Optimizing MRT data processing via comparative analysis of SA and PSO algorithms: a simulation and numerical study

Kaihua Zhang; Yanfen Xu; Weiling Gao; Kun Yu; Kun Yu; Yufang Liu; Yufang Liu

doi:10.1364/OE.493697

1. Introduction

Non-contact temperature measurement is a fast, safe, and versatile method that has many applications in various fields. A promising method is fluorescent nanodiamonds for optical biosensing and disease diagnosis, which are nanosized diamond particles containing nitrogen-vacancy centers that emit stable fluorescence under excitation [1]. Another promising method is multispectral radiation thermometry (MRT) that relies on the radiation characteristics of objects [2–4]. This technique involves measuring and analyzing the radiation energy emitted from an object's surface at different wavelengths to calculate its temperature [5]. To obtain accurate temperature data, it is typically necessary to measure the radiation energy of the surface at multiple wavelengths. MRT is widely used in various fields, including industrial, scientific, and medical applications, particularly in scenarios where high-temperature, non-contact, or high-precision temperature measurement is required [6–8]. As a result, this technique holds significant practical value.

The target emissivity is a key factor for temperature measurement in MRT, as it affects the adjustment of algorithm parameters and the selection of emissivity models. However, the unknown emissivity still poses a significant challenge in the data processing of MRT. The accurate determination of target temperature relies heavily on the emissivity, which is typically unknown and can vary with wavelength [9,10]. As a result, advanced data processing methods such as inversion algorithms and numerical simulations are often required to estimate and correct for emissivity variations, and to achieve accurate temperature measurements.

In recent years, researchers have made a great deal of effort to optimize the algorithms of MRT technology. The advances in this area can be categorized into several aspects, including algorithm model optimization, feature extraction and selection, data augmentation, and noise removal. In MRT, selecting an appropriate algorithm model is crucial for improving measurement accuracy and reducing errors. Therefore, researchers have proposed new algorithm models based on convolutional neural networks to achieve good results in theory and experiment. Zhang Z. et al. proposed a data processing method based on double-stage emissivity neural network, which introduces a novel structure with two stages of neural network for estimating the emissivity and temperature, and achieves high accuracy and stability in practical applications [11]. Xing J. et al. proposed a data processing algorithm based on generalized inverse matrix combined with a long short-term memory for MRT, introduced a nonlinear transformation layer and adaptive activation function, and adopted adaptive weight decay algorithm to improve the accuracy and stability of temperature measurement [12]. Later that year, they proposed a data processing algorithm based on generalized inverse matrix-recurrent neural network, which combines generalized inverse matrix and recurrent neural network, and an adaptive weight decay algorithm is used to improve the accuracy and stability of temperature measurement [13]. Feature extraction and selection play important roles in the accuracy and precision of MRT. Over the past few years, novel methods based on adaptive Bayesian optimization have been proposed by researchers for feature extraction and selection. These methods have achieved good results on different datasets. Zou Z et al. utilized Bayesian optimization to optimize the hyper parameters of mixed kernel support vector regression for MRT, which improves the accuracy and efficiency of temperature measurement and shows the potential of Bayesian optimization in optimization problems of complex models [14]. The application of Bayesian optimization to improve the performance of deterministic techniques for simultaneous temperature and emissivity estimation of metals around their melting points is demonstrated by Pierre T. et al., which shows the effectiveness of Bayesian optimization in optimizing complex parameters and reducing computational cost [15]. Data augmentation involves transforming and expanding the training data to increase its diversity and quantity, thereby improving the model's generalization ability and robustness. Chen L. et al. presented an innovative approach to improve the accuracy of MRT in challenging measurement scenarios by adding noise to the measurement data. The authors demonstrated that this approach can effectively mitigate the impact of measurement errors caused by background radiation, leading to significantly improved temperature measurements. The proposed method provided a practical solution for accurate temperature measurement in high temperature and intense reflection environments [16]. MRT is subject to various types of noise, such as system noise, environmental noise, and image noise, which can affect measurement accuracy and precision. Therefore, researchers have proposed new methods for noise removal, such as those based on wavelet transform and machine learning. Wang Y. et al. applied wavelet transform to extract features from the basic oxygen furnace (BOF) temperature signal and input them into a wavelet network for modeling and prediction. By reducing the dimensions of wavelet coefficients, the model's accuracy and generalization ability can be improved. Additionally, the method can adaptively adjust the structure and parameters of the wavelet network, further enhancing prediction accuracy [17]. A MRT data processing method based on an adaptive emissivity model that improves measurement accuracy under high-temperature backgrounds was proposed by Chen L. et al [18]. This method combined radiation intensity from multiple spectral bands with an adaptive emissivity model, and determined model parameters through optimization algorithms to achieve accurate measurement of target surface temperature. Han J. et al. proposed a machine learning-based approach to recognize tool wear status in turning by measuring cutting temperature using fiber-optic MRT [19]. The method utilized support vector machines and random forests to classify tool wear states based on cutting temperature features. In summary, researchers have proposed various algorithms for MRT, which have improved the accuracy and stability of temperature measurement to varying degrees, and achieved certain research results. However, these algorithms still have some problems, such as local optimal solutions and slow convergence speed. In addition, there is no comprehensive comparison of the algorithms for MRT temperature inversion in the literature.

This paper proposes a data processing method based on PSO and SA algorithms to achieve simultaneous inversion of temperature and emissivity, which can achieve global optimal solution with fast convergence speed and strong robustness. This method can be applied to some possible industrial applications, such as annealing, welding, flame, and high-speed rotating objects. Six materials with hypothetical emissivity distributions are simulated, and the effectiveness of the algorithms is verified using measured data from a rocket engine nozzle. Finally, the performance of two algorithms in terms of accuracy, robustness, and computational efficiency is compared and analyzed. Some guidelines on how to choose the most suitable technique for different scenarios is also provided.

2. Principles of MRT

2.1 Theory of temperature measurement

For a multi-wavelength radiation thermometer with n spectral channels, the output signal V_i of the ith spectral channel according to Planck's formula is

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

where A_λi is a wavelength-dependent but temperature-independent calibration factor influenced by the spectral responsivity of the detector, the optical element and instrument geometry and the radiation constant, ɛ(λ_i, T) the spectral emissivity at temperature T, λ_i the effective wavelength of the measured target in the ith spectral channel, C₂ the second radiation constant.

Via Wien approximation, the output signal V_i can be simplified as

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

The output signal V_i^’ of blackbody in the ith spectral channel can be expressed as

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

where ɛ(λ_i, T^’) is the spectral emissivity of blackbody at temperature T^’, which usually be equal to 1.

Combining Eq. (2) with Eq. (3), the ratio of V_i to V_i^’ can be written as

(4)$$\frac{{{V_i}}}{{V_i^{\prime}}}\textrm{ = }\varepsilon \textrm{(}{\lambda _i}\textrm{,}T\textrm{)}{e^{\frac{{{C_\textrm{2}}}}{{{\lambda _i}{T^{\prime}}}} - \frac{{{C_\textrm{2}}}}{{{\lambda _i}T}}}}\textrm{ (}i\textrm{ = 1, 2, 3}, \textrm{ }\ldots n\textrm{)}$$

After taking the logarithms of both sides on Eq. (4) and then operations, Eq. (4) can be deformed into

(5)$$\ln \left( {\frac{{{V_i}}}{{V_i^{\prime}}}} \right) - \frac{{{C_\textrm{2}}}}{{{\lambda _i}{T^{\prime}}}}\textrm{ = ln}\varepsilon \textrm{(}{\lambda _i}\textrm{,}T\textrm{)} - \frac{{{C_\textrm{2}}}}{{{\lambda _i}T}}\textrm{ (}i\textrm{ = 1, 2, 3}, \textrm{ }\ldots n\textrm{)}$$

Based on the mathematic model of reference temperature, the influence of calibration factor can be ignored. The output signal V_i^’ in the ith spectral channel only needs to be measured at arbitrary reference temperature T^’ to obtain the true temperature T and emissivity ɛ(λ_i, T) of sample, which will not be affected by the reference temperature T^’ as long as it is stable enough. Comparing to the mathematic models of the calibration constant and the bright temperature, this model has the advantages of simple, user-friendly and the measured results are independent of external factors.

2.2 Constrained optimization of MRT

The definition of optimization algorithm is to find the maximum or minimum values under certain constraints.

(6)$$\left\{ \begin{array}{l} \min f(x)\\ \;Ax \ge b \end{array} \right.$$

where f(x) is the objective function, A the constraint vector coefficient, b the restraint vector.

According to Eq. (4), the essence of MRT is to obtain a set of emissivity to solve for the true temperature of the target in each channel, and the measured values of true temperature are consistent. In practice measurement, the measured temperature is different from the true temperature in each channel. If the deviation is infinitely close to zero, the error between the measured temperature and the true temperature will also be infinitely close. Thus, the following optimization equation is constructed as

(7)$$\min f(x) = \sum\limits_{i = 1}^n {{{[{{T_i} - E({{T_i}} )} ]}^2}} \to 0$$

where T_i is the true temperature, E(T_i) the average temperature of all channels. The true temperature T_i and the average temperature of all channels E(T_i) can be expressed as

(8)$${T_i} = \frac{{{c_2}}}{{{\lambda _i}}} \cdot \frac{1}{{{x_i} + {D_i}}}$$

(9)$$E({T_i}) = \frac{1}{n}\sum\limits_{i = 1}^n {\frac{{{c_2}}}{{{\lambda _i}}} \cdot \frac{1}{{{x_i} + {D_i}}}}$$

where x_i= lnɛ(λ_i, T), D_i= c₂/ λ_iT’- lny_i.

Then, Eq. (7) can be converted to

(10)$$\min f(x) = \sum\limits_{i = 1}^n {{{\left( {\frac{{{c_2}}}{{{\lambda_i}}} \cdot \frac{1}{{{x_i} + {D_i}}} - \frac{1}{n}\sum\limits_{i = 1}^n {\frac{{{c_2}}}{{{\lambda_i}}} \cdot \frac{1}{{{x_i} + {D_i}}}} } \right)}^2}}$$

In multispectral radiometry, the value of spectral emissivity is in [0,1], so the constraint range of x_i is not more than 0. So, Eq. (6) can be written as

(11)$$\left\{ \begin{array}{l} \min f(x) = \sum\limits_{i = 1}^n {{{\left( {\frac{{{c_2}}}{{{\lambda_i}}} \cdot \frac{1}{{{x_i} + {D_i}}} - \frac{1}{n}\sum\limits_{i = 1}^n {\frac{{{c_2}}}{{{\lambda_i}}} \cdot \frac{1}{{{x_i} + {D_i}}}} } \right)}^2}} \\ {x_i} \le 0 \end{array} \right.$$

This is a standard constrained optimization problem, where f(x) is the objective function, x_i is the equality constraint. The constrained problem can be resolved by the PSO algorithm, which is a synthetical algorithm of two kinds of intelligent optimization algorithms. The basic thought of the PSO algorithm is to start from a group of initial points satisfying the constraint conditions, find new feasible points according to the optimal suitability value of the objective function and iterate until the optimal value is found.

3. SA algorithms

3.1 Principle of the SA algorithm

The simulated annealing (SA) algorithm is a stochastic search algorithm that simulates the annealing cooling of a solid in thermodynamics [20]. The SA algorithm can simulate the internal energy optimization problem as an objective function and the temperature T as a control parameter, starting from an initial solution large enough to generate a new solution by random perturbation in its domain, and judging whether to accept the solution that worsens the value of the objective function within a limited range according to a criterion. The algorithm iterates through the process of ‘generate new solution – calculate objective function difference – accept or discard’ until the termination rule is satisfied, resulting in a globally optimal solution to the combinatorial optimization problem.

The SA is essentially a random searching . In a practical combinatorial optimization problem, the process of finding the optimal solution is similar to the solid annealing process described above: the object function f(X_i) of the optimization problem is modelled as the energy of the material system in the state X_i, the control parameter t is modelled as the temperature and then slowly decreases from a sufficiently high temperature; modelling the thermal equilibrium state of the solid at this control parameter t according to the Metropolis criterion is a random perturbation of the current state X_i in its field to produce a new state X_i; the acceptance of the new state is judged using the probability P corresponding to the Metropolis criterion; The iterates through the process of ‘generate new solution - calculate objective function difference - accept or discard’ until the termination rule is satisfied, and finally the global optimal solution of the combinatorial optimization problem is obtained.

For solid cooling, to facilitate temperature regulation the temperature at i + 1 is α times the temperature at i, as shown as

(12)$${T_{i + 1}} = \alpha {T_i}$$

where T_i is the current annealing temperature, α is the attenuation coefficient of temperature. The current solution generates a new solution in its domain as

(13)$${X_j} = {[{{X_1} + \Delta {X_1},{X_2} + \Delta {X_2}, \cdots ,{X_n} + \Delta {X_n}} ]^T}$$

where ΔX_i (i = 1, 2,…, n) is the span of each variable T_k change, which is decreases with decreasing T. X_j is the new solution, X_i (i = 1, 2,…, n) is the current solution.

The judging criterion for acceptance of the new state is the Metropolis criterion, which can be expressed as

(14)$$P = \left\{ \begin{array}{l} 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f({X_j}) \le f({X_i})\;\;\;\;\;\;\\ \textrm{exp} \left\{ {\frac{{ - [{f({X_i}) - f({X_j})} ]}}{{{T_i}}}} \right\}\;\;\;\;f({X_j}), f({X_i}) \end{array} \right.$$

where f(X_i) is the objective function corresponding to the current solution, f(X_j) is the objective function corresponding to the new solution, the acceptance of the new state is judged using the probability P. A Markov chain is a set of discrete random variables with Markov properties, which can be described by the mathematical formula as follows:

(15)$$p({X_{t + 1}}|{X_t}, \ldots ,{X_1}) = p({X_{t + 1}}|{X_t})$$

The calculation steps for simulated annealing are as follows:

Step 1: Initializing parameters and setting initial solutions including the initial temperature T₀, the final temperature T and the cooling rate α (α∈ [0,1]). The length of Markov chains L₀ represents the number of new solutions produced at each temperature. A solution is randomly selected in the feasible domain as the initial solution of the optimization problem.

Step 2: A new neighborhood solution produced by the state function centered on the current solution X_i, and the objective function is calculated.

Step 3: The Metropolis criterion is used to compare the objective function of the two states and determine whether to accept the new solution X_j. When f(X_j) < f(X_i), the system will accept directly X_j as the new solution. When f(X_j) ≥ f(X_i), the system will accept directly X_i as the new solution.

Step 4: Steps 2 and 3 are updated L₀ times, a new temperature value can be obtained based on Eq. (11).

Step 5: When the temperature reaches the stop temperature, the step 6 is executed in sequence. Otherwise, the step 2 is executed renewedly.

Step 6: The current solution is output as the optimal solution.

3.2 Simulation experiment of the SA algorithm

Based on the principle of the SA , simulation experiments are carried on for six materials A-F with different emissivity tendency. The spectral emissivity at different wavelengths and the temperature of 800 K for six materials A-F is shown in Table 1. In this experiment, the temperature of reference blackbody is at 700 K. For the selection of spectral channels, the accuracy of inversion results can be affected if the number of spectral channels is small. Conversely, the computational efficiency of the will reduce. Thus, eight effective spectral channels are selected in this article, and the effective wavelengths of different channels are respectively at 3.3, 3.5, 3.7, 3.9, 4.1, 4.3, 4.5 and 4.7 µm.

Table 1. The spectral emissivity of six different materials at 800 K

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The initial solution is set according to Eq. (11), and the feasible domain X is set to a continuous space where the lower limit of the search is 0 and the upper limit is 1. Because the search of the objective function in the feasible solution space is random, it will affect the search accuracy and prolong the computation time when the feasible region is too large. In order to obtain accurate inversion results, the feasible domain can be reduced appropriately. After several experimental simulations, the ideal emissivity range for samples A-F is from 0.4 to 0.9.

According to the initialization requirements for the SA algorithm, the initialization requirement T₀, the final temperature T and the length of Markov chains L₀ are severally set to 100 K, 1 K and 100 during the calculation process. The results simulated by the SA algorithm at 800 K are shown in Table 2.

Table 2. The results of temperatures simulated by the SA algorithm

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In order to verify the anti-noise ability of the SA algorithm, the 5% random noise is added to the voltage signal in Eq. (4). The simulation results of temperature, the absolute error and relative error at 800 K are shown in Table 3.

Table 3. The results of temperatures simulated by the SA algorithm (5% random noise)

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From Tables 2 and 3, the SA algorithm has a high accuracy. The results indicate that the maximum absolute error is 12.9 K and the maximum relative error is 1.61%. When the 5% random noise is added to the voltage signal, the maximum absolute error is less than 16 K and the maximum relative error is less than 2%. It can be found that the trend of emissivity with wavelength obtained by inversion is consistent with the actual value in Fig. 3, which indicates that this method can be used for multi-spectral emissivity inversion. What’s more, the maximum calculation time with random noise is 0.43 s, and the maximum calculation time without random noise is 0.4 s. (Simulation environment: Python 3.6; Windows 7; Intel Core i7-9700 CPU @3.00 GHz; 8 G RAM). Above all, the maximum relative error of temperature with random noise is 1.2 times that without random noise. Thus, the SA algorithm cannot meet the requirement of industrial measurement.

4. PSO algorithms

4.1 Principle of the PSO algorithm

Particle swarm optimization algorithm (PSO) is a function optimization technique inspired by the regularity of bird swarm activity [21]. The algorithm can be described mathematically as following: A population x = (x₁, x_2, …_, x_n) is formed by N particles in a D-dimensional space. For each particle i in the D-dimensional space is represented by a one-dimensional vector x_i, which direction and distance of spatial movement is represented by the velocity vector v_i. The total number of particle iterations in the population is T, the position and velocity vectors of the particle i at the tth iteration are denoted as x_i= [(x_i₁, x_i₂, x_i_3, …_, x_iD)]^t and v_i= [(v_i₁, v_i₂, v_i_3, …_, v_iD)]^t. By generation t, the optimal position found by the ith particle is denoted as Pbest(t) = [(Pbest_i₁, Pbest_i₂, …, Pbest_id)]^t, and the optimal position of whole population is denoted as gbest. According to individual historical experience and information sharing among populations, the ‘flight’ speed of particle is dynamically changed to update the spatial position of each particle for optimization. The position and velocity of particle can be mathematically expressed as

(16)$${x_{id}}({t + 1} )= {x_{id}}(t )+ {v_{id}}({t + 1} )$$

(17)$${v_{id}}({t + 1} )= w \cdot {v_{id}}(t )+ {c_1}{r_1} \cdot ({Pbes{t_{id}}(t )- {x_{id}}(t )} )+ {c_2}{r_2} \cdot ({gbest(t )- {x_{id}}(t )} )$$

where 1 ≤ i ≤ N and 1 ≤ d ≤ D. t is the number of current iteration, D the space dimension of search, w is the inertial weight, c₁ and c₂ (generally c₁= c₂= 2) the learning factors, r₁ and r₂ the random numbers between 0 and 1. Pbest and gbest are the individual and overall optimal position respectively. The program flow chart of the PSO algorithm are shown in Fig. 1, and the calculation steps of the PSO algorithm are as follows:

Step 1: Initializing the relevant parameters: the number of population P, the maximum number of particle iteration T, the learning factors c₁, c₂ and determine the objective function.

Fig. 1. The program flow chart of the PSO algorithm.

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Step 2: The particle and particle velocity are randomly initialized within a certain feasibility area, and the individual Pbest and overall optimal position gbest are determined by the fitness values of all particles calculated by the objective function.

Step 3: When the current number of iteration is less than the maximum number of iteration, the step 4 is executed in sequence. Otherwise, step 8 is executed renewedly.

Step 4: The position and velocity of particle are updated based on Eqs. (16) and (17), and the current fitness value is calculated.

Step 5: The position and velocity of the particle are processed transgressively.

Step 6: The fitness value of the optimal position of the individual particle is compared with the fitness value of the current particle. If the fitness value of the current particle is better than the optimal value of the individual particle history, the current particle position is set to Pbest, while Pbest is unchanged.

Step 7: The fitness value of the global particle optimal position is compared with the current particle fitness value. If the fitness value of the current particle is better than the fitness value of the optimal position of the global particle, the current particle position is set as gbest, while gbest is unchanged.

Step 8: Stop the iteration, and output the global optimal position and the optimal value.

4.2 Simulation experiment of the PSO algorithm

The simulation experiment is carried out using the PSO algorithm, in which population number represents the number of particles in the population, feasible domain is the limited range of emissivity in the temperature inversion process, the objective function is Eq. (11), particles represents the emissivity value to be solved, particle position is the temperature solution obtained by the algorithm inversion, fitness represents the error between the temperature inverted by the PSO algorithm and the actual temperature. Before the simulation experiment, the following parameters should be set: The number of population and iteration is 20 and 650; The initialization of the PSO algorithm is in agreement with the SA algorithm, and the emissivity range is 0.4 to 0.9; The inertial weight w is mainly to balance the global and local search ability of particles in Eq. (12). When w is too large, the global search ability is strong but the local search ability is weak, the algorithm convergence speed is fast but the optimization accuracy is not high. When w is too small, the opposite is true. The classical inertial weight includes three kinds: constant type weight, the inertial weight of dynamic change and adaptive inertia weight. In this paper, four methods are selected, and their mathematical expressions can be written as

(18)$${w_1} = {w_{\max }} - \frac{t}{T}({{w_{\max }} - {w_{\min }}} )$$

(19)$${w_2} = 0.729$$

(20)$${w_3} = 0.5 + rand({0,1} )/2$$

(21)$$\begin{array}{l} IS{A_{ij}} = \frac{{|{{x_{id}} - {\textrm{Pbes}{\textrm{t}_{id}}} |} }}{{|{\textrm{Pbes}{\textrm{t}_{id}} - {g\textrm{bes}{\textrm{t}_i}} |+ \varepsilon } }}\\ {w_4} = 1 - \alpha \left( {\frac{1}{{1 + {e^{ - IS{A_{ij}}}}}}} \right) \end{array}$$

Based on the principle and parameter selection of the PSO algorithm, simulation experiments are carried on for six materials A-F with different emissivity tendency to verify the influence of random noise on the effectiveness of the PSO algorithm.

The inversion results of emissivity with different inertia weights are shown in Fig. 2. The results indicate that the inversion results of emissivity with different inertia weights are very close, so the influence of inertia weight w on the inversion results of emissivity can be ignored.

Fig. 2. Comparison of the spectral emissivity in eight channels for six materials A-F. w_1, w_2, w_3, w₄ represent linear attenuation inertia weight, constant term inertia weight, random inertia weight, and adaptive inertia weight respectively.

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Fig. 3. The simulation results of spectral emissivity for six typical materials with noise.

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The relative errors of temperature with different inertia weight w are shown in Table 4. It can be seen that the inertia weights are less than 2.2%, which indicates that the inversion results are good. And the inertia weights w₁ and w₃ are within 1.6%.

Table 4. The simulation results of the PSO algorithm

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According to Eq. (11), the optimal solution is that the object function is as close as 0 as possible. The optimal solutions with different inertia weight w are shown in Table 5. Comparing the simulation results of six materials, the object function is closest to 0 and most stable with the inertia weights w₁ and w₄.

Table 5. The simulation results of object function under optimal solution simulated by the SA algorithm

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The results of computation time with different inertia weights are shown in Table 6, it can be found that the effect of inertia weight on computation time can ignore.

Table 6. The computation time with different inertia weights simulated by the SA algorithm (unit: s)

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Combining the analysis mentioned above, the accuracy and calculation efficiency of emissivity inversion are relatively high with the inertia weight w₁. The simulation results of temperature, the absolute error and relative error at 800 K are shown in Table 7.

Table 7. The results of temperatures simulated by the PSO algorithm

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In order to verify the anti-noise ability of the PSO , the 5% random noise is added to the voltage signal in Eq. (4). The simulation results of temperature, the absolute error and relative error at 800 K are shown in Table 8.

Table 8. The results of temperatures simulated by the PSO (5% random noise)

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When the 5% random noise is added to the voltage signal, the maximum absolute error is less than 9 K and the maximum relative error is less than 1.06%. The accuracy is higher than that without random noise, indicating that the PSO has certain noise capability. The actual emissivity values of six materials are compared with the inverse emissivity value as shown in Fig. 3. The calculated results of the PSO algorithm are consistent with the actual value distribution, indicating that the PSO algorithm can be used to invert emissivity of different models. The inversion time of the PSO algorithm is fast, and the longest is less than 0.3s (Simulation environment: Python 3.6; Windows 7; Intel Core i7-9700 CPU @3.00 GHz; 8 G RAM). The simulation results show that the PSO algorithm has good inversion accuracy and high inversion efficiency. Therefore, the PSO algorithm can be used to directly process multi-wavelength radiation thermometry data.

5. Comparison of SA and PSO algorithms

In order to visually compare the two algorithms, the emissivity, calculation time and relative error of six typical materials are studied under the same hardware conditions. To verify the practicability of the SA and PSO algorithms, the simulation results of spectral emissivity with noise are shown in Fig. 3.

As can be seen in Fig. 3, the simulation results of spectral emissivity obtained by the PSO algorithm are significantly closer to the true distribution than that obtained by SA algorithm, which indicates that the PSO algorithm has better the anti-noise ability than the SA algorithm. The calculation time and relative error of six typical materials are shown in Fig. 4. It can be seen that the simulation temperatures obtained by the PSO algorithm are close to the true temperature because the relative error of the SA algorithm and PSO algorithm are severally 2% and 1% from Fig. 4(a). The calculation time of the PSO algorithm is nearly twice as fast as that the SA algorithm, which manifests the PSO algorithm has high computational efficiency. In summary, the PSO algorithm is superior to the SA algorithm in all aspects. Therefore, the PSO algorithm is more suitable to solve the data processing problem of multispectral thermometer.

Fig. 4. The calculation time and relative error of six typical materials.

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6. Experiment

In order to verify the reliability of the PSO algorithm in practical application, the algorithm is applied to measure the surface temperature of rocket motor nozzle. The measured data of the surface temperatures in the Ref. [6,22] is used as the data source for the PSO algorithm. The radiation signal of a rocket engine nozzles plume flame (Mainly Al₂O₃) at a set temperature of 2490 K was measured using the photodetector (S4111-16Q). The reference blackbody surface temperature is 2252 K, the emissivity is greater than 0.99 and the temperature accuracy is around 2 K. In order to facilitate the comparison between the measurement results and the theoretical adiabatic calculation results, the measurement point of the ground carrying test is located on the axis of the rocket engine, which is about 3 - 5 cm away from the outlet plane of the engine nozzle. The pyrometer completes 8-channel signal acquisition every 5 ms, and a total of 12 sets of data were measured continuously over a period of time. The radiation signals at eight wavelength channels are displayed in Table 9, and the practical data of rocket engine nozzles at different measuring times is shown in Table 10.

Table 9. The radiation signals of blackbody at eight wavelength channels in Ref. [22]

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Table 10. Practical data of rocket engine nozzles for 12 measuring times in Ref. [22]

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The parameter values in the PSO algorithm are consistent with the simulation experiment. Since the emissivity of the rocket motor nozzle is not high, the search range of the emissivity is from 0.3 to 0.7 to improve the accuracy of the experiment. The inversion results of temperature obtained by the PSO algorithm are given in Table 11. It can be observed that when the temperature of rocket engine is 2490 K, the maximum absolute error and the maximum relative error are 16.27 K and 0.65%, and the calculation time is less than 0.3 s. Therefore, the proposed PSO algorithm can be well applied to real-time online measurement of temperature and emissivity.

Table 11. Experimental simulation results by the PSO algorithm

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7. Conclusions

The paper presented a comparative analysis of two optimization algorithms, particle swarm optimization (PSO) and simulated annealing (SA), for the data processing of MRT. The simulations of six hypothetical emissivity models were compared, and the results indicated that the PSO algorithm outperformed the SA algorithm in terms of accuracy, efficiency, and stability. The PSO algorithm was further applied to process the measured data of the surface temperature of a rocket motor nozzle, and the results showed that the maximum absolute error and maximum relative error were 16.27 K and 0.65%, respectively, with a calculation time of less than 0.3 s. Based on these findings, the paper concluded that the PSO algorithm could be effectively used in the data processing of MRT for accurate temperature measurement, and the proposed method could be applied in real-time temperature measurement in industrial fields. Therefore, this study provides a valuable contribution to the optimization of MRT data processing and can have practical applications in various industrial fields.

Funding

Program for Innovative Research Team (in Science and Technology) in University of Henan Province (23IRTSTHN013); 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).

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.

References

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	3.3 µm	3.5 µm	3.7 µm	3.9 µm	4.1 µm	4.3 µm	4.5 µm	4.7 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

	A	B	C	D	E	F
True temperature (K)	797.4	802.9	790.8	807.7	790.4	787.1
Absolute error (K)	−2.6	2.9	−9.2	7.7	−9.6	−12.9
Relative error (%)	0.32	0.36	1.15	0.96	1.20	1.61

	A	B	C	D	E	F
True temperature (K)	796.7	791.1	815.5	794.5	794.6	791.2
Absolute error (K)	−3.3	−8.9	15.5	−5.5	−5.4	−8.8
Relative error (%)	0.41	1.11	1.94	0.69	0.68	1.01

w (%)	A	B	C	D	E	F
w₁	0.74	1.15	0.53	0.35	0.31	0.50
w₂	0.29	1.63	1.34	1.08	0.93	1.13
w₃	0.75	1.53	0.83	0.1	0.05	1.08
w₄	2.11	0.80	0.61	0.71	0.66	0.02

	A	B	C	D	E	F
w₁	1.77E-2	2.74E-1	2.8E-4	4.42E-2	2.47E-2	3.71E-2
w₂	1.2E-10	3.91E-1	1.02E-2	4.27E-1	2.70E-2	3.10E-3
w₃	3.7E-9	1.56E-2	6.9E-10	3.19E-8	1.37E-10	2.56E-4
w₄	1.05E-1	2.09E-3	2.93E-3	2.51E-3	2.45E-3	1.14E-2

Optimizing MRT data processing via comparative analysis of SA and PSO algorithms: a simulation and numerical study

Abstract

1. Introduction

2. Principles of MRT

2.1 Theory of temperature measurement

2.2 Constrained optimization of MRT

3. SA algorithms

3.1 Principle of the SA algorithm

3.2 Simulation experiment of the SA algorithm

4. PSO algorithms

4.1 Principle of the PSO algorithm

4.2 Simulation experiment of the PSO algorithm

5. Comparison of SA and PSO algorithms

6. Experiment

7. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (11)

Equations (21)

Optics Express

	A	B	C	D	E	F
w₁	0.257	0.261	0.252	0.269	0.270	0.259
w₂	0.250	0.256	0.252	0.258	0.254	0.256
w₃	0.258	0.254	0.251	0.275	0.254	0.283
w₄	0.266	0.269	0.265	0.268	0.263	0.262

	A	B	C	D	E	F
True temperature (K)	805.9	790.8	795.8	802.8	797.5	804
Absolute error (K)	5.9	−9.2	−4.2	2.8	−2.5	4
Relative error (%)	0.74	1.15	0.53	0.35	0.31	0.5

	A	B	C	D	E	F
True temperature (K)	798.5	791.5	807.8	798.7	805.8	795.3
Absolute error (K)	−1.5	−8.5	7.8	−1.3	5.8	−4.7
Relative error (%)	0.19	1.06	0.98	0.16	0.73	0.59

	1	2	3	4	5	6	7	8
Wavelength (µm)	0.574	0.592	0.623	0.654	0.698	0.748	0.826	0.914
Signal (mV)	39.4	139.7	117.5	363.7	345.0	493.9	320.7	406.7

Measuring Time (s)	V_i(mV) of 8 channels
Measuring Time (s)	1	2	3	4	5	6	7	8
1	46.3	254.1	165.3	481.5	367.8	495.0	273.7	323.5
2	46.3	254.1	170.2	476.6	372.7	500.0	278.6	328.4
3	46.3	244.2	165.3	471.6	362.8	495.0	268.7	323.5
4	46.3	244.2	170.2	481.5	372.7	509.9	283.6	333.4
5	46.3	249.1	160.3	471.6	362.8	490.1	268.7	323.5
6	46.3	244.2	160.3	461.7	352.9	480.2	253.9	303.7
7	41.3	234.3	155.4	456.8	343.0	465.3	248.9	293.8
8	46.3	244.2	160.3	461.7	352.9	475.2	253.9	298.7
9	41.3	239.2	155.4	461.7	343.0	470.3	253.9	303.7
10	41.3	239.2	155.4	456.8	343.0	470.3	248.9	298.7
11	41.3	234.3	155.4	451.8	333.1	460.4	239.0	293.8
12	41.3	239.2	155.4	456.8	343.0	470.3	248.9	298.7

Measuring Time (s)	1	2	3	4	5	6
Inversion temperature (K)	2493.28	2486.98	2489.98	2491.58	2491.10	2488.72
Absolute error (K)	3.28	−3.02	−0.02	1.92	1.10	−1.28
Relative error (%)	0.13	0.12	0.39	0.08	0.04	0.05
Measuring Time (s)	7	8	9	10	11	12
Inversion temperature (K)	2476.94	2476.43	2476.94	2473.73	2486.54	2473.73
Absolute error (K)	−13.06	−13.57	−13.06	−16.27	−3.46	−16.27
Relative error (%)	0.52	0.54	0.52	0.65	0.14	0.65