions can be drawn:

(19)$$\left\{ \begin{array}{l} \varepsilon ({\lambda_{i + 1}},T)\frac{{{V_i}V_{i + 1}^{\prime}}}{{V_i^{\prime}{V_{i + 1}}}} - \varepsilon ({\lambda_i},T) > 0\\ \varepsilon ({\lambda_i},T) - \varepsilon ({\lambda_{i + 1}},T){e^{\frac{{{C_2}}}{{{T^{\prime}}}}\left( {\frac{1}{{{\lambda_{i + 1}}}} - \frac{1}{{{\lambda_i}}}} \right)}} > 0\textrm{, when}\frac{{{V_i}}}{{V_i^{\prime}}} \ge \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}} \end{array} \right.$$

According to the above constraints in Eqs. (11) and (19) for the optimization variable ɛ (ɛ = [ɛ(λ₁, T), ɛ(λ₂, T),…, ɛ(λ_n, T)]^T) are solved, the restraint coefficient matrix A and restraint matrix b can be obtained. By combining the objective function Eq. (10), the data processing problem of multispectral thermometer is transformed into a constrained optimization problem:

(20)$$\left\{ \begin{array}{c} {F_{\min }} = \left|{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {\frac{{T_i^2}}{{{E^2}({T_i})}}} } \right) - 1} \right|\\ A\varepsilon \ge b \end{array} \right.$$

This is a non-differentiable and non-linear constrained optimization problem. Some optimization algorithms that do not rely on the gradient such as SRCS algorithm and MPG algorithm can resolve Eq. (20).

3. SRCS data processing algorithm

3.1 Principle of SRCS algorithm

Sequential Randomized Coordinate Shrinking (SRCS) is an optimization algorithm based on the binary classification model and mainly consists of a cycle of two steps [26]: sampling solutions from solution space of target optimization model, and updating the classification model from the sampled solutions along with their evaluation values. Through the cycle, the better solutions are expected to repeatedly found. The main steps of the SRCS algorithm are as follows:

Step 1: Sampling a batch of solutions uniformly from the entire solution space X to generate the initial solution set U={u₁, …, u_r}.
Step 2: Construct the solution-value tuple set U={(u₁, y₁),…,(u_r, y_r)}, ∀u_i∈U: y_i = p(u_i) after querying objective function for each solution in U.
Step 3: Split the solution-value tuple set B into the positive set B⁺ consisting of the tuples of the best k solutions, and the negative set B^- consisting of the rest of tuples.
Step 4: Record the best-so-far solutions and its function values in set H.
Step 5: SRCS classification method infers the region D of positive class solution according to positive and negative class solutions in B.
Step 6: Uniformly samples a solution u from the over D with η probability or over X with 1-η probability.
Step 7: Update B⁺ and B^-, using tuple (u, y) to replace the tuple with worst evaluation value (u´, y´) in B⁺, and using (u´, y´) to randomly replace a tuple in B^-.
Step 8: Repeat from step 4 until the stop criterion is met.
Step 9: Returns the one solution that is optimal in the set H.

3.2 SRCS algorithm simulation

Founded on the principle of SRCS algorithm, simulation experiments are respectively carried out on six kinds of target materials marked A-F with different change trends of emissivity. These six emissivity models of target materials on the temperature of 1800 K for diverse wavelength are displayed in Table 1. The reference temperature of blackbody is 1600 K. In order to balance the accuracy of the inversion results and the computational efficiency of the algorithm, the number of multispectral thermometer effective spectral channels is selected as 8. The wavelengths λ₁∼λ₈ of each channel are 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 and 1.1 µm, separately.

Table 1. Target Emissivity Model

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For Eq. (19), let $\frac{{{V_i}V_{i + 1}^{\prime}}}{{V_i^{\prime}{V_{i + 1}}}} = {a_i}$, ${e^{\frac{{{C_2}}}{{{T^{\prime}}}}\left( {\frac{1}{{{\lambda_{i + 1}}}} - \frac{1}{{{\lambda_i}}}} \right)}} = {c_i}$. By changing the linear inequality Eq. (19) into the form Aɛ ≥ b, the inequality groups then as follows are given: when $\frac{{{V_i}}}{{V_i^{\prime}}} < \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}$,

(21)$$\left\{ \begin{array}{c} - {\varepsilon_1} + {a_1}{\varepsilon_2} > 0\\ - {\varepsilon_2} + {a_2}{\varepsilon_3} > 0\\ \vdots \\ - {\varepsilon_{n - 1}} + {a_{n - 1}}{\varepsilon_n} > 0 \end{array} \right.$$

Order:

A = \left[ {\begin{array}{*{20}{c}} { - 1}&{{a_1}}&0&0&0&0&0&0\\ 0&{ - 1}&{{a_2}}&0&0&0&0&0\\ 0&0&{ - 1}&{{a_3}}&0&0&0&0\\ 0&0&0&{ - 1}&{{a_4}}&0&0&0\\ 0&0&0&0&{ - 1}&{{a_5}}&0&0\\ 0&0&0&0&0&{ - 1}&{{a_6}}&0\\ 0&0&0&0&0&0&{ - 1}&{{a_7}} \end{array}} \right]

\varepsilon = {\left[ {\begin{array}{*{20}{c}} {{\varepsilon_1}}&{{\varepsilon_2}}&{{\varepsilon_3}}&{{\varepsilon_4}}&{{\varepsilon_5}}&{{\varepsilon_6}}&{{\varepsilon_7}}&{{\varepsilon_8}} \end{array}} \right]^T}

b = {\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0 \end{array}} \right]^T}

when $\frac{{{V_i}}}{{V_i^{\prime}}} \ge \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}$,

(22)$$\left\{ \begin{array}{c} - {\varepsilon_1} + {a_1}{\varepsilon_2} > 0\\ {\varepsilon_1} - {c_1}{\varepsilon_2} > 0\\ - {\varepsilon_2} + {a_2}{\varepsilon_3} > 0\\ {\varepsilon_2} - {c_2}{\varepsilon_3} > 0\\ \vdots \\ - {\varepsilon_{n - 1}} + {a_{n - 1}}{\varepsilon_n} > 0\\ {\varepsilon_{n - 1}} - {c_{n - 1}}{\varepsilon_n} > 0 \end{array} \right.$$

Order:

A = \left[ {\begin{array}{*{20}{c}} { - 1}&{{a_1}}&0&0&0&0&0&0\\ 1&{ - {c_1}}&0&0&0&0&0&0\\ 0&{ - 1}&{{a_2}}&0&0&0&0&0\\ 0&1&{ - {c_2}}&0&0&0&0&0\\ 0&0&{ - 1}&{{a_3}}&0&0&0&0\\ 0&0&1&{ - {c_3}}&0&0&0&0\\ 0&0&0&{ - 1}&{{a_4}}&0&0&0\\ 0&0&0&1&{ - {c_4}}&0&0&0\\ 0&0&0&0&{ - 1}&{{a_5}}&0&0\\ 0&0&0&0&1&{ - {c_5}}&0&0\\ 0&0&0&0&0&{ - 1}&{{a_6}}&0\\ 0&0&0&0&0&1&{ - {c_6}}&0\\ 0&0&0&0&0&0&{ - 1}&{{a_7}}\\ 0&0&0&0&0&0&1&{ - {c_7}} \end{array}} \right]

\varepsilon = {\left[ {\begin{array}{*{20}{c}} {{\varepsilon_1}}&{{\varepsilon_2}}&{{\varepsilon_3}}&{{\varepsilon_4}}&{{\varepsilon_5}}&{{\varepsilon_6}}&{{\varepsilon_7}}&{{\varepsilon_8}} \end{array}} \right]^T}

b = {\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&0&0&0&0&0&0 \end{array}} \right]^T}

Generally, constrained optimization problem is addressed by converting it unconstrained optimization problem on a fixed feasible region using penalty function method, where the feasible region is Eq. (11). After the constraints are constructed, the penalty function F_c can be expressed as follows:

(23)$${F_c}(\varepsilon ,\sigma ) = \left\{ \begin{array}{l} F + \sigma \sum\limits_{i = 1}^7 {{{[\min \{ 0,{g_i}(\varepsilon )\} ]}^2}} ,\textrm{ when }\frac{{{V_i}}}{{V_i^{\prime}}} < \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}}\\ F + \sigma \sum\limits_{i = 1}^{14} {{{[\min \{ 0,{g_i}(\varepsilon )\} ]}^2}} ,\textrm{ when }\frac{{{V_i}}}{{V_i^{\prime}}} \ge \frac{{{V_{i + 1}}}}{{V_{i + 1}^{\prime}}} \end{array} \right.$$

where F is the original objective function determined by Eq. (20), σ is the penalty factor, g_i(ɛ)=A_iɛ-b_i, A_i is the element of ith row of the restraint coefficient matrix A, and b_i is the ith element of restraint vector b.

Next, the data processing model Eq. (20) is optimized according to the steps in Fig. 1. The solution space X is set to a continuous space with the lower limit of 0 and the upper limit of 1, i.e., feasible region Eq. (11). But too large a feasible region can reduce the computational speed and influence the emissivity accuracy. The size of feasible region can be properly reduced for obtain accurate emissivity. Thus, the feasible region for the material A, B, D, E, F models are set from 0.1 to 0.9, and that of the material C model is set from 0.1 to 0.8.

Fig. 1. Program flow chart of the SRCS algorithm.

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In the calculation process of the algorithm, the following parameters are also required to set for the SRCS algorithm: Sample size r in each iteration by 12, where the number of positive samples k is 2; Sampling probability η=0.99 from solution space D; The termination condition ρ=1e-6 and the number of iterations δ=1000. SRCS algorithm was applied to repeat the calculation for 100 times. The average results of temperature simulation are shown in Table 2, and the relative errors of calculated temperature are shown in Fig. 2.

Fig. 2. Relative errors of temperature calculation for six models by SRCS algorithm.

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Table 2. Average Results of Temperature Simulation by SRCS Algorithm

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Subsequently, 5% random noise is added to the voltage signal in Eq. (4) for simulation to verify the anti-noise ability of SRCS algorithm. The relative errors of calculated temperature and the average results of temperature simulation are shown in Fig. 3 and Table 3, separately.

Fig. 3. Relative errors of temperature calculation for six models by SRCS algorithm (5% random noise).

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Table 3. Average Results of Temperature Simulation by SRCS Algorithm (5% Random Noise)

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The stability of the SRCS algorithm can be visually seen from Figs. 2 and 3. In the noise-free state at true temperature 1800 K, the maximum relative error for temperature calculation was 0.9%. When voltage signal with noise, the results are almost the same as those without noise. The average inversion results in Tables 2 and 3 show that SRCS algorithm has high accuracy. When voltage signal without noise, the average absolute error of temperature is below 9 K, the mean relative error is within 0.5%, and the maximum sample standard deviation for single calculated temperature is 3.15. After adding 5% random noise, the overall average absolute error of temperature is about 6 K, the holistic mean relative error does not exceed 0.35%, and the average sample standard deviation for single calculated temperature of six models is 2.8. The inversion results are basically consistent with those without noise, indicating that SRCS algorithm has good anti-noise ability. The trend of spectral emissivity inversion results with wavelength (as shown in Fig. 7) is basically consistent with the true distribution, which shows that this algorithm can be used for the inversion of spectral emissivity. However, the number of repeated measurements in the actual thermometry is limited, and only more stable algorithms can obtain more reliable results. The stability of SRCS algorithm is slightly low and cannot be applied to real measurement. In addition, running time of SRCS algorithm is long with the calculation time about 0.65 s, which cannot meet the demands of real-time industrial thermometry (Simulation environment: Python 3.8; Windows 10; AMD Ryzen Threadripper 3960X 24-Core Processor @3.79 GHz; 64G RAM). The algorithm is written in the Windows 10 environment using the Zoopt 0.3.0 framework.

4. MPG data processing algorithm

4.1 Genetic algorithm and Multi-Population genetic structure

Genetic algorithm (GA) is a classic function optimization technique that simulate the living biological evolution based on the principle of survival of the fittest in nature. Since GA does not depend on the gradient in optimization calculation, it is particularly suitable for solving non-differentiable optimization problem. However, in the search process of GA, when there are some super-individuals with too high fitness in the early evolution of population, it may lead to premature convergence. At the same time, the setting methods of crossover probability and mutation probability affect the balance of global search ability and local search ability of GA, inappropriate probability values may influence the overall performance of the algorithm. In view of the above problems of GA, a multi-population genetic structure model can be used to replace the conventional standard GA (SGA) model [27]. The structure of MPG algorithm is shown in Fig. 4.

Fig. 4. The structure chart for MPG algorithm.

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In Fig. 4, the evolution mechanism of population 1-N is routine SGA, mainly including population initialization, selection, crossover and mutation. Each population is independent of each other and conducts evolutionary search in parallel. The crossover probability P_c and mutation probability P_m of population 1-N increase in turn. The population with smaller values of P_c and P_m focus on local refinement search, while those with larger values are used for global search. In the process of population evolution, the migration operator periodically replaces a certain number of the worst individuals in each population with the same number of the best individuals in adjacent population to realize the co-evolution between populations. In each generation of evolution, the optimal individuals of population 1-N are selected by artificial selection operator and stored in the elite population. Then, the optimal individual in the elite population is chose based on the fitness ranking, and its evolution stagnation is judged. If the set optimal individual maintenance generation is reached, the evolution is terminated.

4.2 MPG algorithm simulation

For test the performance of MPG algorithm in solving the data processing model Eq. (20), some preparatory operation needs to be set up before setting simulation experiments:

(1) Determine encoding mode
There are two types of encoding for individuals in MPG algorithm: binary encoding and non-binary encoding. For the requirements of this article, the most widely used real number encoding in non-binary class encoding is chosen, which is more efficient and more suitable for multi-dimensional high-precision numerical optimization problems.
(2) Setting MPG algorithm parameters
The number of populations N is set to 5, and the number of individuals in each population is 500, 400, 300, 200, 100, respectively. The optimal population retention generation is 55. The crossover probability P_c of each population is equispaced generated in the range of 0.8-0.95, and the mutation probability P_m of each population is equispaced generated in the range of 0.1-0.2. The individual migration rate of the population is set to 2%, and the interval generation of population immigration is 3.
(3) Construct fitness function
As with the SRCS, MPG algorithm also converts the data processing model Eq. (20) to an unconstrained optimization problem within fixed feasible region by penalty function. The feasible region, the restraint coefficient matrix A and the restraint vector b in the algorithm calculation process is the same as set in the simulation experiment by using SRCS algorithm. For the infeasible solution individuals which violate the constraint condition Eq. (19), these are punished when calculating the individual fitness, so as to reduce the fitness of the individual and reduce the opportunity to be inherited to the next generation. After several generations of evolution, the population finally converges to the feasible solution. The fitness function Eq. (24 of MPG algorithm is as follows:
$(24)$${F_S}[exIdx] = {F_S}[exIdx] + \alpha \cdot \{{Max({F_S}) - Min({F_S}) + \beta } \}\textrm{ }(S = 1,\ldots ,N)$$$ where F_S is a matrix of fitness function values for the Sth population, exIdx is the index of individual in the population that violates constraint condition Eq. (19), $\textrm{Max(}{\textrm{F}_\textrm{S}}\textrm{)}$ is the value of the fitness function of the individual with the highest fitness in the Sth population, $\textrm{Min(}{\textrm{F}_\textrm{S}}\textrm{)}$ is the value of the fitness function of the individual with the lowest fitness in the Sth population, α and β are the penalty factor and the penalty minimum offset, respectively. After many simulation experiments, the values of α are set to 2 and β to 1.

Based on the principle of MPG algorithm and the above steps, six kinds of emissivity models which are the same as the SRCS algorithm is applied to simulate for verifying effectiveness of MPG algorithm with and without noise. MPG algorithm is used to repeat the calculation for 100 times. The relative errors of calculated temperature and the average results of temperature simulation are shown in Fig. 5 and Table 4 for six kinds of material models, respectively.

Fig. 5. Relative errors of temperature calculation for six models by MPG algorithm.

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Table 4. Average Results of Temperature Simulation by MPG Algorithm

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To test the anti-noise capability of MPG algorithm, 5% random noise is added to the voltage signal in Eq. (4). The relative errors of calculated temperature are shown in Fig. 6, and the average results of temperature simulation are shown in Table 5.

Fig. 6. Relative errors of temperature calculation for six models by MPG algorithm (5% random noise).

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Table 5. Average Results of Temperature Simulation by MPG Algorithm (5% Random Noise)

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As can be seen intuitively from Tables 4 and 5, the simulation results of MPG algorithm with noise are basically the same or even better than those without noise, the maximum relative error for single calculated temperature is below 0.7%. Thus, MPG algorithm has excellent anti-noise capability and accuracy. When 5% random noise is added, the maximum average absolute error is about 6 K, the mean relative error is less than 0.35%, and the average sample standard deviation is about 2.1. As shown in Fig. 7, the inversion results for spectral emissivity of changing with the wavelength exactly agree with true distribution of emissivity for the six emissivity models in Table 1, seeing that MPG algorithm can be used for inversion of some complex emissivity models. The calculation time of MPG algorithm is short, and the average cost is about 0.36 s. Therefore, MPG algorithm is another method to optimization model of data processing for multispectral thermometer. (Simulation environment: Python 3.8; Windows 10; AMD Ryzen Threadripper 3960X 24-Core Processor @3.79 GHz; 64G RAM). The algorithm is written in the Windows 10 environment using the Geatpy 2.6.0 framework.

5. Comparison of SRCS and MPG algorithm

For contrast the performance of MPG algorithm and SRCS algorithm more intuitively, the two algorithms are compared profoundly under the same simulation environment for six emissivity models in Table 1 including inversion emissivity, calculation time, temperature error and algorithm stability. Since the actual measured signal value will inevitably have certain noise, the following simulation results are obtained with noise to verify proposed algorithm practicability.

It can be seen from Fig. 7 that the emissivity inversion results of MPG algorithm are obviously more consistent with the real distribution than that of SRCS algorithm. From Fig. 8, the average calculation speed of MPG algorithm is nearly twice faster than SRCS algorithm, showing that MPG algorithm has higher efficient. The mean temperature inversion results of MPG algorithm has less errors than SRCS algorithm, which accuracy is about twice times more than SRCS algorithm. The average sample variance of MPG algorithm is also less than SRCS algorithm, so MPG algorithm has better stability. In summary MPG algorithm is obviously better than SRCS algorithm in all the above aspects, so MPG algorithm is more suitable for solving data processing for multispectral thermometer.

Fig. 7. Emissivity comparison of two algorithms for six models (‘set’ represents real value).

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Fig. 8. Calculation time, temperature relative error, sample standard deviation comparison of two algorithms for six models.

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6. Experiment

In order to verify the effectiveness of MPG algorithm in practical application, the algorithm is applied to measure the temperature of two samples silicon carbide and tungsten. In experiment, the temperature of blackbody was set to 973 K, while the silicon carbide was heated to 1023, 1073 and 1123 K respectively. The surface temperature of silicon carbide was monitored by an S-type thermocouple with the measurement error less than 1 K. When the surface temperature of silicon carbide was stabilized sufficient, the radiation intensities were collected by the optical fiber spectrometer (Idea Optics NIR900-2500). The radiation signals from eight wavelength channels are given in Table 6.

Table 6. Output Signal of Silicon Carbide Measured by the Optical Fiber Spectrometer (unit: Counts)

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The emissivity values of silicon carbide measured by optical fiber spectrometer and tungsten referenced from literature [28] at certain temperatures are listed in Tables 7 and 8 respectively. Then the temperature and emissivity values of two samples were calculated by the MPG algorithm, with the derived emissivity for diverse wavelengths at three different temperatures to compare with the true values.

Table 7. Spectral Emissivity Data for Silicon Carbide (Measured by Optical Fiber Spectrometer)

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Table 8. Spectral Emissivity Data for Tungsten (From [28])

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Each time the temperature values calculated by MPG algorithm are different, but the calculated results are close enough to the true values. The following inversion results are the average after 100 repeated calculations. The parameters of MPG algorithm are consistent with those in the simulation experiment for the above hypothetical material model. By consulting the literature [28–31], the emissivity range of silicon carbide and tungsten in the above wavelength and temperature ranges can be roughly determined. In order to improve the calculation speed and inversion accuracy, the feasible region of the two sample is set to (0.5, 0.99) and (0.01, 0.5) respectively during the calculation process of the algorithm.

Temperature inversion results obtained by the MPG algorithm are shown in Table 9. It can be seen that for silicon carbide, the absolute and relative error of calculated temperature are less than 3 K and 0.3% respectively, and the average calculation time is 0.35 s; for tungsten, the absolute error is below 10 K, the relative error does not exceed 0.4%, and the average calculation time is 0.36 s. As shown in Fig. 9, the trends of spectral emissivity of the two samples obtained by inversion also excellent agree with real distribution. Therefore, the proposed MPG algorithm can be well applied to the actual measurement of temperature and emissivity.

Fig. 9. Comparison between measured and calculated spectral emissivity for two sample materials. (a), (b), (c) are silicon carbide; (d), (e), (f) are tungsten.

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Table 9. Temperature Inversion Results by MPG Algorithm

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7. Conclusions

Based on the relationship between the ratio of radiation signal of adjacent wavelength and its emissivity, a novel data processing model of multispectral thermometer is proposed. Founded on the non-gradient optimization theory, two new multispectral thermometer data processing algorithms, SRCS algorithm and MPG algorithm, each of which requires neither the assuming of spectral emissivity model nor the selecting of appropriate initial emissivity solution, are proposed. The simulation results display that although the two algorithms can achieve the effective inversion of true temperature and spectral emissivity simultaneously, the MPG algorithm is superior to SRCS algorithm in all aspects, including accuracy, efficiency and stability. Excellent performance shows the superiority of MPG algorithm in the novel data processing model. The practicability of MPG algorithm is also verified by the two samples with the calculated temperature relative error does not exceed 0.4%. In summary, the established data processing method ensures accurate and efficient temperature estimation without needing to assume the spectral emissivity model, the method can be thus applied to real-time measurement of high-temperature in industrial fields. Meanwhile the proposed two algorithms do not have special require the form of function, and provide an optimization idea for the subsequent research to propose more complex and reliable multispectral thermometry models.

Funding

National Natural Science Foundation of China (62075058, U1804261); Innovation Scientists and Technicians Troop Construction Projects of Henan Province (224000510007); Natural Science Foundation of Henan Province (222300420011, 222300420209); Key Scientific Research Project of Colleges and Universities in Henan Province (22A140021); Key Scientific and Technological Project of Xinxiang City (GG2020002); Outstanding Youth Foundation of Henan Normal University (20200171).

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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Target	0.4 µm	0.5 µm	0.6 µm	0.7 µm	0.8 µm	0.9 µm	1.0 µm	1.1 µm
A	0.85	0.75	0.67	0.60	0.55	0.50	0.48	0.45
B	0.45	0.48	0.50	0.55	0.60	0.67	0.75	0.85
C	0.45	0.55	0.65	0.75	0.74	0.65	0.55	0.45
D	0.85	0.75	0.65	0.55	0.54	0.65	0.75	0.80
E	0.85	0.65	0.55	0.65	0.84	0.65	0.55	0.50
F	0.50	0.55	0.65	0.84	0.65	0.55	0.65	0.85

T = 1800K	A	B	C	D	E	F
Inversion Temperature (K)	1801.32	1795.60	1806.39	1808.97	1806.21	1809.02
Absolute Error (K)	1.32	4.40	6.39	8.97	6.21	9.02
Relative Error (%)	0.07	0.24	0.36	0.50	0.35	0.50
Sample Standard Deviation	3.06	2.56	2.63	2.95	2.89	2.80

T = 1800 K	A	B	C	D	E	F
Inversion Temperature (K)	1801.16	1795.97	1806.36	1808.94	1806.25	1810.36
Absolute Error (K)	1.16	4.03	6.36	8.94	6.25	10.36
Relative Error (%)	0.06	0.22	0.35	0.50	0.35	0.58
Sample Standard Deviation	2.67	2.75	2.56	3.15	2.85	2.84

T = 1800K	A	B	C	D	E	F
Inversion Temperature (K)	1796.21	1801.37	1793.75	1806.03	1798.91	1805.41
Absolute Error (K)	3.79	1.37	6.25	6.03	1.08	5.41
Relative Error (%)	0.21	0.08	0.35	0.34	0.06	0.30
Sample Standard Deviation	1.82	2.74	1.70	2.35	3.07	2.91

T = 1800 K	A	B	C	D	E	F
Inversion Temperature (K)	1797.55	1801.47	1794.58	1805.77	1800.92	1804.76
Absolute Error (K)	2.46	1.47	5.42	5.77	0.92	4.76
Relative Error (%)	0.14	0.08	0.30	0.32	0.05	0.26
Sample Standard Deviation	1.63	2.71	1.76	2.60	1.59	2.54

Data processing method for simultaneous estimation of temperature and emissivity in multispectral thermometry

Abstract

1. Introduction

2. Principles of multispectral thermometry

2.1 Reference temperature model

2.2 Model of data processing for multispectral thermometry

2.2.1 Establishment of objective function

2.2.2 Constructing constraints of optimization problem

3. SRCS data processing algorithm

3.1 Principle of SRCS algorithm

3.2 SRCS algorithm simulation

4. MPG data processing algorithm

4.1 Genetic algorithm and Multi-Population genetic structure

4.2 MPG algorithm simulation

5. Comparison of SRCS and MPG algorithm

6. Experiment

7. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (9)

Equations (30)

Optics Express

Temperature (K)		Wavelength (µm)
		1.441	1.538	1.635	1.733	1.830	1.928	2.027	2.121
Blackbody	973	1509.7	2556.4	4061.5	5961.8	7907.9	10458.6	12123.0	12882.7
Sample	1023	2005.8	3294.1	5122.9	7343.9	9495.7	12382.7	14088.3	14785.6
	1073	3160.4	5057.6	7663.9	10744.5	13628.5	17435.6	19506.4	20202.5
	1123	4732.9	7376.5	10928.8	15022.3	18751.3	23590.1	25979.2	26595.8

Temperature (K)	Wavelength (µm)
Temperature (K)	1.441	1.538	1.635	1.733	1.830	1.928	2.027	2.121
1023	0.8096	0.8088	0.8133	0.8141	0.8107	0.8150	0.8141	0.8162
1073	0.8129	0.8125	0.8148	0.8165	0.8164	0.8183	0.8159	0.8174
1123	0.8069	0.8053	0.8084	0.8104	0.8103	0.8136	0.8120	0.8138

Temperature (K)	Wavelength (µm)
Temperature (K)	0.300	0.476	0.665	0.800	1.000	1.500	1.800	2.000
2000	0.498	0.469	0.435	0.414	0.380	0.287	0.215	0.191
2600	0.492	0.460	0.423	0.403	0.373	0.297	0.245	0.222
3200	0.486	0.452	0.411	0.392	0.366	0.305	0.273	0.255

Sample	Sample Temperature (K)	Inversion Temperature (K)	Absolute Error (K)	Relative Error (%)	Calculation Time (s)
Silicon Carbide	1023	1025.56	2.51	0.25	0.34
	1073	1075.44	1.76	0.16	0.35
	1123	1120.89	1.46	0.13	0.35
Tungsten	2000	2004.54	4.54	0.23	0.36
	2600	2609.83	9.83	0.38	0.36
	3200	3208.85	8.85	0.28	0.37