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Abstract
The data processing of multi-wavelength pyrometry (MWP) is faced with the problem of solving N equations and N+1 unknown underdetermined equations. The traditional iterative optimization methods are difficult to meet the actual measurement requirements in terms of accuracy and efficiency. With the development of artificial intelligence technology in the field of data processing, it is expected to solve this problem. A generalized inverse matrix (GIM) is combined with a long short-term memory (LSTM) neural network algorithm for data processing of MWP is proposed, which emissivity influence is dispensed completely. Firstly, GIM is used for classification of the emissivity. Furthermore, inputting to the LSTM network not only ensures the accuracy of temperature measurement but also greatly improves the efficiency. The simulation results demonstrated that the accuracy of the GIM-LSTM algorithm was superior to that of the GIM-EPF and BP methods. After random noise was added, the relative error was still less than that for the GIM-EPF and BP methods, and the algorithm exhibited excellent anti-noise performance. Publicly available temperature data for the exhaust plume of a rocket engine were processed by the GIM-LSTM method, and the average relative error was less than the traditional method. Especially, in terms of inversion speed, the operational time of the GIM-LSTM algorithm was at the millisecond level, which is of great significance for the real-time monitoring of rocket exhaust plumes. The proposed GIM-LSTM data processing algorithm affords high accuracy and speed and is suitable for practical measurement of high-emissivity objects in real-time via MWP.
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1. Introduction
Information on the spectral radiation emitted from an object [1,2] is used in multi-wavelength pyrometry (MWP) to obtain the real temperature of the object. The difficulty of data processing in MWP is that n spectral channels require n radiation equations based on Planck’s formula, but there are n + 1 unknown parameters (n unknown spectral emissivities and unknown true temperature). Mathematical methods are used to solve nonlinear under-determined equations. The data processing methods used in multispectral thermometry has always been a research hotspot. Gardner proposed a logarithmic linear hypothesis model to characterize the surface of oxidized or non-oxidized materials [3]. Coates pointed out for metallic materials that a rough estimate of emissivity would yield more accurate results than the logarithmic linear hypothesis model [4]. Khan proposed a nonlinear fitting method for the spectral emissivity-linear hypothesis model and described an experimental verification method for the multi-wavelength radiation theory on a blackbody and an attenuator [5]. Dai proposed data processing methods for a hypothetical emissivity model which included an automatic sorting method, a stepwise regression method [6], and a square error method [7]. Sun described a secondary measurement method [8], which assumed a linear relationship between emissivity and temperature at a selected wavelength. Hagqvist proposed a spectral temperature measurement method with compensation for the emissivity. This method may be used for metals having different emissivities [9]. Araújo explored a polynomial differential algorithm for obtaining the temperature data of multi-wavelength radiation [10]. Wang proposed a constrained optimization algorithm for the calculation of the spectral emissivity in multi-wavelength temperature measurement [11]. Yu proposed a spectral emissivity measurement method based on Fourier transformation [12]. Li used the least-squares method to measure accurately the temperature of a transparent smooth surface using multiple wavelengths via multi-angle polarization [13]. An algorithm for multi-wavelength radiation temperature measurement for turbine blades based on a moving narrowband spectral window was proposed by Zhao, and this did not depend on the assumed emissivity model [14]. Xing proposed the use of a generalized inverse matrix-normalization (GIM-NOR) data processing algorithm [15], an emissivity range constraint optimization algorithm [16], a gradient projection (GP) algorithm, an internal penalty function (IPF) algorithm [17,18], and a GIM external penalty function (GIM-EPF) algorithm [19] for data processing of MWP. The aforementioned methods may only be applied to the data processing of spectral data on specific materials and not to materials in general.
With the development of artificial intelligence technology, the neural network has become one of the most powerful tools for solving nonlinear black-box mapping problems. Sun solved the multi-wavelength radiation temperature data processing problem by using a conventional BP neural network [20]. Xi established an infrared radiation model based on a RBF neural network and estimated the spectral emissivity of the target [21]. Chen explored a multi-wavelength temperature measurement method based on an adaptive emissivity model under high temperature [22]. Most materials used in engineering can be measured by this algorithm without the need for models that contain information on emissivity as a function of wavelength. However, due to the shortcomings of the BP neural network, such as easily falling into a local minimum, slow convergence speed, and weak generalization ability, it is difficult for this method to meet the requirements of practical applications. In recent years, transformer neural network (TNN) [23] and informer neural network (INN) [24] models have been shown to have high predictive power for long-series time series forecasting (LSTF). MWP is often used to measure high temperatures in extreme conditions in a short time, so the efficiency is key to data processing algorithm [25]. However, the occurrence of unknown emissivities in the measurement system is a big obstacle which affects the accuracy of temperature measurement. The iterative data processing method based on generalized inverse constraint optimization is inefficient and difficult to apply to real-time measurement scenarios. The BP based neural network data processing method does not consider the temporal correlation of the temperature data and the method requires much training, hence it is of low accuracy. Therefore, research needs to focus on the latest developments and progress in neural network models with the view to realizing improvements in MWP on a continual basis.
In this paper a data processing algorithm, the GIM, is combined with the long short-term memory (LSTM) neural network algorithm and applied to high-temperature MWP, this approach being suitable for studying unknown emissivity targets with less tested data efficiently. The accuracy of simulation inversion for the emissivity models for various applications is discussed, and the anti-noise and generalization capabilities are explored. In addition, the feasibility of the algorithm is verified by processing the MWP measurement data for the temperature of the exhaust plume of a rocket engine.
2. Basic principles
The overall design of the GIM-LSTM algorithm is that the target temperature can be calculated through the corresponding LSTM subnet after a set of emissivities is classified via the GIM. The overall structure is shown in the flow diagram of Fig. 1.
2.1 Generalized inverse matrix algorithm
Based on MWP theory, each spectral channel can have an equation with 2 unknown parameters, one is tested temperature T, the other is spectral emissivity ε(λi, T) [6]. The output signal Vi of the ith channel is:
The emissivity of a blackbody is 1. When the blackbody temperature is T’, the output signal Vi’ of the ith channel is:
The above four equations are known as the Moore-Penrose equation (M-P equation). Let A∈Cm × n, and if there is a G∈Cn × m satisfying all or part of Eq. (7) to Eq. (10) of the M-P equation, then G is called the GIM of A. The GIM satisfying the M-P equations is denoted as A{1,2,3,4}. For a given A, this GIM has a unique solution, which is called A + . Therefore,
A set of spectral emissivities and a temperature value can be obtained by Eq. (11).
The emissivity model of an object has many possibilities. A eight-wavelength thermometer is used in this study. There are six emissivity values for each model. The emissivity model is classified according to the emissivity relationship between two adjacent channels. There are three possibilities between every two channels (greater, equal, less), thus the six-channel thermometer has a total of 243 emissivity models (1*3*3*3*3*3).
The simulation data are as follows: the real temperature is 1800K, and the blackbody reference temperature is 1600 K. There are 243 emissivity models (1*3*3*3*3*3). Twelve emissivity models are listed in Table 1 as examples. According to Eq. (11), the calculated results from the generalized inverse matrix are obtained as shown in Fig. 2.
Fig. 2. Emissivity results from generalized inversion compared with the actual target emissivities (12 of 243 models).
Table 1. Target material emissivity model (12 of 243 models)
As can be seen from Fig. 2, the trend for the distribution of the emissivity obtained from GIM is consistent with the actual situation. According to this trend characteristic of the emissivity from generalized inverse, an automatic classification function can be set to classify the data into the corresponding emissivity model. If we use -1,0,1 to represent the size relationship between the channels, then the value of the six-channel thermometer can be represented by a 5*1 vector. The emissivity model can also be represented by a 5*1 vector, for example, the specified vector [1,1,1,1,1] represents the class B emissivity model. By automatic comparison, the resulting values are quickly classified into the corresponding emissivity models. Therefore, the classification of the emissivity models can be realized quickly by the generalized inverse method. The classification model is shown in Fig. 3.
2.2 LSTM network
The voltage data collected by the multi-wave pyrometer are derived from the radiation emitted from the surface of the object. The process whereby the radiation intensity (temperature) changes is a continuous process, and can be seen as an orderly sequence. Therefore, compared with a general neural network and a convolution neural network, the recursive neural network is more suitable for processing multi-wavelength spectral radiation data. The long short-term memory (LSTM) network is a typical recurrent neural network and its network structure is shown in Fig. 4. In the schematic, h represents the input, x represents the output, and c represents the cell status, which contains information from previous nodes. The subscript t represents the current time, and t-1 represents the correlation value from the previous time. w and b are parameters of the network itself, and they are optimized during the continuous iteration training of the network.
The LSTM neural network is designed specifically to solve long-term dependencies problems. Different from a single layer neural network, interaction is specifically conducted. The gate structure used by the LSTM enables selective transmission of information through the neural layer of a sigmoid function for point-by-point multiplication. Each element output of the sigmoid function layer (a vector) is a real number between 0 and 1. This number represents the weight (or proportion) that allows the transmission of the corresponding information. For example, “0” means no information is allowed to pass, and “1” means all information is allowed to pass. The LSTM realizes the protection and control of the information through the input gate, the forget gate, and the output gate.
The forget gate is the structure that determines which information is discarded. The gate reads ht-1 and xt, and outputs a value ft between 0∼1 to the cell state ct-1. “1” means complete reservation, and “0” means complete abandonment. So ft is as follows:
The input gate determines the amount of new information which is added to the cells. First, the information that needs to be updated it is determined by a sigmoid layer, and a vector ćt is generated by a tanh layer, which is the optional updated content. Then the two parts are combined to update the status of the cell. The output gate is used to determine the output value. The output cell status is determined by a sigmoid layer. The A value is obtained by treating the cell state with a tanh function. The output of the output gate is determined by multiplying the value with the output of the sigmoid gate ot. The error of the LSTM training model is evaluated by the cross entropy loss function, and the parameters are updated by the gradient descent algorithm. The theory of the stochastic gradient descent (SGD) iterative method is the method of choice because the SGD theory is easy to accept, while other new methods are less interpretable. Also the SGD method relies on an algorithm that can run efficiently in the case of information redundancy. Compared with the non-random algorithm, the SGD method performs better in early cycle training. Therefore, using the SGD method can reduce the number of training rounds, reduce the training time, and the accuracy will not degrade.The training process of the LSTM neural network is divided into the following steps.
- (1) Forward propagation: The sample is input into the LSTM network and information passes sequentially passes through the input layer, all hidden layers, and the output layer to obtain an output value.
- (2) Calculation error: the output value is compared with the expected output to obtain the error value of the network output.
- (3) Back propagation: The output error passes through the hidden layer and the input layer in turn. The purpose is to return the output error and assign the error to all units of each layer, thereby obtaining an error signal for each layer of units, and then continuously correcting the weight of each unit.
- (4) Iterative training: the above steps are repeated to make the network error less than the expected error or to make the accuracy higher than the expected results, then the training is terminated.
3. Simulation experiments
3.1 Preparation of dataset for training and testing
In the experiment, 243 emissivity models (1*3*3*3*3*3) were used to train the network. Each model has 252 emissivity groups. These models represent all the trends of the emissivity, twelve kinds of emissivity models are shown in the wavelength functions in Fig. 5. In order to compare with the experimental results in Ref. [8], the effective wavelength and reference temperature (2252 K) are selected as same to Ref. [8]. In the simulation, the temperature rose from 2300 K to 3000 K in steps of 25 K. After comparing the trained output with the actual temperature, the network was optimized until the accuracy requirements were met. Thirty temperature data were selected as the test set, and their data were used as the classification training set for each model. The emissivity values for general objects range between 0 to 1, while the emissivity values of real objects typically are between 0.3 and 0.8, thus the emissivity values selected in this study were between 0.3 and 0.8.
3.2 Generalized inverse-LSTM simulation results
The data after GIM classification were input into the corresponding LSTM subnet for training. Theoretically, any function can be approximated by a three-layer neural network. According to training experience, 400 training sessions were conducted using the 8-8-8-1 network structure. Network performance was validated by the test dataset. The network parameters are shown in Table 2.
Table 2. LSTM network parameters and values
To facilitate interpretation, the prediction results of six common emissivity models of 243 models are listed as simulation experimental results. The results are shown in Fig. 6 and Fig. 7.
From the training set simulation results in Fig. 6, it can be seen that in the temperature range 2300∼3000 K, the six kinds of material objects temperature average relative error of the training set after GIM classification was less than 1.0%, the relative error without GIM classification was about 2.0%. From the testing set simulation results in Fig. 7, the six kinds of material objects temperature relative error of the testing set after GIM classification was less than 1.0%, and the relative error without GIM classification was about 2.5%. Therefore, the classification by GIM can improve significantly the accuracy of the inversion.
A comparison of the performances of the GIM-LSTM, the GIM-EPF and the BP algorithms is presented in Fig. 8. Temperature average relative error of the six kinds of material objects was 1.5%, 2.1% and 0.6% by BP, GIM-EPF and GIM-LSTM algorithm respectively, hence the performance of the GIM-LSTM algorithm was superior to the other traditional methods.
3.3 Simulation of anti-noise performance
In practice, MWP is affected by noise, resulting in data distortion. Therefore, to investigate the influence of random noise on the method, Gaussian random noise at the 5% level was added to the input data of the dataset. The samples with the Gaussian random noise were input into the aggregated neural network model of the above experimental design. The training results were as follows.
It can be seen from inspection of Fig. 9 and Fig. 10 that the temperature average relative error of the six kinds of material objects by GIM-LSTM was less than the unclassified dataset after the addition of random Gaussian noise (5% level). Moreover, after the addition of Gaussian random noise to the input data, the average relative error was still less than 3.0% for the six kinds of material objects, hence the algorithm exhibited excellent anti-noise performance characteristics.
A comparison of the performance of the GIM-LSTM algorithm, GIM-EPF algorithm and BP algorithm is presented in Fig. 11. The six kinds of material objects temperature average relative error of the BP algorithm was 3.2%, the GIM-EPF algorithm was 4.5%, whereas the GIM-LSTM algorithm was 1.5%. Thus, it has been shown that the GIM-LSTM has better adaptability to an noise environment than the other two traditional methods.
3.4 Validation of experimental data
The GIM-LSTM algorithm was used to process the temperature data for the exhaust plume of a rocket engine by multispectral pyrometer, which are published in Ref. [8]. In this way, it was possible to verify the practicality of GIM-LSTM algorithm. A set of spectral data every second was collected, encompassing a total of 12 groups. The measurement data are input to GIM-LSTM, which has been trained by the above simulation data. Temperature results from different algorithms, GIM-EPF, BP, GIM-LSTM is shown in Fig. 12.
The results showed that the temperature average error is less than 1.8% for the theoretical temperature 2490 K of rocket plume, whereas the average error of the BP algorithm was 2.7%, and ones of the GIM-EPF algorithm was 2.3%; thus all methods reflected good inversion accuracy but the GIM-LSTM method exhibited the lowest average error. It can be seen from Fig. 12 that the temperature obtained by the GIM-LSTM algorithm was affected by fluctuations in time. This finding is also consistent with the trend for the measured voltage data over time as shown in Ref. [8]. Based on the above results, it can be concluded that the GIM-LSTM algorithm has good versatility and wide applicability. In addition, the run time of the new method is much faster than those for the other algorithms. As can be seen in Fig. 13, the operational time of the GIM-LSTM algorithm was only 0.13 seconds, while the average run time of the BP algorithm was 0.34 seconds, and that for the GIM-EPF algorithm was 3.4 seconds. This is because the training times and the size of the dataset for the LSTM network after GIM classification are much less than the traditional BP neural network, thus the efficiency is improved greatly.
4. Conclusions
In this study, the GIM and LSTM have been combined for the processing of MWP data. It was demonstrated that the integrated approach ensures not only good accuracy in temperature measurement but also greatly improves the efficiency of data processing, thus the impact of unknown emissivities on the quality of the measured temperature data is reduced for much different material objects. The simulation results showed that the accuracy of the GIM-LSTM algorithm was superior 60% to the GIM-EPF and BP methods. Also, the relative error of the method was less than 53% that of the GIM-EPF and BP methods after the addition of 5% random noise to the input data. It was further demonstrated that the proposed algorithm exhibited excellent anti-noise performance. Publicly available data for the temperature of the exhaust plume of a rocket engine were processed by the GIM-LSTM method, and it was found that the temperature results were more accurate than those for the traditional method, especially in terms of inversion speed, where the operational time of the new algorithm was at the millisecond level. Moreover, the efficiency of the new method was some 30 times better than that of the traditional method, which is of great significance for the real-time measurement and diagnostics of rocket engine exhaust plumes. In summary, the proposed algorithm offers the advantages of high accuracy and high speed with less data and is suitable for practical high-temperature and real-time measurement scenarios via MWP.
Funding
Heilongjiang Provincial Postdoctoral Science Foundation (2022); Fundamental Research Funds for the Central Universities (2572022BH02); National Natural Science Foundation of China (61975028).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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