Theoretical and experimental verification of wide-spectrum thermometry based on Taylor series de-integration method

Qiansong Yu; Shengyu Gu; Yuan Dong

doi:10.1364/OE.512126

1. Introduction

Calculating and measuring surface temperature distribution has always been an important foundation for conducting research in areas such as material properties, metal processing, combustion analysis [1,2], and other fields. The temperature measurement technology based on an array-type wide-spectrum detector is the main method of simultaneously obtaining temperature and its distribution. This method uses Planck’s law to relate the radiative behavior of a thermally emitting surface to temperature, thereby constructing integrated equations about temperature [3], and then obtaining the temperature value by solving it. Therefore, the processing and solving of integral equations have become the major obstacles for multi-channel wide-spectrum temperature measurement technology.

Following a greater depth of research into dual-wavelength temperature measurement methods [4–8], a novel wide-spectrum temperature measurement methodology focusing on the construction of a dual-spectrum temperature measurement algorithm [9,10] has been introduced and subsequently referred to as the dual-color temperature measurement method. The core principle of this technique is to eliminate emissivity and instrument parameters through the utilization of dual-band radiative data ratios, thus effectively reducing the influence of unknown emissivity on temperature measurement [5]. Thus, to eliminate the spectral emissivity term, this term should be moved outside the integral sign in the processed equation. In light of this fact, de-integration techniques have been proposed for the three temperature measurement equations to address this issue. First, a de-integration method based on the Lagrange mean value theorem is proposed to convert the integral equation into a product of the integrated function values at the center wavelength position of the device filter and the proportionality factor (an experimentally calibrated constant) [11,12]. It is also currently the most widely used method in applications. The second approach involves extracting the emissivity term from the integral equation by defining the average spectral emissivity, which must also be calibrated using the same surface [13,14]. The third approach involves approximating the object as a gray body and extracting the emissivity as a constant outside the integral. The multi-band integral equation can then be solved using the least squares method [15,16].

However, as research has progressed, it has become clear that there are drawbacks to all three equation processing methods. Savino et al. conducted a study on the applicable spectral range and the influence of the spacing between the center wavelengths of the filters on measurements using the Lagrange mean value theorem method [17]. The results showed that this processing method is not suitable for long-wavelength positions and that the selection of the center wavelengths of the two bands has a significant impact on measurement results. Wang et al. conducted a study on three-channel pyrometer and found that the measurement error of the processing method varied greatly under different channel ratios [18]. This indicates that there is still insufficient discussion on the selection of the center wavelength and channels in this method. The application of the Lagrange intermediate value theorem using the center wavelength of the filters also lacks theoretical support. In contrast, the second and third equation processing methods preserve the integration operation, which directly increases the computational complexity of subsequent calculations. Additionally, the use of wavelength ratios instead of spectral emissivity ratios, or directly removing spectral emissivity ratios, completely decouples spectral emissivity from temperature calculation, directly hindering the development of multi-channel wide-spectrum technology towards higher measurement accuracy. To fundamentally enhance the measurement capability of this technology and obtain a more reliable approach to temperature determination, changing the traditional treatment of temperature measurement integral equations and providing a more effective algebraic approach to the equations is the starting point for solving the problem.

For this reason, a de-integration method of wide-spectrum temperature measurement integral equation based on Taylor series is proposed. The emissivity applicability of the method and the influence of the emissivity on the characteristic wavelength (optimal expansion wavelength, OEW) are discussed. Three forms of algebraic equations derived from this method are constructed, and the theoretical calculation errors of each method are statistically analyzed. Further, a multi-parameter, multi-sample temperature measurement experiment is conducted, and the measurement results of the three calculation forms are discussed and demonstrated. This method eliminates the constraints of the traditional two-color temperature measurement mode and has strong theoretical support for the selection of characteristic wavelengths (optimal expansion wavelength, OEW), providing stability for the solution of multi-channel wide-spectrum temperature measurement.

2. Taylor series expansion of the integral equation system

By combining Planck’s law with the measurement theory of a three-channel wide-spectrum detector, it is possible to obtain the relationship between the pixel channel gray level and the temperature of a hot surface that satisfies the temperature measurement integral equation system [19,20].

(1)$$\left\{ {\begin{array}{{c}} {{\textrm{H}_\textrm{r}}\textrm{ = }\eta {\kappa_r}\left( {\Phi \int_{{\lambda_1}}^{{\lambda_2}} {{s_r}(\lambda )\varepsilon (\lambda ){I_B}(\lambda ,\textrm{T})d\lambda \textrm{ + }{D_r}} } \right)}\\ {\textrm{ }{\textrm{H}_\textrm{g}}\textrm{ = }\eta {\kappa_g}\left( {\Phi \int_{{\lambda_1}}^{{\lambda_2}} {{s_g}(\lambda )\varepsilon (\lambda ){I_B}(\lambda ,\textrm{T})d\lambda \textrm{ + }{D_g}} } \right)}\\ {{\textrm{H}_\textrm{b}}\textrm{ = }\eta {\kappa_b}\left( {\Phi \int_{{\lambda_1}}^{{\lambda_2}} {{s_b}(\lambda )\varepsilon (\lambda ){I_B}(\lambda ,\textrm{T})d\lambda \textrm{ + }{D_b}} } \right)} \end{array}} \right.$$

where H_i corresponds to the grayscale of each channel of the pixel; s_i represents the spectral response curve of each channel; D_i is the dark noise of each channel; ε(λ) represents spectral emissivity; η is the global gain coefficient of the detector; κ_i is the transfer function of radiation intensity and grayscale; $\varPhi$ is the geometric factor related to the acquisition aperture and exposure time. I_B is the directional radiation intensity of blackbody. Considering that the research is mainly carried out in the shortwave region, it is expressed by Wien approximation [21].

(2)$${I_B}(\lambda ,T) = {c_1}{\lambda ^{ - 5}}\exp \left( { - \frac{{{c_2}}}{{\lambda T}}} \right)$$

To simplify computational processes, the reduced channel signal K_i is defined as the net radiation intensity that can be received by the detector pixel and converted into an electrical signal. Combined with the relationship between the grayscale of the special channel and the target temperature in Eq. (1), K_i can be expressed by the following formula.

(3)$${K_i}(T) = \frac{1}{\Phi }\left( {\kappa_\textrm{i}^{ - 1}\left( {\frac{{{\textrm{H}_\textrm{i}}(T)}}{\eta }} \right) - {\textrm{D}_\textrm{i}}} \right) = \int_{{\lambda _1}}^{{\lambda _2}} {{s_i}(\lambda )\varepsilon (\lambda ){I_B}(\lambda ,T)d\lambda }$$

where κ_i^-1 is the inverse function of the transfer function κ_i of each channel. Solving this equation is the key to wide-spectrum temperature measurement. According to the Taylor expansion relation of multivariate function [22].If the integrand in Eq. (3) is defined as f (λ, T), then after second-order Taylor expansion, Eq. (3) can be rewritten as:

(4)$$\begin{aligned} {K_i}({\lambda _i},T) &= \frac{{\partial \textrm{f}({\lambda _i},T)}}{{\partial \lambda }}\int_{{\lambda _1}}^{{\lambda _2}} {({\lambda - {\lambda_i}} )d\lambda \textrm{ + }} \frac{{{\partial ^2}\textrm{f}({\lambda _{i,0}},T)}}{{2 \cdot \partial {\lambda ^2}}}\int_{{\lambda _1}}^{{\lambda _2}} {{{({\lambda - {\lambda_i}} )}^2}d\lambda } \\& \textrm{ + }f({\lambda _i},T)\int_{{\lambda _1}}^{{\lambda _2}} {d\lambda } \textrm{ + }\int_{{\lambda _1}}^{{\lambda _2}} {\textrm{o}({{{({\lambda - {\lambda_i}} )}^3}} )\textrm{d}\lambda \textrm{ }({i = r,g,b} )} \end{aligned}$$

As mentioned by Eq. (4), in the Taylor expansion, the integrand f (λ, T) and its partial derivative at the position of the expansion wavelength λ_i can be proposed outside the integral sign in the form of a constant. k_i represents the approximated data obtained by neglecting higher-order infinitesimal terms in Eq. (4). Subsequently, it can be rearranged as the following equation:

(5)$${{k}_{i}}({\lambda _i},T) = {f_i}({\lambda _i},T)\varDelta \lambda + \frac{{\partial {\textrm{f}_i}({\lambda _i},T)}}{{\partial \lambda }}{C_{i,1}}({{\lambda_i}} )+ \frac{{{\partial ^2}{\textrm{f}_i}({\lambda _i},T)}}{{2 \cdot \partial {\lambda ^2}}}{C_{i,2}}({{\lambda_i}} )$$

where C_i,1 and C_i,2 are fixed values after the upper and lower limits of the integral and the expansion position are determined, and the specific forms of the first and second derivatives of the integrand are as follows:

(6)$$\begin{array}{cc} {{C_{i,1}}({{\lambda_i}} )= \int_{{\lambda _1}}^{{\lambda _2}} {({\lambda - {\lambda_i}} )d\lambda } }&{{C_{i,2}}({{\lambda_i}} )= \int_{{\lambda _1}}^{{\lambda _2}} {{{({\lambda - {\lambda_i}} )}^2}d\lambda } } \end{array}$$

(7)$$\frac{{\partial {{f}_{i}}({\lambda _i},T)}}{{\partial \lambda }} = \frac{{\partial {s_i}({\lambda _i})}}{{\partial \lambda }}\varepsilon ({\lambda _i}){I_B}({\lambda _i},T)\textrm{ + }{s_i}({\lambda _i})\frac{{\partial \varepsilon ({\lambda _i})}}{{\partial \lambda }}{I_B}({\lambda _i},T)\textrm{ + }{s_i}({\lambda _i})\varepsilon ({\lambda _i},)\frac{{\partial {I_B}({\lambda _i},T)}}{{\partial \lambda }}$$

(8)$$\begin{aligned} \frac{{{\partial ^2}{{f}_i}({\lambda _i},T)}}{{\partial {\lambda ^2}}} &= {I_B}({\lambda _i},T)\left( {\frac{{{\partial^2}\varepsilon ({\lambda_i})}}{{\partial {\lambda^2}}}{s_i}({\lambda_i})\textrm{ + }\varepsilon ({\lambda_i})\frac{{{\partial^2}{s_i}({\lambda_i})}}{{\partial {\lambda^2}}}\textrm{ + 2}\frac{{\partial \varepsilon ({\lambda_i})}}{{\partial \lambda }}\frac{{\partial {s_i}({\lambda_i})}}{{\partial \lambda }}} \right)\\& \textrm{ + }\varepsilon ({\lambda _i})\left( {{s_i}({\lambda_i})\frac{{{\partial^2}{I_B}({\lambda_i},T)}}{{\partial {\lambda^2}}}\textrm{2}\frac{{\partial {s_i}({\lambda_i})}}{{\partial \lambda }}\frac{{\partial {I_B}({\lambda_i},T)}}{{\partial \lambda }}} \right)\textrm{ + 2}\frac{{\partial \varepsilon ({\lambda _i})}}{{\partial \lambda }}{s_i}({\lambda _i})\frac{{\partial {I_B}({\lambda _i},T)}}{{\partial \lambda }} \end{aligned}$$

The applicability of second-order Taylor expansion in equation manipulation is subject to the precondition that the approximated data k_i is either proximate to or coincident with the primary data K_i. In fact, the approximation data k_i and the primary data K_i exhibit disparities since higher-order terms are disregarded. Such disparities are contingent upon the expansion position λ_i[23]. Therefore, the assessment criteria for approximation level and the examination of optimal expansion position (wavelength) are of utmost importance in this methodology.

3. Discussion on approximation level

To facilitate a discussion on the approximation level, a Gaussian-type spectral response function (SRF) [3,17], which satisfies the spectral response characteristics of both the Bayer filter and narrowband filter, is employed in this study. In particular, the standard visible light three-channel wide-spectral detector charge-coupled device (CCD) is taken as an example. The Gaussian function with central wavelengths of 0.45µm, 0.54µm, and 0.63µm, and FWHM of 0.05µm is applied as a spectral response function in all theoretical simulations of this study.

In line with the depiction of Eq. (5), the approximated value k_i of the reduced channel signal can be regarded as a function of expansion wavelength λ_i and temperature T. The reduced channel signal K_i in Eq. (3) is dependent only on the temperature T. Therefore, the approximation level of the Taylor series should be evaluated based on the correlation between the source signal values K_i and the approximated values k_i at different positions λ_i over an identical temperature range. Under blackbody conditions (where the spectral emittance ε(λ_i) equals 1 and the first and second derivatives equal 0), the difference between the source signal value K_i and the approximated value k_i at different expansion wavelength λ_i for each channel is shown in Fig. 1.

Fig. 1. (a) The approximation level of different expansion wavelengths in R channel. (b) The approximation level of different expansion wavelengths in G channel. (c) The approximation level of different expansion wavelengths in B channel.

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The influence of the expansion position λ_i on the approximation level is noteworthy. For any given channel data, there exists an expansion position that can achieve the optimal level of approximation. To quantitatively evaluate the performance of each expansion position, the determination coefficient R²[24], which describes the level of approximation, is introduced into this study. It is defined as the ratio of the regression sum of squares (SSR) to the total sum of squares (SST), and its expression is as follows.

(9)$${R^2} = \frac{{SSR}}{{SST}} = 1 - \frac{{SSE}}{{SST}} = 1 - {\left[ {\frac{{{{||{{{\mathbf k}_i} - {{\mathbf K}_i}} ||}_2}}}{{{{||{{{\mathbf K}_i} - {{\bar{K}}_i}} ||}_2}}}} \right]^2}$$

where k_i is the row vector composed of the approximation data of different temperatures, K_i is the row vector of the primary data of different temperatures, and`K_i is the average value of the primary data. The value range of R² is [1, -∞), and as R² approaches 1, it indicates a higher correlation between k_i and K_i. Conversely, a lower R² value indicates a lower correlation between the two sets of data.

Table 1 presents the determination coefficient R² of different expansion position data for each channel’s data, which corresponds to Fig. 1 The highest R² for each channel were observed at expansion positions of 0.591, 0.493, and 0.398, respectively, signifying that these are the positions with the best approximation for each channel. Considering that these expansion positions with the best approximation degree have a dimension of wavelength, these positions are called the optimal expansion wavelength λ_{i, OEW} (OEW), which is defined as the expansion position λ_i with R² closest to 1. The optimal expansion wavelengths of each channel differ because they are mainly affected by the spectral response curve.

Table 1. Determination coefficient of approximation data at different expansion positions

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The selection of optimal expansion wavelength (OEW) is crucial for this processing method. Equation (5) demonstrates that the integration equation can only be properly de-integrated when the OEW has been determined. In actual measurements, heat sources are typically non-gray bodies whose thermal radiation behavior differs from that of blackbodies due to the presence of spectral emissivity. The impact of this situation on the selection of the OEW should be discussed to evaluate the practical suitability of this processing method.

4. Effect of spectral emissivity on OEW

Compared with blackbody situations, the spectral emissivity of non-gray thermal surfaces will cause changes in Eq. (7) and Eq. (8), thus affecting the selection of the OEW. The methodology for selecting the OEW is critical to de-integrating the temperature measurement integration equation for non-gray heat sources under different spectral emissivity conditions and is fundamental for the practical application of this approach.

To thoroughly investigate the influence of spectral emissivity on the selection of the OEW, an emissivity model library was constructed, which consists of the spectral emissivity models that have been reported [25–29], as shown in Table 2.

Table 2. Comparison table of spectral emissivity model

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By combining the eight emissivity expressions above with their distinct trends, a total of 16 emissivity models were built. The amplitude and line shape of each emissivity model is controlled by parameters A and B. To facilitate a more complete discussion, ten sets of A and B parameters were selected for each spectral emissivity model. The Appendix provides the selected parameters and spectral emissivity images for each model.

The 16 emissivity models with different parameters were inserted into Eq. (3) and Eq. (5) to estimate the original data values (K_i) and the approximated data (k_i). The closeness of the R² value to 1 obtained from the two sets of data was used as a criterion to determine the optimal expansion wavelength (OEW) for each emissivity model under different parameters. For instance, using Mod11 as an example, the approximated effect at the optimal expansion wavelength (OEW) under various sets of parameters a-j is presented in Fig. 2.

Fig. 2. Approximation effects of Mod11 spectral emissivity at the optimal expansion wavelength under different parameters. ‘K_i @ Para x’ represents the primary data of i channel calculated by the Mod11 spectral emissivity model using the parameter set x. ‘x TE@ λ_{i, OEW}’ represents the expansion result of the spectral emissivity Mod11 using parameter set x, at the location λ_{i, OEW}.

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The spectral emissivity modulates the thermal radiation signal differently under different sets of parameters, resulting in the K_i of each channel being at different numerical levels. Nevertheless, irrespective of the level at which K_i is situated, there exists a k_i and OEW that correspond to it and provide the optimal approximation effect. This phenomenon can be observed in simulations of all emissivity models. Furthermore, the OEWs corresponding to the signals of each channel at different levels are generally similar in Fig. 2. Under all parameter groups, the OEWs of the G channel and B channel stabilize at 0.493 µm and 0.398 µm, respectively, which is identical to the OEWs of the black body listed in Table 1. The OEW of the R channel is predominantly centered at 0.591 µm, which is the same as the black body condition. However, discrepancies in the OEWs arise at parameter groups i and j, where they are at 0.590 µm. To ensure the effectiveness of the Taylor series in the de-integration process of equations, it is necessary to discuss these discrepancies in OEW.

When using a known emissivity model for measurements, the determination of OEW can be obtained directly based on the degree of R² approaching 1. Conversely, when the emissivity model is unknown, determining OEW poses a challenge. Theoretically, the OEW of non-gray surfaces λ_i,n,OEW may differ from the OEW of blackbody λ_i,b,OEW due to the presence of spectral emissivity. This deviation caused by spectral emissivity is called the OEW deviation δ_i, which is defined as follows.

(10)$${\delta _\textrm{i}}\textrm{ = }|{{\lambda_{i,b,OEW}} - {\lambda_{i,n,OEW}}} |$$

Based on Fig. 2 and the OEW results of black surfaces in Section 3, the λ_i,n,OEW of each channel in the Mod11 emissivity model are similar to the λ_i,b,OEW under the black surface conditions for different parameter sets. Specifically, in channels B and G, both OEW values are identical, resulting in an OEW deviation δ_i of 0. The deviation only exists in the i and j parameter sets of the R channel, and its value is 0.001um. The OEW deviations of all emissivity models under different parameter sets were obtained, and Fig. 3(a) illustrates the average OEW deviation for each spectral emissivity model under different parameter sets.

Fig. 3. (a) The average OEW deviation of each spectral emissivity model under different parameter groups. (b)-(d) The relationship between the approximation effect of the spectral emissivity model with OEW deviation and the expansion wavelength.

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In the simulations of all parameter groups for each emissivity model, the obtained OEW deviation (δ_i) results are limited to two values: 0 µm and 0.001 µm. To comprehensively illustrate the quantity of parameter groups exhibiting OEW deviation in each model, the simulation results are displayed in the form of average OEW deviation in Fig. 3(a). The average OEW deviation is obtained by calculating the sum of OEW deviations for all parameter groups under the nth model and dividing it by the total number of parameter groups. Given that the maximum δ_i in the simulation is 0.001µm and the total number of parameter groups is 10, the value of the average δ_i is a multiple of 1 × 10⁻⁴µm, representing the quantity of parameter groups exhibiting δ_i in this model. For instance, as shown in Fig. 3(a), Mod11 demonstrates an average deviation of 2 × 10⁻⁴µm in the R channel, indicating the presence of δ_i in two out of the ten parameter groups within this model. This corresponds to the parameter sets i and j as shown in Fig. 2. If the average OEW deviation of a model is 1 × 10⁻³µm, it indicates that the model shows OEW deviations across all parameter groups.

Within Fig. 3(b)-(d), certain models display deviations in both the R channel and B channel, while the G channel has no deviation in all emissivity models. The R channel shows seemingly irregular quantities of OEW deviation within Mod1-6 and Mod9-16. Conversely, the B channel demonstrates prominent regularity, with deviations present across all parameter groups of even-numbered models. To facilitate a more comprehensive analysis of the impact of OEW deviation on the approximation effect, these emissivity models with OEW deviation are further discussed. Figure 3(b)-(d) illustrates how the approximation effect is affected by the expansion wavelength under the conditions where OEW deviation exists.

The approximation effect of the emissivity model for the increasing and decreasing trends on channel R is depicted in Fig. 3(b) and (c), respectively. The calculated results of odd (increasing) spectral emissivity model except Mod7 are shown in Fig. 3(b). In this case, there is a negative OEW deviation of 0.001µm (OEW of each model λ_{r,n, OEW} is 0.590µm, blackbody OEW λ_r,b,OEW is 0.591µm). In Fig. 3(c), the calculation results of the even (decreasing) spectral emissivity model except Mod8 are shown. In this result, all OEW deviations are positive deviations of 0.001µm (OEW of each model λ_{r,n, OEW} is 0.592µm, OEW of blackbody λ_r,b,OEW is 0.591µm). That is, the OEW deviation direction is related to the trend of spectral emissivity. This is consistent with the conclusion for channel B described in Fig. 3(a).

Considering that the approximation effect of expansion wavelengths on K_i can be evaluated by R², the approximation effect of using the dashed line position (λ_i,b,OEW) as the expansion wavelength is very close to that of using the solid line position (λ_i,n,OEW). The maximum disparity in determination coefficient R² between the two is situated within the range of 0.0227-0.0313. This indicates that, regardless of the presence of OEW deviation in the emissivity model, using blackbody OEW as the expansion wavelength in the process of approximating integral equations with Taylor series can achieve satisfactory approximation effects. Put differently, OEW shows an inert response to spectral emissivity, and when the emissivity is unknown, using blackbody OEW for algebraic transformation of equations also results in a high approximation accuracy. This conclusion constitutes the essential theoretical basis for the practical implementation of the method in temperature measurements.

Based on the mathematical principle of Taylor series. Theoretically, the OEW deviation of each channel is caused by the difference between the changing rate of the reduced signal intensity of different spectral emissivity in the channel spectral range and that of the blackbody condition. OEW deviation occurs when the rate of change is significantly higher than that of blackbody condition, and the Taylor expansion takes the expansion wavelength which is more consistent with the actual rate of change as the best expansion wavelength, which leads to the phenomenon of OEW deviation. Therefore, when the K_i is modulated by the spectral emissivity, the spectral emissivity models with the opposite trend to the changing rate of blackbody thermal radiation signal are more prone to generating OEW deviations. Furthermore, the high spectral emissivity with high modulation effect is also more likely to cause the change of K_i’s changing rate, thereby resulting in the generation of OEW deviation. The higher ordinal parameter group in the Appendix I can produce higher spectral emissivity. To explore the maximum disparity in the approximation effect caused by deviations, the highest ordinal parameter group (Para j) is utilized in the calculations for Fig. 3.

5. Steps for de-integrating processing

Assuming that the OEW for each channel has been determined, Eq. (3) can be de-integrated using Taylor series. The resultant transformed expression of Eq. (3) is as follows.

(11)$$\begin{aligned} {K_i} = \varepsilon ({\lambda _{i,b,{OEW}}})\left( {{I_B}({\lambda_{i,b,{OEW}}},T){{N}_{i,1}} + \frac{{\partial {I_B}({\lambda_{i,b,{OEW}}},T)}}{{\partial \lambda }}{{N}_{i,2}} + \frac{{{\partial^2}{I_B}({\lambda_{i,b,{OEW}}},T)}}{{\partial {\lambda^2}}}\frac{{{{N}_{i,3}}}}{2}} \right)\\ + \varepsilon ^{\prime}({\lambda _{i,b,{OEW}}})\left( {{I_B}({\lambda_{i,b,{OEW}}},T){{N}_{i,2}} + \frac{{\partial {I_B}({\lambda_{i,b,{OEW}}},T)}}{{\partial \lambda }}{{N}_{i,3}}} \right){ + }\varepsilon ^{\prime\prime}({\lambda _{i,b,{OEW}}}){I_B}({\lambda _{i,b,{OEW}}},T)\frac{{{{N}_{i,3}}}}{2} \end{aligned}$$

where N_i_,1, N_i_,2, and N_i_,3 is a constant term containing C_i,1 and C_i,2, which is in the form of:

(12)$$\begin{aligned} {N_{i,1}} &= {s_i}({{\lambda_{i,b,OEW}}} )\cdot \Delta \lambda \textrm{ + }{C_{i,1}}\frac{{\partial {s_i}({\lambda _{i,b,OEW}})}}{{\partial \lambda }}\textrm{ + }{C_{i,2}}\frac{{{\partial ^2}{s_i}({\lambda _{i,b,OEW}})}}{{\partial {\lambda ^2}}}\\ {N_{i,2}} &= {C_{i,1}}{s_i}({\lambda _{i,b,OEW}}) + 2{C_{i,2}}\frac{{\partial {s_i}({\lambda _{i,b,OEW}})}}{{\partial \lambda }}\\ {N_{i,3}} &= 2{C_{i,2}}{s_i}({\lambda _{i,b,OEW}}) \end{aligned}$$

The more difficult integral equation in Eq. (3) is transformed into an algebraic equation in Eq. (11), in which the first and second derivatives of blackbody radiation I_B(λ_i,b,OEW, T) are as follows:

(13)$${\left. {\frac{{\partial {I_B}(\lambda ,T)}}{{\partial \lambda }}} \right|_{\lambda { = }{\lambda _{{i,b,OEW}}}}} = \left( {\frac{{{c_2}}}{{{c_1}T\lambda_{i,b,OEW}^2}} - \frac{1}{{5{\lambda_{i,b,OEW}}{c_1}}}} \right){I_B}({\lambda _{i,b,OEW}},T)$$

(14)$${\left. {\frac{{{\partial^2}{I_B}(\lambda ,T)}}{{\partial {\lambda^2}}}} \right|_{\lambda \textrm{ = }{\lambda _{i,b,OEW}}}} = \left( {\frac{{30}}{{\lambda_{i,b,OEW}^2}} - \frac{{12{c_2}}}{{T\lambda_{i,b,OEW}^3}} + \frac{{\textrm{c}_2^2}}{{{\textrm{T}^2}\lambda_{i,b,OEW}^4}}} \right){I_B}({\lambda _{i,b,OEW}},T)$$

Although Eq. (11) may appear complex, it can be interpreted as a classical equation for multi-wavelength temperature measurement, which includes wide-spectrum compensation terms. As the upper and lower limits of the integration wavelength in Eq. (6) approach each other, the integrated function on the right-hand side of both equations tends to zero, indicating that the constants C_i,1 and C_i,2 tend towards zero. Following Eq. (12), the equation constants N_i,2 and N_i,3 described by these two constants will be eliminated, and only the first term of N_i,1 will be retained. At this point, Eq. (11) consists only of calibration parameters, which include spectral response s_i(λ_i,b,OEW) and a small spectral bandwidth Δλ, multiplied by emissivity ε(λ_i,b,OEW) and blackbody radiation intensity I(λ_i,b,OEW,T), the equation simplifies into the classical form [30–32] of multi-wavelength thermometry. Therefore, the first term of Eq. (11) can be understood as the multi-wavelength thermometry form with a detection wavelength of OEW, while the remaining terms are the wide-to-narrow spectral compensation caused by the wide detection spectral range.

Furthermore, based on the Eq. (11)-(14), the solving equations of the existing wide-spectrum pyrometers are changed, and three forms of algebraic equations satisfying the closed conditions for solving the equations can be constructed. This includes gray body method for gray objects, two-color temperature measurement method and three-channel fusion temperature measurement method for non-gray objects (spectral emissivity is not constant).

a. The form of the gray body temperature measurement method is as follows. $(15)$${K_{i}} = \varepsilon \left( {\frac{{\partial {I_B}({\lambda_{{i,}b,{OEW}}},T)}}{{\partial \lambda }}{{N}_{{i},2}} + \frac{{{\partial^2}{I_B}({\lambda_{{i},b,{OEW}}},T)}}{{\partial {\lambda^2}}}\frac{{{{N}_{{i},3}}}}{2}} \right) + \varepsilon {I_B}({\lambda _{{i},b,{OEW}}},T){{N}_{{i},1}}$$$

Eq. 15 is a typical system of contradictory (overdetermined) equations, and the least square method should be applied to the process of solving the equations. Based on Eq. (15), the least squares relationship Q can be constructed as follows:

(16)$${Q = }{\sum\limits_{i}^{r,g,b} {\left[ {\varepsilon \left( {\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial \lambda }}{{N}_{{i},2}} + \frac{{{\partial^2}{I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial {\lambda^2}}}\frac{{{{N}_{{i},3}}}}{2}} \right) + \varepsilon {I_B}({\lambda_{{i},{OEW}}},T){{N}_{{i},1}} - {{K}_{i}}} \right]} ^2}$$

By using this relationship, the solution of the overdetermined equation Eq. (15) can be transformed into an optimization problem of the following equations.

(17)$$\begin{array}{l} \frac{{\partial Q}}{{\partial T}} = \sum\limits_{i}^{r,g,b} {2\left[ {\varepsilon \left( {{{N}_{{i},2}}\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial \lambda }} + \frac{{{{N}_{{i},3}}}}{2}\frac{{{\partial^2}{I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial {\lambda^2}}} + {{N}_{{i},1}}{I_B}({\lambda_{{i},{OEW}}},T)} \right) - {{K}_{i}}} \right]} \\ \cdot \varepsilon \left[ {{{N}_{{i},2}}\frac{\partial }{{\partial T}}\left( {\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial \lambda }}} \right) + \frac{{{{N}_{{i},3}}}}{2}\frac{\partial }{{\partial T}}\left( {\frac{{{\partial^2}{I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial {\lambda^2}}}} \right) + {{N}_{{i},1}}\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial T}}} \right] = 0\\ \frac{{\partial Q}}{{\partial \varepsilon }} = \sum\limits_{i}^{r,g,b} {2\left[ {\varepsilon \left( {{{N}_{{i},2}}\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial \lambda }} + \frac{{{{N}_{{i},3}}}}{2}\frac{{{\partial^2}{I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial {\lambda^2}}} + {{N}_{{i},1}}{I_B}({\lambda_{{i},{OEW}}},T)} \right) - {{K}_{i}}} \right]} \\ \cdot \left( {{{N}_{{i},2}}\frac{{\partial {I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial \lambda }} + \frac{{{{N}_{{i},3}}}}{2}\frac{{{\partial^2}{I_B}({\lambda_{{i},{OEW}}},T)}}{{\partial {\lambda^2}}} + {{N}_{{i},1}}{I_B}({\lambda_{{i},{OEW}}},T)} \right) = 0 \end{array}$$

When the emissivity of gray body target is known, theoretically, the target temperature can be obtained by solving any equation in Eq. (15). In addition, the simultaneous numerical solution of the three equations is also allowed, which can obtain accurate temperature results. When the emissivity of the gray body is unknown, the temperature T and emissivity ε can also be obtained by solving Eq. (17).

b. The form of the two-color temperature measurement method is as follows. $(18)$$\frac{{{K_i}}}{{{K_j}}} = \frac{{{I_B}({\lambda _{i,b,OEW}},T){N_{i,1}} + \frac{{\partial {I_B}({\lambda _{i,b,OEW}},T)}}{{\partial \lambda }}{N_{i,2}} + \frac{{{\partial ^2}{I_B}({\lambda _{i,b,OEW}},T)}}{{\partial {\lambda ^2}}}\frac{{{N_{i,3}}}}{2}}}{{{I_B}({\lambda _{j,b,OEW}},T){N_{j,1}} + \frac{{\partial {I_B}({\lambda _{j,b,OEW}},T)}}{{\partial \lambda }}{N_{j,2}} + \frac{{{\partial ^2}{I_B}({\lambda _{j,b,OEW}},T)}}{{\partial {\lambda ^2}}}\frac{{{N_{j,3}}}}{2}}}$$$

The two-color temperature measurement equation of Eq. (18) can be constructed with any two channels, in which the unknown quantity is only the target temperature T. When the number of channels is greater than two, an ill-conditioned problem may arise as multiple equations correspond to a single unknown variable. As a result, temperature estimation is achieved by constructing the mean difference minimization relationship [33,34] or the least squares relationship of the equations. Taking the example of constructing the least squares relationship, the numerical solution of the two-color temperature measurement can be obtained using the following termination conditions (i ≠ j).

(19)$$\sum\limits_{i,j}^{r,g,b} {{{\left[ {\frac{{{I_B}({\lambda_{i,b,OEW}},T){N_{i,1}} + \frac{{\partial {I_B}({\lambda_{i,b,OEW}},T)}}{{\partial \lambda }}{N_{i,2}} + \frac{{{\partial^2}{I_B}({\lambda_{i,b,OEW}},T)}}{{\partial {\lambda^2}}}\frac{{{N_{i,3}}}}{2}}}{{{I_B}({\lambda_{j,b,OEW}},T){N_{g,1}} + \frac{{\partial {I_B}({\lambda_{j,b,OEW}},T)}}{{\partial \lambda }}{N_{j,2}} + \frac{{{\partial^2}{I_B}({\lambda_{j,b,OEW}},T)}}{{\partial {\lambda^2}}}\frac{{{N_{j,3}}}}{2}}} - \frac{{{K_i}}}{{{K_j}}}} \right]}^2} = 0}$$

In addition, in previous studies, the reciprocal relationship between spectral emissivity ratio and characteristic wavelength ratio can also be applied [4,5]. The additional calibration of the proportionality coefficient, which is required in traditional methods, is no longer necessary in this dual-color temperature measurement form. Moreover, the wavelength selection in this method is more scientifically grounded than in conventional two-color temperature measurement approaches.

c. The form of the three-channel fusion temperature measurement method is as follows. $(20)$$\begin{aligned} {K_{i}} &= \frac{{\partial {I_B}({\lambda _{{i},b,{OEW}}},T)}}{{\partial \lambda }}({\varepsilon {{N}_{{i},2}} + \varepsilon^{\prime}{{N}_{{i},3}}} )+ \varepsilon \frac{{{\partial ^2}{I_B}({\lambda _{{i},b,{OEW}}},T)}}{{\partial {\lambda ^2}}}\frac{{{{N}_{{i},3}}}}{2}\\ &\quad{ + }{I_B}({\lambda _{{i},b,{OEW}}},T)({\varepsilon {{N}_{{i},1}}{ + }\varepsilon^{\prime}{{N}_{{i},2}}} )\end{aligned}$$$

The unknown terms in the formula are spectral emissivity, spectral emissivity derivative and target temperature. The equation is a typical nonlinear algebraic equation, and its numerical solution can be obtained. Temperature calculation errors arising from the same-color different-temperature phenomenon (i.e., encountering different temperature targets with the same numerical value at the spectral position) during two-color form can be effectively mitigated using a three-channel fusion form. Additionally, the differences in results caused by the selection of dual-color channels can also be effectively avoided.

In summary, the specific steps for de-integrating the three-channel wide-spectrum temperature equation system using Taylor series are shown in Fig. 4. The de-integration process can be divided into four stages, namely the calibration stage, the OEW determination stage, the parameter computation stage, and the temperature calculation stage. The calibration stage primarily focuses on acquiring the parameters of the wide-spectrum detector. This requires obtaining the spectral response function (SRF) of the acquisition equipment, as well as determining the conversion relationship κ_i between radiance and gray level. In the OEW determination stage, the wavelength range within the spectral response range of each detector channel is traversed to calculate the corresponding k_i, which is then used to obtain the determination coefficient R² for each wavelength. Considering that the relationship between R² and the expansion wavelength is parabolic (Fig. 3), the OEW results can be obtained by comparing the R² values in the continuous iterative process. The expansion wavelength corresponding to the last iteration before the decline of R² is outputted as OEW. Where v represents the number of iterations. To ensure the accuracy of OEW, the iterative step is limited to 0.001µm. In the parameter calculation stage, the determined OEW is used along with Eq. (6) and Eq. (12) to calculate all the constant terms in the equations. The process concludes with the construction of Eq. (11) in algebraic form, thereby completing the de-integration process of the temperature measurement equation.

Fig. 4. De-integration process based on second-order Taylor expansion.

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In the Temperature calculation stage, the grayscale of the collected pixels is processed by Eq. (3) into reduced channel signals K_r,K_g and K_b. The thermometry equations in the form of Eq. (17), Eq. (19) and Eq. (20) are constructed. Equation (17), which describes the solving method of gray body form, is the optimal approximate form of Eq. (15) constructed through the least-squares relationship. All the three equations satisfy the closed conditions, and their numerical solutions can be obtained by Levenberg-Marquardt, trust-region and other methods.

To effectively illustrate the feasibility of this de-integration method in the application of temperature measurement, and to prove that the above three forms of equations have the ability of temperature solution, the theoretical simulation of temperature measurement in the form of gray body, two-color and three-channel fusion needs to be carried out.

6. Theoretical error analysis

In order to prove the temperature measurement capability of the method under the condition of all spectral emittance, the spectral emissivity model in Table 2 and its corresponding parameter groups are applied in the theoretical error simulation of the method. In fact, the method has different errors under different temperature, different spectral emissivity and different parameters of the same spectral emissivity. The theoretical errors of the three methods are analyzed horizontally and longitudinally in order to comprehensively demonstrate the measurement capabilities of the three methods.

The comparison of the temperature calculation ability of each method is the content of horizontal analysis. To comprehensively reflect the changes of multi-parameters, the average relative error characterizing the temperature solving ability is defined as follows.

(21)$${\gamma _{H}}{ = }\frac{1}{{{MN }}}\sum\limits_{\alpha { = }1}^{M} {\sum\limits_{\beta { = }1}^{N} {\frac{{|{{{t}_{\alpha \beta }}({T} )- {T}} |}}{{T}}} }$$

where β characterizes the ordinal number of the parameter group, the upper limit N = 10; α characterizes the spectral emissivity model ordinal number, and the upper limit M = 16. T is the real temperature and t_αβ is the solution temperature obtained by using the α-th spectral emissivity of the β-th parameter group. The blackbody OEW in Table 1 is applied in theoretical error simulation. In the horizontal analysis, the average relative errors of the three methods are shown in Fig. 5.

Fig. 5. The theoretical errors of the three methods at different temperatures under different spectral emissivity and parameters. The solution error in the case of each spectral emissivity model is shown by the subgraph. The corresponding shaded part is composed of subgraph data.

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Figure 5(a) shows the temperature measurement ability of the gray body method, Fig. 5(b), (c) shows the theoretical simulation results of the three-channel fusion method and the two-color method, respectively. The shadow part can be understood as the continuous variation range of the solution error corresponding to the continuous variation of spectral emissivity, which can be used as a representation of solving ability and stability. The variation trend of the theoretical error of the increasing spectral emissivity model (odd number model) with temperature is different from that of the decreasing model, which is revealed in the results of the three methods. The error of the three methods for the decreasing trend spectral emissivity models are generally higher than that of the increasing trend models, which can be explained by the simulation results of OEW deviation (as shown by Fig. 3(a)).

The increasing trend spectral emissivity model generally exhibits OEW deviations only in the R channel, thus resulting in smaller calculation errors. For the decreasing trend model, almost all models have OEW deviations in R channel and B channel, which makes the solution error at a relatively high level. That is, the more channels with OEW deviation, the more obvious the influence on the temperature solving ability.

The average error performance of three-channel fusion method and two-color method is essentially the same under the decreasing trend spectral emissivity model, all with the range of (1.26%, 5.78%). The gray body form exhibits a relatively higher average error, ranging from 2.98% to 6.14%. In the solution of the increasing trend model, the three-channel fusion method has the best theoretical solution ability. Most areas have an error of less than 1%, with an overall error of (0.076%, 3.663%). For the two-color temperature measurement form, in the Fig. 5(c) subgraph, the solution error of the Mod1-Mod14 spectral emissivity model is in a similar stable state, while the results of Mod15 and Mod16 are quite different from those of other models. Although the parabolic spectral emissivity (Mod15, Mod16) model affects the stability of the solution, no matter what the trend is, most of the errors are still less than 3%. In addition, the simulation results of Fig. 5(a) show that the method designed for gray body targets can also play a role in the temperature measurement of non-gray body targets. While the two-color form Eq. (18) is derived from the gray body form Eq. (15) through arbitrary ratios of two channels, the termination conditions Eq. (17) and Eq. (19) in their respective optimization solutions are different. This is also the main reason for the different distribution of theoretical error results between the two forms.

The statistical analysis of the accuracy of each emissivity model under the three solving methods is the content of longitudinal comparison, which is used to characterize the solving stability and suitability of the method for specific spectral emission targets. In the longitudinal comparison, the average relative error is defined as follows.

(22)$${\gamma _{L}}{ = }\frac{1}{{{N}({{{T}_2} - {{T}_1}} ){ }}}\sum\limits_{{T = }{{T}_1}}^{{{T}_2}} {\sum\limits_{\beta { = }1}^{N} {\frac{{|{{{t}_\beta }(T )- {T}} |}}{{T}}} }$$

T₂ and T₁ represent the upper and lower limits of temperature T in the simulation, respectively. For each spectral emissivity model, a longitudinal average relative error can be obtained by Eq. (22). The results of the longitudinal comparison analysis are shown in Fig. 6.

Fig. 6. Average relative errors of three methods under different spectral emissivity models. The shadow part is the range of the mean value add-subtract standard deviation (SD), 68.2% of all error data are concentrated in this range, which is used to characterize the solution stability of each method under the emissivity model.

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According to the distribution of the maximum and minimum error, the error of the gray body method is relatively high for most spectral emissivity models. With the exception of mod8 and mod16, the three-channel fusion method shows the lowest error performance across all emissivity models, while its high stability is observed in the majority of models. The measurement effect of the two-color method is moderate. in the case of each emissivity model, the average relative error distribution is relatively stable, and the error stable results can be obtained in the temperature measurement of all spectral emissivity targets.

In terms of overall error performance, the measurement results of the two-color method and the three-channel fusion method can be trusted in most cases. With the use of the de-integration method, the proportional coefficient calibration process of the two-color method is eliminated without losing the measurement accuracy. Meanwhile, the measurement ability of the gray body method under non-gray conditions is also revealed. The theoretical contribution of the method to the existing wide-spectral thermometry technology can be proved accordingly.

7. Experiments

To verify the universality and measurement performance of the three calculation methods, a temperature measurement experiment with multiple acquisition parameters and multiple samples is designed and conducted. The tube furnace, widely used in the thermal radiation performance analysis of materials [35,36], serves as the primary heat source for this experiment, and the experimental equipment is shown in Fig. 7.

Fig. 7. Schematic diagram of selection of experimental equipment and acquisition parameters.

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The wafer-shaped samples of copper and the 304 stainless steel with thickness of 0.8 mm are placed in the furnace of tubular furnace as objects to be tested. The K-type thermocouple is inserted into the furnace and directly contacted with the furnace wall, which is the main feedback device of the furnace temperature. The MER2-051 (DAHENG Image) camera, with a known radiance-gray transfer relationship, is used as a three-channel thermal radiation acquisition device, which collects the visible band thermal radiation signal of the sample through the tube furnace window. The spectral response of each channel and the radiation-gray transfer relationship are shown in Fig. 8(a),(b).

Fig. 8. Camera parameters and actual acquisition effect picture. (a) Camera spectral response curve. (b) Camera radiation-grayscale transfer relationship. (c) Thermal radiation signal acquisition results of two samples under different lenses.

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Two different fixed-focus lenses were used in the experiment, with focal lengths of 25 mm and 35 mm respectively. Considering the varying thermal radiation intensity at different temperatures, the settings of the acquisition parameters need to be coordinated with this variation to ensure that the captured images remain within the dynamic range of the camera.The selection of acquisition parameters for each temperature zone in the experiment follows the two tables in Fig. 7. The table on the left is the exposure time selection table controlled by the camera, and the ‘x: y: z’ indicates that the exposure time increases in steps of y from x until all exposure times of z are obtained. The table on the right is the aperture selection table controlled by the lens. The three apertures with the maximum light flux for each focal length are applied in the experiment to provide a smaller depth of field and to homogenize background noise for image acquisition. Taking the temperature T_R= 1273 K aperture as f /2.8 and the exposure time of 1 × 10⁴µm as an example, Fig. 8(c) shows the acquisition results of the two materials under different lenses.

Due to different emissivities, there are slight differences in the thermal radiation images of the two samples, and the radiation image of 304 stainless steel tends to be dark yellow at the same temperature. The sample is heated uniformly by the tube furnace, and the surface thermal radiation chromaticity of each sample is highly uniform. Also, it is precisely due to the exceptional temperature uniformity of the tube furnace that, under conditions of minimal temperature fluctuations, it can be assumed that the sample temperature is consistent with the internal furnace temperature, i.e., the feedback temperature T_R from the thermocouple represents the actual sample temperature. The temperature of the tube furnace can be adjusted through programming. When the set temperature in the program matches the stable feedback temperature T_R from the internal thermocouple, the actual temperature of the sample is in a stable and known state. All collections are carried out in this state. By using the three aforementioned calculation methods, the central pixel of the captured image is calculated, and comparing the results of each method with the feedback temperature T_R yields the relative measurement errors for each method.

8. Results and discussion

The experiments of multiple collection parameters of each sample at different temperatures are conducted. The selection of temperature points started from 1173 K, where weak visible thermal radiation appeared, and increased in increments of 10 K until 1323 K. The radiation images of 16 temperature points were collected. For each temperature point, an acquisition was conducted under each corresponding parameter of the exposure parameter selection table and the aperture selection table in Fig. 7. For example, for the temperature point of 900 K, when using a 25 mm focal length lens, 24 captures are performed with 8 different exposure times (5 × 10³: 5 × 10³: 4 × 10⁴µs) combined with 3 different apertures (f/1.4, f/2, f/2.8), based on the parameter selection table in Fig. 7. A similar approach is used for the capture of the other temperature points. The collection results from all temperature points are calculated by the three methods and compared with the actual temperature T_R. The error statistics of the copper sample calculations for each method are presented in Fig. 9. The three-channel OEW in the temperature range of [1100 K, 1400 K] is used in the solving process, which is obtained by the OEW determination stage in Fig. 4, which is 0.597, 0.503 and 0.396µm, respectively. Moreover, the iterative initial value of the numerical operation is 1400 K.

Fig. 9. The calculation results of copper samples by three methods under different collection conditions. (a) Measurement error when the focal length is 25 mm. (b) Measurement error when focal length is 35 mm.

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The box plots in Fig. 9 are used to describe the relative calculation errors of the three methods at each temperature point, which is obtained by dividing the temperature deviation by T_R. The dot chart is used to describe the most probable results obtained from the measurement. It is defined as the average of the solution temperatures within the range of the mean error plus or minus one standard deviation. It refers to the average of the results within the range where the probability in the calculated outcomes falls within the top 68.2%.

The measurement errors of the three methods under different focal lengths show similar effects in statistics, and the most probable solutions also show a trend similar to the actual temperature. The gray body solving method shows an increasing trend in measurement errors within the temperature range, and each temperature point exhibits a relatively stable distribution. In order of magnitude, it is mainly distributed in the error range of less than 8%, which is close to the simulation results (Fig. 5(a)). In terms of the most probable solution, the method performs better at lower temperature and deviates greatly at high temperature.

Compared with other methods, the three-color method maintains an overall lower solving error and shows a trend of decreasing at first and then fluctuating. Its stability is slightly lower than the gray method, with the majority of values spanning below 7% in magnitude. In the whole temperature range, the minimum error of the method is lower than 2% for a 25 mm focal length, and lower than 3% for a 35 mm focal length, which is consistent with the simulation results (Fig. 5(b)). Compared with other methods, the most probable measurement results are most consistent with the actual heating process, but the growth rate displays nonlinearity, which deviates from reality. A good effect is exhibited in the intermediate temperature range.

The calculation error of the two-color method shows a trend of decreasing first and then increasing. The reason for the same solution in the position of the yellow box is that the ratio operation with the participation of the B component tends to be zero or infinite due to the low amount of radiation of the B component of the camera at the low temperature position, so that the solution of the two methods stops after the first iteration. As a result, the two-color method shows a meaningless error solution in the temperature range of 1173-1193 K. The stability of this method is also lower than that of the gray method, and its most probable solution is consistent with the reality at higher temperature. Except for the temperature points where iteration termination occurs, the magnitude of relative error can be controlled within a range of 8%, which is consistent with the simulated results (Fig. 5(c)). The theoretical error simulation results of the three methods are verified, and the universality of each method to different imaging parameters is demonstrated.

To further illustrate the universality of the methods for different measurement objects, the same calculation process is applied in the temperature calculation of 304 stainless steel, and the results are shown in the Fig. 10.

Fig. 10. The calculation results of 304 stainless steel samples by three methods under different collection conditions. (a) Measurement error when the focal length is 25 mm. (b) Measurement error when focal length is 35 mm.

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The measurement results of the three methods for 304 stainless steel also exhibit similar trends under different focal lengths. The error of the gray method still shows an increasing trend and remains below 8% overall. The three-color method shows a fluctuation form with an error of less than 7%, and the two-color temperature measurement method shows a trend of decreasing error followed by increasing error. This parallels the experimental findings for copper material, and the most probable results also exhibit similar trends. The only difference is that the iterative termination occurs only at 1173 K and 1183 K temperature points (yellow box position) in the process of solving steel by two-color method. This is attributed to the spectral emissivity disparity between 304 stainless steel and copper materials [37,38], allowing the B component to reach a solvable level earlier. Even when there are clear emissivity differences between the two materials, the similar results can still be achieved by three methods, and the universality of each method to the material is proved. Furthermore, the practical applicability of the de-integration method has been substantiated in actual measurements.

9. Conclusion

In this work, a de-integration method of thermometry equations based on Taylor series and multiple derived algebraic wide-spectrum temperature calculation forms are proposed. The approximation ability of Taylor series for wide-spectrum radiation integral equations has been theoretically validated under different emissivity conditions. However, the approximation effect varies at different expansion positions (wavelengths). Hence, the expansion position (wavelength) with the best approximation effectiveness is defined as the optimal expansion wavelength (OEW). The determination method and selection steps for this parameter have been defined, and the influence of OEW under different spectral emissivity line shapes and amplitudes has also been discussed simultaneously. In this process, the inertness of the OEW to the spectral emissivity has been discovered. That is, in the case of unknown spectral emissivity, using the blackbody OEW can still achieve a satisfactory approximation effect. Based on this characteristic, several algebraic wide-spectrum thermometry equation forms oriented towards actual measurement processes have been proposed, identified as the gray body form, two-color form, and three-channel fusion form. The theoretical errors of the three forms have been statistically analyzed, showing that the average relative errors within the continuous temperature range are less than 7%. To validate the practical application effectiveness of each form, experiments involving wide-spectrum thermometry with multiple materials and varied collected parameters are conducted. In the statistical analysis of the experimental results, the error distribution trend and the average error range in the fixed interval temperature range are consistent with the theoretical results. The introduction of Taylor series addresses the theoretical gap in wavelength selection of the traditional wide-spectrum thermometry methods, optimizing the process of calibrating proportionality coefficients in traditional approaches, streamlining and improving measurement modes, and expanding the development space for wide-spectrum thermometry.

Appendix I. Parameter group of emissivity

In this study, 10 different sets of parameters for all the spectral emissivity models listed in Table 2 were discussed. These parameter sets were denoted by the alphabetical characters a through j, and the parameter selection for each of the 16 emissivity models was determined according to Table 3.

Table 3. Selection of spectral emissivity parameters and comparison table of parameter set serial number

View Table | View all tables in this article

Under this parameter selection, the overall situation of the spectral emissivity used in this study is shown in Figure a. The upward trend of the first seven emissivity expressions is shown in Fig. 11(a), and the downward trend of the first seven emissivity expressions is shown in Fig. 11(b). Because the eighth expression is parabolic, it has no obvious upward and downward trend, the spectral emissivity modes of the expression in different parameter groups are shown in Fig. 11(c) according to the difference of opening direction.

Fig. 11. The overall performance of each spectral emissivity under different parameter sets.

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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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R Channel		G Channel		B Channel
Wavelength(λ_r)	R²	Wavelength(λ_g)	R²	Wavelength(λ_b)	R²
0.585µm	0.5226	0.487µm	0.4272	0.392µm	0.4318
0.587µm	0.7471	0.489µm	0.6744	0.394µm	0.7018
0.589µm	0.9208	0.491µm	0.8847	0.396µm	0.9258
0.591µm	0.9969	0.493µm	0.9969	0.398µm	0.9907
0.593µm	0.9129	0.495µm	0.9163	0.400µm	0.6929
0.595µm	0.5889	0.497µm	0.5049	0.402µm	-0.3138
0.597µm	-0.0734	0.499µm	-0.4307	0.404µm	-2.5906

Number	Trend	Expression	Number	Trend	Expression
Mod1	Increase	$ε (λ) = A λ + B$	Mod 9	Increase	$ε (λ) = \exp (A + B \sqrt{λ})$
Mod2	Decrease	$ε (λ) = A λ + B$	Mod 10	Decrease	$ε (λ) = \exp (A + B \sqrt{λ})$
Mod3	Increase	$ε (λ) = \exp (A + B λ)$	Mod 11	Increase	$ε (λ) = \exp (A + \frac{B}{\sqrt{λ}})$
Mod4	Decrease	$ε (λ) = \exp (A + B λ)$	Mod 12	Decrease	$ε (λ) = \exp (A + \frac{B}{\sqrt{λ}})$
Mod5	Increase	$ε (λ) = A + B λ^{3}$	Mod 13	Increase	$ε (λ) = 1 - \exp (\frac{- B}{λ^{A}})$
Mod6	Decrease	$ε (λ) = A + B λ^{3}$	Mod 14	Decrease	$ε (λ) = 1 - \exp (\frac{- B}{λ^{A}})$
Mod7	Increase	$ε (λ) = \frac{1}{A + \frac{B}{λ} \ln (λ)}$	Mod 15	Increase	$ε (λ) = A λ^{2} + B λ + C$
Mod8	Decrease	$ε (λ) = \frac{1}{A + \frac{B}{λ} \ln (λ)}$	Mod 16	Decrease	$ε (λ) = A λ^{2} + B λ + C$

Parameter		a	b	c	d	e	f	g	h	i	j
Mod1	A	0.303	0.364	0.424	0.485	0.545	0.606	0.667	0.727	0.788	0.848
Mod1	B	0.085	0.082	0.079	0.076	0.073	0.070	0.067	0.064	0.061	0.058
Mod2	A	-0.303	-0.364	-0.424	-0.485	-0.545	-0.606	-0.667	-0.727	-0.788	-0.848
Mod2	B	0.415	0.478	0.541	0.604	0.667	0.730	0.793	0.856	0.919	0.982
Mod3	A	-2.076	-2.015	-1.956	-1.899	-1.845	-1.792	-1.742	-1.694	-1.648	-1.603
Mod3	B	1.229	1.319	1.393	1.453	1.504	1.548	1.586	1.618	1.647	1.673
Mod4	A	-0.737	-0.578	-0.438	-0.315	-0.205	-0.105	-0.014	0.070	0.148	0.220
Mod4	B	-1.229	-1.319	-1.393	-1.453	-1.504	-1.548	-1.586	-1.618	-1.647	-1.673
Mod5	A	0.182	0.198	0.215	0.231	0.247	0.264	0.280	0.297	0.313	0.329
Mod5	B	0.330	0.396	0.462	0.528	0.594	0.660	0.726	0.792	0.858	0.924
Mod6	A	0.318	0.362	0.405	0.449	0.493	0.536	0.580	0.623	0.667	0.711
Mod6	B	-0.330	-0.396	-0.462	-0.528	-0.594	-0.660	-0.726	-0.792	-0.858	-0.924
Mod7	A	2.944	2.566	2.273	2.039	1.847	1.688	1.554	1.440	1.341	1.254
Mod7	B	-0.808	-0.777	-0.744	-0.710	-0.677	-0.646	-0.617	-0.590	-0.564	-0.541
Mod8	A	5.390	4.920	4.525	4.189	3.898	3.645	3.423	3.226	3.050	2.893
Mod8	B	0.808	0.777	0.744	0.710	0.677	0.646	0.617	0.590	0.564	0.541
Mod9	A	-2.715	-2.701	-2.680	-2.654	-2.626	-2.596	-2.566	-2.534	-2.503	-2.472
Mod9	B	1.793	1.925	2.032	2.120	2.195	2.259	2.313	2.361	2.404	2.441
Mod10	A	-0.099	0.108	0.285	0.440	0.577	0.699	0.810	0.911	1.004	1.089
Mod10	B	-1.793	-1.925	-2.032	-2.120	-2.195	-2.259	-2.313	-2.361	-2.404	-2.441
Mod11	A	-0.099	0.108	0.285	0.440	0.577	0.699	0.810	0.911	1.004	1.089
Mod11	B	-0.931	-1	-1.055	-1.101	-1.140	-1.173	-1.202	-1.227	-1.248	-1.268
Mod12	A	-2.715	-2.701	-2.680	-2.654	-2.626	-2.596	-2.566	-2.534	-2.503	-2.472
Mod12	B	0.931	1	1.055	1.101	1.140	1.173	1.202	1.227	1.248	1.268
Mod13	A	-0.750	-0.823	-0.888	-0.948	-1.006	-1.063	-1.120	-1.178	-1.238	-1.302
Mod13	B	0.461	0.551	0.648	0.754	0.870	0.998	1.139	1.298	1.479	1.685
Mod14	A	0.750	0.823	0.888	0.948	1.006	1.063	1.120	1.178	1.238	1.302
Mod14	B	0.173	0.187	0.202	0.218	0.233	0.248	0.263	0.278	0.292	0.306
Mod15	A	2.496	2.583	2.688	2.813	2.963	3.143	3.361	3.627	3.957	4.375
	B	-2.496	-2.583	-2.688	-2.813	-2.963	-3.143	-3.361	-3.627	-3.957	-4.375
	C	0.014	0.026	0.042	0.063	0.091	0.126	0.170	0.227	0.299	0.394
Mod16	A	-2.496	-2.583	-2.688	-2.813	-2.963	-3.143	-3.361	-3.627	-3.957	-4.375
	B	2.496	2.583	2.688	2.813	2.963	3.143	3.361	3.627	3.957	4.375
	C	-0.014	-0.026	-0.042	-0.063	-0.091	-0.126	-0.170	-0.227	-0.299	-0.394

R Channel		G Channel		B Channel
Wavelength(λ_r)	R²	Wavelength(λ_g)	R²	Wavelength(λ_b)	R²
0.585µm	0.5226	0.487µm	0.4272	0.392µm	0.4318
0.587µm	0.7471	0.489µm	0.6744	0.394µm	0.7018
0.589µm	0.9208	0.491µm	0.8847	0.396µm	0.9258
0.591µm	0.9969	0.493µm	0.9969	0.398µm	0.9907
0.593µm	0.9129	0.495µm	0.9163	0.400µm	0.6929
0.595µm	0.5889	0.497µm	0.5049	0.402µm	-0.3138
0.597µm	-0.0734	0.499µm	-0.4307	0.404µm	-2.5906

Number	Trend	Expression	Number	Trend	Expression
Mod1	Increase	$ε (λ) = A λ + B$	Mod 9	Increase	$ε (λ) = \exp (A + B \sqrt{λ})$
Mod2	Decrease	$ε (λ) = A λ + B$	Mod 10	Decrease	$ε (λ) = \exp (A + B \sqrt{λ})$
Mod3	Increase	$ε (λ) = \exp (A + B λ)$	Mod 11	Increase	$ε (λ) = \exp (A + \frac{B}{\sqrt{λ}})$
Mod4	Decrease	$ε (λ) = \exp (A + B λ)$	Mod 12	Decrease	$ε (λ) = \exp (A + \frac{B}{\sqrt{λ}})$
Mod5	Increase	$ε (λ) = A + B λ^{3}$	Mod 13	Increase	$ε (λ) = 1 - \exp (\frac{- B}{λ^{A}})$
Mod6	Decrease	$ε (λ) = A + B λ^{3}$	Mod 14	Decrease	$ε (λ) = 1 - \exp (\frac{- B}{λ^{A}})$
Mod7	Increase	$ε (λ) = \frac{1}{A + \frac{B}{λ} \ln (λ)}$	Mod 15	Increase	$ε (λ) = A λ^{2} + B λ + C$
Mod8	Decrease	$ε (λ) = \frac{1}{A + \frac{B}{λ} \ln (λ)}$	Mod 16	Decrease	$ε (λ) = A λ^{2} + B λ + C$

Theoretical and experimental verification of wide-spectrum thermometry based on Taylor series de-integration method

Abstract

1. Introduction

2. Taylor series expansion of the integral equation system

3. Discussion on approximation level

4. Effect of spectral emissivity on OEW

5. Steps for de-integrating processing

6. Theoretical error analysis

7. Experiments

8. Results and discussion

9. Conclusion

Appendix I. Parameter group of emissivity

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (3)

Equations (22)

Optics Express