A Multi-Spectral Temperature Field Reconstruction Technology under a Sparse Projection

Abstract

In optical sparse projection reconstruction, the reconstruction of the tested field often requires the utilization of a priori knowledge to compensate for the lack of information due to the sparse projection angle. For situations where the radiation field of unknown materials is reconstructed or prior knowledge cannot be obtained, this paper proposes a multi-spectral temperature field reconstruction technology under a sparse projection. This technology utilizes the principles of multi-spectral temperature measurement technology, takes the correlation of radiation information between sub-regions of the temperature field as the optimization objective, and establishes statistical rules between the missing information by combining the equation constraint optimization algorithm and multi-spectral temperature measurement technology. Finally, the temperature field to be measured is reconstructed. The simulation and experimental tests show that, without any prior knowledge, the proposed method can reconstruct the temperature field under two projection angles, with an accuracy of 1.64~12.25%. Moreover, the projection angle is lower, and the robustness is stronger than that of the other methods.

1. Introduction

Optical tomography is a diagnostic technology that does not interfere with the tested field, and it has shown great advantages in physical measurements, temperature field measurements, plasma diagnosis, and other applications [1,2,3,4].

To explore beyond the constant, known, and uniform pressure during temperature measurement, Weiwei Cai et al. proposed a new optical tomography technique. The technique achieves synchronous distribution of temperature and concentration by optimizing the cost function between the reconstructed region and the actual region and obtains accurate spectra by introducing the regularization terms of temperature, concentration, and pressure (the average absolute deviation of the parameters of the region from the parameters of the adjacent region). The effectiveness and robustness of the proposed method are verified by the simulation of a typical flame phantom [5]. Min-Gyu Jeon et al. have recently proposed three new reconstruction algorithms: the MART (multiplicative algebraic reconstruction technique), SART (simultaneous algebraic reconstruction technique) and SMART (simultaneous multiplicative algebraic reconstruction technique) for CT-TDLAS (computed tomography–tunable diode laser absorption spectroscopy). By improving the contribution of relaxation factor to iteration quantity and iteration function in ART algorithm, the algorithm can be iterated in a more effective way, and the precision of temperature field reconstruction results is improved. The performance of the new algorithm is verified by comparing with the reconstruction results of the existing algorithms. The experimental results show that SMART reconstruction speed is the fastest, convergence after 50 iterations, the highest accuracy, up to 1.4% [6]. To provide high-speed imaging technology for fuel mixing, Paul Wright et al. developed a low-noise optoelectronic system coupled with an optical access layer (OPAL) to obtain the tomography of 27 angles of highly dynamic chemical reaction processes through offline processing under the condition of a uniform fuel distribution [7]. Chang Liu et al. used a fan-beam laser instead of a parallel beam laser to simplify the optical structure of TDLAS. Meanwhile, combined with the onion skin deconvolution algorithm, the regularization parameter λ is selected from the L-curve curvature criterion to adaptively damping σ_j, so that TDLAS can be easily applied to the actual flame measurement, improving the accuracy and robustness of the measurement. Numerical simulation and experiments under 40 projection angles demonstrate the effectiveness of the proposed method [8]. To solve the problem of the slow speed of algebraic reconstruction algorithms in clinical applications, Klaus Mueller et al. proposed a scheme that extends the precision of a given framebuffer by four bits, using the color channels. With this extension, a 12-bit framebuffer delivers useful reconstructions for 0.5% tissue contrast, while an 8-bit framebuffer requires 4%. Finally, 3D algebraic fast reconstruction was realized under 80 projection angles [9].

The studies above show the performance advantages of optical tomography. In general, it is not possible to obtain test data from all angles. For most optical computed tomography (OpCT) methods, due to environmental limitations and the necessity for complex test equipment, such as beam deflection, interferometry, light emission computed tomography (LECT), and emission spectral tomography (EST), it is not possible to obtain test data from all angles. The amount of data for each angle is limited (the data are incomplete) [10,11]. Therefore, a CT algorithm with limited data is needed in OpCT reconstruction. When the maximum-entropy (ME) algorithm is used for tomography reconstruction in the case of limited projection views, it is difficult to accurately reconstruct test fields with a complex distribution. To solve this problem, Xiong Wan et al. proposed a fusion entropy algorithm, which combines maximum-entropy (ME) and cross entropy (CE) and realizes the tomography reconstruction under six projection angles by adaptively adjusting the weight factors of ME and CE. The results of the numerical simulation showed that the proposed algorithm has high reconstruction accuracy in both symmetric and asymmetric fields [12]. Haimiao Zhang et al. propose a new sparse Angle CT image reconstruction network that only learns the empirical part of the traditional network, leaving the rest unchanged, and it uses the CNN module as a supernetwork to predict the initialization of the u-subproblem, which is the only trainable component. While maintaining the superior performance of traditional networks in CT image reconstruction, this method greatly reduces the design of trainable parameters. Simulation and experimental verification show that the network can reconstruct CT images with high precision under only 15 projection angles [13]. Yiqing Gao et al. proposed a new chromatographic reconstruction method for a three-dimensional plasma temperature field. This method uses a camera to obtain the spectral information of the temperature field, adopts the improved CT reconstruction algorithm, and combines the maximum-entropy and least-square methods to process the spectral data, finally achieving three-dimensional temperature field reconstruction through the use of prior knowledge. The experimental results show that this method has high accuracy and real-time performance in the case of 2–4 projection angles [14].

In optical tomography reconstruction, although the research methods above can reconstruct the field with a smaller projection angle, the premise is that prior knowledge must be used to make up for the information loss caused by a sparse projection angle. When reconstructing the radiation field of unknown materials or without prior knowledge, it is difficult to achieve optical tomography reconstruction using the methods above. To solve this problem, this research proposed the use of multi-spectral temperature field reconstruction technology under the conditions of sparse projection. This technology utilizes the principles of multi-spectral temperature measurement technology, taking the correlation of radiation information between sub-regions of the temperature field as the optimization objective. The statistical rule of missing information is established by combining the equation constraint optimization algorithm and multi-spectral thermometry technology, indirectly compensating for the lack of information caused by a sparse projection angle, which realizes the reconstruction of the temperature field.

2. Basic Principles

The basic idea of the optical tomography reconstruction algorithm is using a detector to collect the radiation projection distribution of a tested field at an appropriate angle, reconstructing the field according to the projection distribution; furthermore, the reconstruction of the field is also the reconstruction of the original function from the line integral. As shown in Figure 1, $f\left(x,y\right)$ is the tested field, x-y is the right-angle coordinate system, s-t is the coordinate system after rotating the x-y coordinate axis along the counterclockwise direction by an angle of θ_k, which is also the projection angle, and the projection P(k,t) of f(x,y) along the k th angle is expressed using the following equation:

$$P\left(k,t\right)={\int }_{\left(t,{\theta }_k\right)line}f\left(x,y\right)ds={\int }_{-\infty }^{\infty }f\left(t\mathit{cos}{\theta }_k-s\mathit{sin}{\theta }_k,t\mathit{sin}{\theta }_k+s\mathit{cos}{\theta }_k\right)ds,$$

(1)

where $P\left(k,t\right)$ is the projection data, $f\left(x,y\right)$ is the tested field, and the transformation relationship between the two coordinate systems is $x=t\mathit{cos}{\theta }_k-s\mathit{sin}{\theta }_k$, $y=t\mathit{sin}{\theta }_k+s\mathit{cos}{\theta }_k$.

For the convenience of measurement and calculation, the tested field is divided into sub-regions and numbered, as shown in Figure 2. The reconstruction of the field to be measured and the representation of the projection data are changed from a line integration to the form of summation; here, Equation (1) transforms into the following equation:

$$P\left(k,t\right)={\sum }_{s=-\infty }^{\infty }f\left(t\mathit{cos}{\theta }_k-s\mathit{sin}{\theta }_k,t\mathit{sin}{\theta }_k+s\mathit{cos}{\theta }_k\right),$$

(2)

From our knowledge of mathematics, it is known that, in the case of few projection angles, the reconstruction of the tested field with the projection data is converted into a problem of solving underdetermined system of equations; i.e., the reconstruction of the temperature field becomes a numerical optimization problem under the equational constraints. The basic form of the equationally constrained problem is as follows:

$$\left\{\begin{array}{c}minF\\ s.t. c_i=0\left(i=\mathrm{1,2},\cdots ,n\right)\end{array}\right.,$$

(3)

where F denotes the objective function to be optimized and c_i denotes the equation constraints.

2.1. Objective Function

The spectral characteristic of materials relates to the radiation value curve of materials at different wavelengths, and the key to solving the temperature by using the spectral characteristics of materials is to assume the spectral emissivity model by using the spectral characteristic of materials. However, as outlined in [15], the multi-spectral temperature measurement method based on the optimization idea can realize multi-spectral high-precision temperature measurements without assuming spectral emissivity. In this process, the spectral characteristics of materials are unknown and not used. Therefore, a multi-spectral temperature measurement method based on optimization ideas can overcome the dependence of measurement technology on material spectral characteristics. Therefore, this section constructs an objective function according to the measurement method described in [15]. According to [15], for the temperature at the same point of the object to be measured at the same time, F in the following formula can be optimized through combination with multi-spectral temperature measurement technology:

$$F={\left(T_{{\lambda }_1}-T_{{\lambda }_3}\right)}^2+{\left(T_{{\lambda }_2}-T_{{\lambda }_4}\right)}^2+\cdots +{\left(T_{{\lambda }_{n-2}}-T_{{\lambda }_n}\right)}^2+{\left(T_{{\lambda }_{n-1}}-T_{{\lambda }_1}\right)}^2,$$

(4)

where F represents the temperature measurement model, and $T_{{\lambda }_i}$ represents the temperature at the same point measured by different spectral channels. Specifically, $T_{{\lambda }_i}$ is

$$T_{{\lambda }_i}=1/\left({\displaystyle \frac{1}{T_{{\lambda }_b}+{\eta }_1\mathsf{\Delta }{T}_{{\lambda }_b}}}+{\displaystyle \frac{{\lambda }_i+{\eta }_2\mathsf{\Delta }{\lambda }_i}{C_2}}ln\left({\displaystyle \frac{{\epsilon }_{{\lambda }_i}\left(V_{{\lambda }_{i}b}+{\eta }_3\mathsf{\Delta }{V}_{{\lambda }_{i}b}\right)}{V_{{\lambda }_i}+{\eta }_4\mathsf{\Delta }{V}_{{\lambda }_i}}}\right)\right),$$

(5)

where ${\eta }_1$, ${\eta }_2$, ${\eta }_3$, and ${\eta }_4$ are correction factors within the value range [−1,1] and ${∆T}_{rb}$, ${∆\lambda }_i$, ${∆V}_{{\lambda }_{i}b}$, and ${∆V}_{{\lambda }_i}$ are the maximum measurement errors. They are used to correct measurement errors. In this article, ${∆V}_{{\lambda }_{i}b}$ and ${∆V}_{{\lambda }_i}$ are represented by ${∆f\left(x,y\right)}_{{\lambda }_{1}b }$ and ${∆f\left(x,y\right)}_{{\lambda }_1}$, respectively. C₂ stands for Planck’s second constant, $C_2=1.43879\times {10}^4 {\mathsf{\mu }m}^4·K$. ${\epsilon }_{{\lambda }_i}$ represents the spectral emissivity at wavelength ${\lambda }_i$, and it is the only unknown in the formula. It can be seen from [15] that by optimizing F in the formula and constraining the range of spectral emissivity, ${\epsilon }_{{\lambda }_i}$ can be obtained, thus obtaining temperature $T_{{\lambda }_i}$.

Similarly, the temperature field with m sub-regions shown in Figure 2 can also be reconstructed by using m temperature-measuring equipment and temperature-measuring models F₁, F₂, ⋯, F_m (as shown below).

$$\left\{\begin{array}{c}{F}_1={\left(T_{{1 \lambda }_1}-T_{{1 \lambda }_3}\right)}^2+{\left(T_{{1 \lambda }_2}-T_{{1 \lambda }_4}\right)}^2+\cdots +{\left(T_{{1 \lambda }_{n-2}}-T_{{1 \lambda }_n}\right)}^2+{\left(T_{{1 \lambda }_{n-1}}-T_{{1 \lambda }_1}\right)}^2\\ F_2={\left(T_{{2 \lambda }_1}-T_{{2 \lambda }_3}\right)}^2+{\left(T_{{2 \lambda }_2}-T_{{2 \lambda }_4}\right)}^2+\cdots +{\left(T_{{2 \lambda }_{n-2}}-T_{{2 \lambda }_n}\right)}^2+{\left(T_{{2 \lambda }_{n-1}}-T_{{2 \lambda }_1}\right)}^2\\ \begin{array}{c}\cdots \\ F_m={\left(T_{{m \lambda }_1}-T_{{m \lambda }_3}\right)}^2+{\left(T_{{m \lambda }_2}-T_{{m \lambda }_4}\right)}^2+\cdots +{\left(T_{{m \lambda }_{n-2}}-T_{{m \lambda }_n}\right)}^2+{\left(T_{{m \lambda }_{n-1}}-T_{{m \lambda }_1}\right)}^2\end{array}\end{array}\right.,$$

(6)

where $T_{{1 \lambda }_1}$, $T_{{2 \lambda }_1}$, and $T_{{m \lambda }_1}$ represent the temperature of the first, second, and m sub-regions at wavelength ${\lambda }_1$, respectively. At present, due to technical limitations, the radiation of each point cannot be directly obtained from the corresponding region of the temperature field via any technical means, and only the line integral of the radiation in the measurement direction can be obtained from one end of the temperature field. Therefore, combining temperature measurement technology with chromatographic reconstruction technology, following [15], the temperature field shown in Figure 2 is divided into a multi-band radiation field, as shown in Figure 3, by adding in the spectral separation technique.

Figure 3 shows that the temperature field $f\left(x,y\right)$ is separated into $f_{{\lambda }_1}\left(x,y\right)$, $f_{{\lambda }_2}\left(x,y\right)$, ⋯, and $f_{{\lambda }_j}\left(x,y\right)$ at λ₁, λ₂, ⋯, and λ_j wavelengths. Therefore, $T_{{1 \lambda }_1}$, $T_{{2 \lambda }_1}$, and $T_{{m \lambda }_1}$, can be expressed as follows:

$$\left\{\begin{array}{c}{T}_{{1 \lambda }_1}={T\left(\mathrm{1,1}\right)}_{{\lambda }_1}=1/\left({\displaystyle \frac{1}{{T\left(\mathrm{1,1}\right)}_{rb}+{\eta }_1\Delta {T\left(\mathrm{1,1}\right)}_{rb}}}+{\displaystyle \frac{{\lambda }_1+{\eta }_2\Delta {\lambda }_1}{C_2}}ln\left({\displaystyle \frac{{\epsilon \left(\mathrm{1,1}\right)}_{{\lambda }_1}·\left({f\left(\mathrm{1,1}\right)}_{{\lambda }_{1}b}+{\eta }_3\Delta {f\left(\mathrm{1,1}\right)}_{{\lambda }_{1}b}\right)}{{f\left(\mathrm{1,1}\right)}_{{\lambda }_1}+{\eta }_4\Delta {f\left(\mathrm{1,1}\right)}_{{\lambda }_1}}}\right)\right)\\ T_{{2 \lambda }_1}={T\left(\mathrm{1,2}\right)}_{{\lambda }_1}=1/\left({\displaystyle \frac{1}{{T\left(\mathrm{1,2}\right)}_{rb}+{\eta }_1\Delta {T\left(\mathrm{1,2}\right)}_{rb}}}+{\displaystyle \frac{{\lambda }_1+{\eta }_2\Delta {\lambda }_1}{C_2}}ln\left({\displaystyle \frac{{\epsilon \left(\mathrm{1,2}\right)}_{{\lambda }_1}·\left({f\left(\mathrm{1,2}\right)}_{{\lambda }_{1}b}+{\eta }_3\Delta {f\left(\mathrm{1,2}\right)}_{{\lambda }_{1}b}\right)}{{f\left(\mathrm{1,2}\right)}_{{\lambda }_1}+{\eta }_4\Delta {f\left(\mathrm{1,2}\right)}_{{\lambda }_1}}}\right)\right)\\ \begin{array}{c}\cdots \\ T_{{m \lambda }_1}={T\left(x,y\right)}_{{\lambda }_1}=1/\left({\displaystyle \frac{1}{T_{\left(x,y\right)rb}+{\eta }_1\Delta T_{\left(x,y\right)rb}}}+{\displaystyle \frac{{\lambda }_1+{\eta }_2\Delta {\lambda }_1}{C_2}}ln\left({\displaystyle \frac{{\epsilon \left(x,y\right)}_{{\lambda }_1}·\left({f\left(x,y\right)}_{{\lambda }_{1}b}+{\eta }_3\Delta {f\left(x,y\right)}_{{\lambda }_{1}b}\right)}{{f\left(x,y\right)}_{{\lambda }_1}+{\eta }_4\Delta {f\left(x,y\right)}_{{\lambda }_1}}}\right)\right)\end{array}\end{array}\right.,$$

(7)

where ${T\left(x,y\right)}_{{\lambda }_1}$ represents the temperature of the radiation field $f_{{\lambda }_1}\left(x,y\right)$ at (x,y), and also the temperature measured by the spectral channel ${\lambda }_1$ at position (x,y) of the temperature field. The temperature measurement model presented in Equation (6) is a multi-spectral temperature measurement model based on a reference temperature; therefore, ${T\left(x,y\right)}_{rb}$ represents the reference temperature of the radiation field f at position (x,y). ${\epsilon \left(x,y\right)}_{{\lambda }_1}$ represents the spectral emissivity of the radiation field $f_{{\lambda }_1}\left(x,y\right)$ at position (x,y).

Similarly, the temperature in the temperature field $f_{{\lambda }_2}\left(x,y\right)$, ⋯, and $f_{{\lambda }_j}\left(x,y\right)$ can be expressed in this way. The objective function of temperature field reconstructed according to Equations (6) and (7) is shown in Equation (8).

$$F=F_1+F_2+\cdots +F_m,$$

(8)

The reconstruction of temperature field can be realized by optimizing the objective function in the equation above.

2.2. Constraints

In order to limit the optimization range of the objective function, it is necessary to constrain the range of values of the radiation (i.e., unknowns) in the field, and the projection data of the multi-band radiation field are shown in Figure 4.

From Figure 4, when the projection angle θ_k takes the value of k₁, the projection data of the field at the wavelengths of λ₁, λ₂, ⋯, and λ_j are, respectively,

$$\left\{\begin{array}{c}{P}_{{\lambda }_1}\left(k_1,t\right)=\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_1}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right)\\ P_{{\lambda }_2}\left(k_1,t\right)=\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_2}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right)\\ \begin{array}{c}\cdots \\ P_{{\lambda }_j}\left(k_1,t\right)=\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_j}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right)\end{array}\end{array}\right.,$$

(9)

Similarly, the multi-band radiation projection data P(k₁,t), P(k₂,t), ⋯ can be obtained by changing θ_k at different angles. The projection data are the key information used to reconstruct the field, which constrains the relationship between the spatial information in the field as well as the radiative information, constituting the constraints shown in Equation (10).

$$\left\{\begin{array}{c}\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_1}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right)-P_{{\lambda }_1}\left(k_1,t\right)=0\\ \sum _{s=-\infty }^{\infty }{f}_{{\lambda }_2}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right){-P}_{{\lambda }_2}\left(k_1,t\right)=0\\ \begin{array}{c}\cdots \\ \begin{array}{c}\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_j}\left(t\mathit{cos}{k}_1-s\mathit{sin}{k}_1,t\mathit{sin}{k}_1+s\mathit{cos}{k}_1\right){-P}_{{\lambda }_j}\left(k_1,t\right)=0\\ \sum _{s=-\infty }^{\infty }{f}_{{\lambda }_1}\left(t\mathit{cos}{k}_2-s\mathit{sin}{k}_2,t\mathit{sin}{k}_2+s\mathit{cos}{k}_2\right){-P}_{{\lambda }_1}\left(k_2,t\right)=0\\ \begin{array}{c}\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_2}\left(t\mathit{cos}{k}_2-s\mathit{sin}{k}_2,t\mathit{sin}{k}_2+s\mathit{cos}{k}_2\right)-P_{{\lambda }_2}\left(k_2,t\right)=0\\ \cdots \\ \begin{array}{c}\sum _{s=-\infty }^{\infty }{f}_{{\lambda }_j}\left(t\mathit{cos}{k}_2-s\mathit{sin}{k}_2,t\mathit{sin}{k}_2+s\mathit{cos}{k}_2\right)-P_{{\lambda }_j}\left(k_2,t\right)=0\\ \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right.,$$

(10)

2.3. Reconstruction of the Field

In order to realize the multi-spectral temperature field measurement and optimization of an objective function, constraint conditions of Equation (10) are used to optimize the objective function of Equation (8); this process is shown in Figure 5. The specific steps are as follows:

(1)

Construct a multi-band radiation field. The radiation fields $f_{{\lambda }_1}\left(x,y\right),f_{{\lambda }_2}\left(x,y\right),\cdots ,f_{{\lambda }_j}\left(x,y\right)$ shown in Figure 5a are constructed at wavelengths of ${\lambda }_1{,\lambda }_2,\cdots {,\lambda }_j$, respectively, and are divided into sub-regions and numbered, as shown in Figure 5b.

(2)

Build a function model. As shown in Figure 5b, according to the temperature measurement method described in [15], the sum of squares of temperature differences in a single sub-region in different bands is established by using the radiation information in $f_{{\lambda }_1}\left(x,y\right),f_{{\lambda }_2}\left(x,y\right),\cdots ,f_{{\lambda }_j}\left(x,y\right)$; that is, the temperature correlation between different bands is used to measure the temperature, and the temperature correlation between different sub-regions is coupled to establish the function model F, as shown in Equation (8).

(3)

Data measurement. As shown in Figure 5c, a distributed network composed of multiple temperature-measuring units is used to collect projection data from multiple angles of the measuring field, and it is used as the constraint condition to measure the reconstruction of the field. The projection data are shown in Figure 5d.

(4)

Function optimization. The equation constraints obtained in step (3) are used to optimize the field reconstruction objective function, and the function solution is judged to reach the optimal value according to the convergence condition of the algorithm; otherwise, return to step (1), and the radiation value of each band of the field to be measured is optimized until the optimal solution is obtained. The optimization algorithm can be realized through the use of gradient descent, particle swarm, and neural networks. In Figure 5, there are no specific requirements for the use of neural networks. In this paper, a BP neural network has been adopted to achieve optimization. The input layer is m × j neurons, the output layer is m × j neurons, and there are three hidden layers.

(5)

Temperature field reconstruction. Combined with Equation (7) and using the temperature measurement method in [15], the optimal solution of the function is substituted, and the temperature value of the sub-region is calculated to realize the reconstruction of the temperature field.

After the principal analysis above, the theoretical abstraction of temperature field reconstruction, as well as the conversion to the equational constraint problem, are completed, and the reconstruction of a sparse tomography multi-spectral temperature field is realized.

3. Experimental Verification

3.1. Experimental Setup

In order to verify the abovementioned theory of sparse tomography multi-spectral temperature field reconstruction, the temperature field in a temperature range of 600–2400 K, shown in Figure 6, is taken as an example, and the reconstruction method above is utilized to complete the experimental verification. The experiment is set up with single-peak and multi-peak temperature fields, as shown in Figure 6, where Figure 6a,c show the spatial distribution of the temperature field, and Figure 6b,d show the temperature distribution of the temperature field and the projection angle. The figures shown in the experimental process are the result of surface fitting.

The spectral data used in the experiments were obtained from the spectral emissivity of CO in the 5066–5659 nm band published by NASA [16], and the specific values are shown in

Table 1. The spectral emissivity of CO.

Temperature (K)	Wavelength (nm)
	5066	5126	5185	5244	5304	5363	5422	5481	5541	5600	5659
600	0.32	0.6	0.52	0.4	0.39	0.52	0.47	0.31	0.21	0.16	0.12
1200	0.19	0.29	0.43	0.35	0.31	0.31	0.37	0.41	0.39	0.24	0.19
2400	0.13	0.28	0.36	0.36	0.32	0.28	0.24	0.24	0.25	0.28	0.29

.

According to the Blackbody radiation law, the radiant brightness of an object with absolute temperature T at wavelength ${\lambda }_i$ is as follows:

$$f_{{\lambda }_i}\left(x,y\right)=C_1{\epsilon }_{{\lambda }_i}{\left(\pi {\lambda }_i^5{e}^{C_2/{\lambda }_{i}T\left(x,y\right)}\right)}^{-1},$$

(11)

where C₁ is the first radiation constant. Equation (11) was used to calculate the radiation data of each sub-region of the experimental temperature field according to the temperature distribution shown in Figure 6 and the spectral data in Table 1, which are the basis of the data processing described in this paper. Combined with Figure 6, projection direction will be substituted into Equation (2) to obtain the projection data required for the experiment; its specific values are shown in

Table 2. Data of projection.

Projection (10−11 W·m−2·μm−1·sr−1)		Wavelength (nm)
		5066	5126	5185	5244	5304	5363	5422	5481	5541	5600	5659
Single-peak	P(0°,1)	5.68	10.00	8.22	5.98	5.46	6.95	5.90	3.67	2.42	1.72	1.23
	P(0°,2)	4.26	6.97	7.35	5.51	4.79	5.24	5.20	4.45	3.61	2.24	1.64
	P(0°,3)	4.08	6.93	7.14	5.55	4.83	5.17	4.86	4.03	3.31	2.33	1.86
	P(0°,4)	4.26	6.97	7.35	5.51	4.79	5.24	5.20	4.45	3.61	2.24	1.64
	P(0°,5)	5.68	10.00	8.22	5.98	5.46	6.95	5.90	3.67	2.42	1.72	1.23
	P(90°,1)	5.68	10.00	8.22	5.98	5.46	6.95	5.90	3.67	2.42	1.72	1.23
	P(90°,2)	4.26	6.97	7.35	5.51	4.79	5.24	5.20	4.45	3.61	2.24	1.64
	P(90°,3)	4.08	6.93	7.14	5.55	4.83	5.17	4.86	4.03	3.31	2.33	1.86
	P(90°,4)	4.26	6.97	7.35	5.51	4.79	5.24	5.20	4.45	3.61	2.24	1.64
	P(90°,5)	5.68	10.00	8.22	5.98	5.46	6.95	5.90	3.67	2.42	1.72	1.23
Multi-peak	P(0°,1)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23
	P(0°,2)	5.70	10.08	8.25	5.99	5.48	6.97	5.92	3.68	2.43	1.73	1.23
	P(0°,3)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23
	P(0°,4)	5.70	10.08	8.25	5.99	5.48	6.97	5.92	3.68	2.43	1.73	1.23
	P(0°,5)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23
	P(90°,1)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23
	P(90°,2)	5.70	10.08	8.25	5.99	5.48	6.97	5.92	3.68	2.43	1.73	1.23
	P(90°,3)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23
	P(90°,4)	5.70	10.08	8.25	5.99	5.48	6.97	5.92	3.68	2.43	1.73	1.23
	P(90°,5)	5.69	10.06	8.23	5.98	5.47	6.95	5.91	3.68	2.42	1.72	1.23

. In the projection data, the total number of detection points for each projection is 5, and the number of spectral lines is 5. Combined with Figure 6b, it can be seen that P(0°,1) in the table represents the first projection from left to right at a projection angle of 0°, and it is the integral of five spectral lines in this projection direction.

3.2. Simulation Results

In the experiment, the radiation characteristics at position (5,5) are selected as the reference radiation, and the projection data are substituted into the function model to realize the reconstruction of the temperature field. The reconstruction results are shown in Figure 7.

It can be seen from Figure 7 that the temperature field can be reconstructed under two projection angles by using the tomography reconstruction method described in this paper, and the temperature distribution is basically consistent with the tested field. In order to evaluate the accuracy of the reconstruction method, the error calculation formula shown in Equation (12) is used to evaluate the accuracy of the method.

$$\left\{\begin{array}{c}Aver={\displaystyle \frac{\sum _{x,y}\left|T\left(x,y\right)-T^\left(x,y\right)\right|}{T_{max}\times M\times N}}\\ Max={\displaystyle \frac{{\left|T\left(x,y\right)-T^\left(x,y\right)\right|}_{max}}{T_{max}\left(x,y\right)}}\\ Rmse={\left({\displaystyle \frac{\sum _{x,y}{\left[T\left(x,y\right)-T^\left(x,y\right)\right]}^2}{\sum _{x,y}{\left[T\left(x,y\right)\right]}^2}}\right)}^{{\displaystyle \frac{1}{2}}}\end{array}\right.,$$

(12)

In the formula, Aver represents the average error of the temperature field reconstruction, M and N represent the number of rows and columns of the field, Max represents the maximum error, and Rmse represents the root-mean-square error.

Table 3. Error of temperature field reconstruction.

Temperature Field	Aver (%)	Max (%)	Rmse (%)
Single-peak	1.64	6.26	5.08
Multi-peak	2.1	12.25	3.99

shows the evaluation results of the reconstruction of the temperature field in Figure 7 achieved using the error calculation method shown in Equation (12).

As can be seen from Table 3, the error distribution of both single-peak and multi-peak temperature fields is $Max>Rmse>Aver$. It indicates that the temperature field reconstruction method in this paper has the same reconstruction quality for temperature fields with different distributions; that is, the robustness of this method is strong. Overall, the proposed method can reconstruct the temperature field with a precision of 1.64–12.25%.

3.3. Experimental Test

In order to verify the feasibility of the temperature field measurement technology in this paper and the authenticity of the simulation results, based on the reconstruction process of a single-peak temperature field in the simulation above, the measurement system in [17] was used to measure the combustion radiation of black powder for the projection data, and the multi-spectral temperature field reconstruction technology described in this paper was used as the data processing technology to measure the temperature field generated by the combustion of black powder. The system can produce a linear spectrum within the working spectrum range of the target and ensure the accurate acquisition of the spectral information of the measured field. In the experiment, the width of the radiation field is 60 cm, and the field of view that a single detector can detect is 12 cm. Therefore, this paper adopted the use of five detectors on the two projection angles, respectively, and divided the temperature field into 25 sub-regions. After the experiment above, the temperature field information was obtained through the measurement system. The measurement results are shown in Figure 8.

Figure 8a,c,e show the combustion conditions of black powder at 1.1 s, 2.3 s, and 4.5 s after ignition, respectively; Figure 8b,d,f show the results of the corresponding measurement data after data processing and data fitting. An infrared thermometer with a temperature-measuring range of −25~3000 °C, adjustable spectral emissivity, and a measuring error of ±2% was used to measure the central peak temperature at 1.1 s, 2.3 s, and 4.5 s during the combustion of black powder. The accurate error of the method in this paper was calibrated, and the results were 12.1%, 5.7%, and 13.4%. It can be seen that the experimental measurement results are consistent with the combustion of black powder and basically record the whole process of black powder combustion. Therefore, the temperature field measurement method described in this paper can realize the actual measurement of the temperature field under two projection angles.

3.4. Advantages of This Method

In order to explore the advantages of the proposed method in terms of temperature field reconstruction, the performance of the proposed method was compared with that of [12,13,14]; the comparison results are shown in

Table 4. Advantages of this method.

Method	Prior Knowledge	Error Distribution (%)	Projection Numbers
Textual method	×	1.64–12.25	2
Method in [12]	√	0.5–11.87	6
Method in [13]	√	None	15–180
Method in [14]	√	0.21–21.57	2–6

‘×’ indicates that this method requires no prior knowledge. ‘√’ indicates that this method requires prior knowledge.

. “None” indicates that no relevant data are published in [13].

As can be seen from Table 4, compared with other methods [12,13,14], the method described in this paper does not require any prior knowledge and can achieve temperature field reconstruction from just two angles. In addition, although the minimum error of the method in [14] is smaller than that of the proposed method, the maximum error is much higher than that of the proposed method, indicating that the proposed method is more robust than the method outlined in [14].

Through the experiments above, it can be seen that without any prior knowledge, the proposed method realizes the temperature field reconstruction with an accuracy of 1.64–12.25% through the projection data from two angles, which has been verified as feasible and has certain advantages compared with other methods.

4. Discussion

4.1. The Influence of the Angle between Two Projections on the Reconstruction Results

In order to explore the influence of different angles between two projections on the reconstruction results, this paper set the angles as 0°, 30°, 60°, 90°, 120°, 150°, and 180° to acquire projection data for the measured field and used the temperature field reconstruction method in this paper to process the acquired results. The reconstruction results are shown in the figure below.

As can be seen from the figure, when the angle between two projections is between 60° and 120°, the reconstruction error distribution is in a small range. When the angle exceeds this range, the reconstruction error rapidly increases, and the error distribution is relatively concentrated, indicating that when the angle exceeds a certain value, the two-projection data will become very redundant, and the algorithm in this paper will quickly converge in the place where the error is large. This phenomenon shows that the method should obtain as much information as possible from two-projection data when selecting a projection angle so that the redundancy between data becomes smaller.

In addition, the symmetry of the temperature field is related to the reconstruction effect. When the field is symmetric, the 0° projection data are the same as the 180° projection data, and the 90° projection data are the same as the 270° projection data. The symmetry of the field helps to reduce the number of projections. However, this does not affect the superiority of this method without prior knowledge, and even if the number of projections increases, the projections are still sparse. From the analysis in Figure 9, we can see that in the case of a large object shadowing a small object, we can still reconstruct the temperature field under projection angles of 60°, −120°, so 90° is not strict.

4.2. The Influence of the Number of Sub-Regions on the Reconstruction Results

In order to explore the influence of the number of sub-regions on the reconstruction results, the field to be measured was divided into 1, 9, 25, 49, 81, 121, and 169 sub-regions, respectively, and the temperature field was reconstructed using the method described in this paper. The reconstruction results are shown in the figure below.

It can be seen from the abovementioned experimental setup that the figures shown in the experimental process are the results of surface fitting, so there are interpolations in the reconstructed temperature field results. When the number of temperature field sub-regions is small, there is a large error in interpolation, so the error is larger. When the number of sub-regions of the temperature field increases, interpolation will smooth the surface without introducing errors, so the errors will decrease sharply when the number of sub-regions exceeds 9; when the number exceeds 49, the errors begin to increase, because when the number of sub-regions exceeds a certain amount, the calculation amount of the optimization algorithm will increase sharply, and the efficiency and accuracy will be affected to a certain extent. Therefore, it can be seen from the Figure 10 that when the number of sub-regions changes from 1 to 169, the reconstruction accuracy first becomes smaller and then larger and tends to be stable when the number of sub-regions exceeds 121, with the highest accuracy obtained at 49.

The division of the sub-region is designed according to the actual measurement requirements; specifically, the number of detectors is determined according to the size of the field to be measured and the width of radiation that a single detector can detect. In the case of ensuring that the radiation information of a single angle is not omitted, the division of the sub-region is determined according to the number of detectors used at a single projection angle. Compared with the optimal number of 49, when the number of sub-regions in this study is 25, the accuracy of temperature field reconstruction is not much different.

5. Conclusions

This study proposed the use of multi-spectral temperature field reconstruction technology under the conditions of sparse projection. Without any prior knowledge, this method can realize the reconstruction of the temperature field with an accuracy of 1.64–12.25% under two projection angles. Moreover, the influence of the angle between the two projections on the reconstruction results and the influence of the number of sub-regions on the reconstruction results are discussed in this paper to provide technical support for field reconstruction in complex temperature fields and harsh environments.