Two-Dimensional Temperature Field Distribution Reconstruction Based on Least Square Method and Radial Basis Function Approximation

The information of temperature field distribution is complex but quite important for theoretical study and industrial applications, such as burning, drying, and heating, because it can reflect the internal running state of industrial equipment, assist to develop control strategy, and ensure safety in operation. Acoustic imaging is a kind of noncontact temperature measurement technique which is able to achieve temperature field distribution reconstruction.The process of temperature field distribution reconstruction using acoustic imaging is an inverse problem and its reconstruction performance depends on reconstruction algorithm. In this paper, a modified reconstruction algorithm is investigated. Compared with other reconstructionmethods of least square algorithm and radial basis function approximation, the reconstruction results indicate that the modified reconstruction algorithm occupies the best information reflection of temperature field distribution and the reconstruction accuracy is improved.


Introduction
The information of temperature field distribution is complex but of crucial important for some industrial applications, such as burning, drying, and heating, because it can reflect the internal running state of industrial equipment, assist to develop control strategy, and ensure safety in operation [1][2][3].Thus, it becomes a critical issue to explore a proper temperature measurement method to reconstruct temperature field distribution and acquire temperature information in industrial environment.
Common temperature measurement techniques include contact and noncontact measurement methods.Contact measurement techniques, for instance, thermal thermocouples, thermal resistances, mercury thermometers, and fiber pyrometer, are hardly applied to accomplish temperature field distribution reconstruction, since these methods are only suitable for single-point temperature measurement [4][5][6][7].
Noncontact measurements mainly consist of laser spectroscopy, infrared temperature measurement, and optical temperature measurement.Though, noncontact measurements surmount the limitation of contact measurements in single-point temperature measurement, these techniques are only applied to the temperature measurement of material surface [8].In addition, the measurement accuracy of infrared radiations may be deteriorated because of different emissivity and reflectivity of infrared radiations from other sources [9,10].The measurement result of optical temperature measurement could be affected because the camera lens is disturbed by dust.
Acoustic imaging is a kind of noncontact temperature measurement technique which is able to achieve temperature field distribution reconstruction.The theory of this technique is in accordance with the relevance between acoustic velocity and temperature [11].Reconstructing temperature field distribution using acoustic imaging is an inverse process which depends on reconstruction algorithm.One of the most widely used reconstruction algorithms is least square algorithm (LSA) [12][13][14].But, this algorithm could not complete an entire temperature field distribution reconstruction due to its flaw of interpolation range.So, temperature message on measuring edge may be ignored.In order to overcome the shortage of LSA, [4] proposes a reconstruction algorithm based on radial basis function approximation (RBF) and singular values decomposition.However, LSA has advantages of relatively high stability and reconstruction accuracy and RBF is capable of mathematic fitting.Reconstruction results would be a breakthrough if the advantages of LSA and RBF are considered and combined reasonably.In this paper, a modified reconstruction algorithm based on LAS and RBF is investigated.Compared with LSA and RBF, the reconstruction results indicate that the modified reconstruction algorithm occupies the best information reflection of temperature field distribution and the reconstruction accuracy is improved.
This paper is organized as follows.Section 2 describes the basic principle of acoustic imaging.Section 3 presents the modified temperature field distribution reconstruction algorithm.Section 4 uses three temperature field distribution models to validate the performance of proposed reconstruction algorithm and shows simulation experiment results and analysis.Section 5 focuses on conclusions and future research.

The Basic Principle of Acoustic Imaging
The fundamental theory of acoustic imaging is that the acoustic speed in a medium is a function of the medium temperature.The simplest acoustic imaging is single path temperature measurement.It consists of two acoustic transducers installed on both sides of a path where acoustic travels.The acoustic transducers are used to transmit acoustic signal and detect it, respectively.The relationship between acoustic velocity and temperature is expressed as [14] where  is acoustic velocity,  is universal gas constant,  is absolute temperature (K), and  and   are ratio and average molecular weight of gas.For different gas, , , and   are fixed constants.So, they are replaced by a coefficient .If medium is air, , , and   are 1.40, 8314, and 29.Thus,  equals 20.03.On the contrary, if acoustic speed  and the type of gas are known, then the temperature  is expressed as The distance  between acoustic transducers could be measured and acquired.According to the correlativity of acoustic speed , acoustic flying time (AFT)   , and distance , we can get Combining ( 2) and (3),  could be rewritten as From (4), the average temperature ( ∘ C) of the measuring path is gained.

The Modified Temperature Field Distribution Reconstruction Algorithm
3.1.Least Square Algorithm.The single path temperature measurement of acoustic imaging barely fulfills temperature field distribution.But, if several acoustic transducers are installed appropriately in a measuring area, multiple paths acoustic travels will form and more temperature information will be gained.Then, proper reconstruction algorithm can reconstruct the temperature field distribution of measuring area.
Figure 1 shows a specific acoustic transducer layout in a square measuring area (10 m × 10 m) isometrically divided into 100 blocks by dashed lines.The geometric center of square measuring area is defined as the origin of coordinate.The acoustic transducers  0 - 11 are represented by black symbols.The working procedure of acoustic transducers is as follows: first, transducer  0 acts as a transmitter and launches acoustic signal and transducers  4 ,  5 , . . .,  8 receive the signal.Then, transducer  1 launches acoustic signal and transducers  4 ,  5 , . . .,  11 detect the signal.This process will not end until all the transducers launch acoustic signal and receive the signal.There are 42 acoustic flying routes (AFR).Temperature field distribution in the measuring field could be reconstructed with estimated AFTs in 42 AFRs and reconstruction algorithm.
Theoretically, the th AFT    in the th AFR   is described as [13]    = ∫ where   is the reciprocal of acoustic velocity in the th specific block.(, ) is temperature field distribution model which is artificially designed.The function of reconstruction algorithm is to reconstruct the model (, ).
Assuming that the measuring area in Figure 1 is divided into  blocks, and temperature distribution in each block is uniform, ( 5) is turned into where  and  represent the number of paths and the number of blocks.Δ  indicates the length of the th AFR passing through the th block.If   is the practical measured value of AFT in the th AFR, the difference   between   and the theoretical AFT    is expressed as Supposing  is the amount of AFRs, thus, there are  differences   in total.In order to make    and   the same as far as possible, according to LSA,   should be minimized, which means minimizing the sum of the squares of these  differences   , given by Then, canonical equation is gained as where So, the reciprocal matrix  of average acoustic velocity in each block can be calculated as As described above, when the geometric features of measuring area and the arrangement of acoustic transducers are set,  becomes a constant matrix which can be determined beforehand.Moreover, when the temperature field distribution model is designed,  can be obtained from (5).Thus, the average temperature array of the blocks  is estimated as where  = [ 1  1 ⋅ ⋅ ⋅   ]  and  is 100.
When the average temperature of each block is worked out, temperature field distribution of the measuring area could be reconstructed by interpolation among  values in matrix .However, temperature information over the border of the measuring field (Figure 1) could be missing because of the interpolation limitation of LSA.

Modified Temperature Field Distribution Reconstruction
Algorithm Based on LSA and RBF.To make up for deficiencies of LSA in interpolation and meanwhile make use of its advantages in stability and reconstruction accuracy, a modified temperature field distribution reconstruction algorithm based on LSA and RBF is proposed.
In Figure 1, assuming that (  ,   ) is the coordinate of the central point in the th block, the RBF of this block is expressed as where  is the shape parameter of RBF and  equals 0.0001.Temperature field distribution model (, ) could be indicated with a linear combination of  RBFs and shown as If each element of matrix  in ( 14) is the central point temperature of corresponding block, then, matrix  could be displayed as Equation ( 15) is rewritten as where When the coordinate of measuring area is set, matrix Φ would be calculated ahead of time and the matrix  could be gained by LSA.Thus, the unknown matrix  is acquired as In the end, the reconstructed temperature field distribution could be accomplished from (14) with  estimated .The procedure of modified reconstruction algorithm based on LSA and RBF is demonstrated in Figure 2.

Simulation Experiment Results and Analysis
A reconstruction algorithm should reconstruct multifarious temperature field distributions with complexity levels.Using MATLAB software and air as simulation experiment platform and medium, three temperature field distribution models with different hot zones distributed over the measuring area in Figure 1 are artificially made and expressed as one-peak symmetrical temperature field distribution model with one hot zone: two-peak asymmetrical temperature field distribution model with two hot zones: and three-peak asymmetrical temperature field distribution model with three hot zones: + 1000 [−20(/9−2/5) 2 −14(/9−1/4) 2 ] + 1100 [−14(/10−4/5) 2 −13(/11−4/7) 2 ] . ( The simulation experiment results are displayed using LSA, RBF, and the modified algorithm with two-dimensional isothermal contour figures in Figures 3-5.
From Figures 3-5, RBF and the modified algorithm overcome the disadvantage of LSA in interpolation limitation and offer entire temperature field information of the measuring area.But, the modified algorithm could present a better and more precise reconstruction performance over the border of the measuring area than LSA.
When the reconstruction process is finished, quantitative analysis of reconstruction performance which consists of mean relative error   and root-mean-square percent error   is computed.  and   are defined as where  is the number of calculating points in the measuring area.At the coordinate (  ,   ),   is from reconstructed temperature field distribution with LSA, RBF, and the modified algorithm and   is from model. mean is the mean temperature value of the temperature field distribution model.
It is important to monitor the hotspots in hot zones because they may lead to unexpected and major accidents.Hotspot temperature error  ℎ is given as  where  ℎ and  ℎ are the temperature values of hotspots from the temperature field distribution models and the reconstructed ones.
Judging from Tables 1 and 2, for simple temperature field distribution, like one-symmetrical, though   and   with LSA and RBF are small (  : 4.28% and 4.50%,   : 5.01%, and 4.92%), the modified algorithm owns the Mathematical Problems in Engineering  In a word, comparisons of reconstructed two-dimensional isothermal contour figures and error analysis of   ,   , and  ℎ indicate that the modified algorithm occupies the best information reflection of temperature field distribution and the highest reconstruction accuracy.

Conclusions and Future Research
The information of temperature field distribution is complex but quite important for theoretical study and industrial applications.Temperature field distribution reconstruction using acoustic imaging is an inverse problem and its reconstruction performance depends on reconstruction algorithm.In this paper, a modified reconstruction algorithm based on least square method and radial basis function approximation is investigated.Firstly, the basic principle of acoustic is described.Secondly, the modified algorithm is expressed.Finally, simulation experiments are carried out to study the reconstruction performance of the proposed algorithm.Compared with LSA and RBF, the results of simulation experiments show that the modified reconstruction algorithm occupies the best information reflection of temperature field distribution and the highest reconstruction accuracy.
The major work of future research should focus on achievement of actual experimental system.The measuring area could be a square cavity with acoustic transducers installed in a specific plane of it.The hot zones maybe replaced by candles or heater source.Simultaneously, (5) is not fit for calculating AFT because temperature field distribution of the practical experiment environment is hardly artificially made.Therefore, a method for estimating AFT based on acoustic transmitted and received waveforms should be studied.Thus, with actual experimental system and AFT estimation method, the advantages of the proposed reconstruction algorithm over LSA and RBF will be verified.

11 Figure 1 :
Figure 1: A specific acoustic transducer layout in a square measuring area.

Figure 2 :
Figure 2: The procedure of modified reconstruction algorithm based on LSA and RBF.

Figure 3 :
Figure 3: The comparison of simulation experiment results of one-peak symmetrical temperature field distribution: (a) model; (b) with LSA; (c) with RBF; (d) with modified algorithm.

Figure 4 :
Figure 4: The comparison of simulation experiment results of two-peak asymmetrical temperature field distribution: (a) model; (b) with LSA; (c) with RBF; (d) with modified algorithm.

Figure 5 :
Figure 5: The comparison of simulation experiment results of three-peak asymmetrical temperature field distribution: (a) model; (b) with LSA; (c) with RBF; (d) with modified algorithm.