A two-dimensional temperature and concentration reconstruction method based on hybrid radial basis iteration solution

Abstract

The application relates to a two-dimensional temperature concentration reconstruction method based on mixed radial basis iteration solving, mainly including the following steps: uniformly discretizing an imaging area into N grids, taking a grid center point as a radial basis function center and obtaining a radial basis function vector with the point as the center, combining different types of basis functions to obtain a mixed radial basis function model, forming a mixed basis function matrix of the mixed basis function vectors of different grid points in the imaging area, approximating spectral line absorption rate values of each discrete point, iteratively solving basis function vector coefficients according to M laser absorption spectrum data received by a detector array, in the iteration process, using prior information of smoothness and non-negativity to correct the iteration solving result, and updating the mixed radial basis function matrix, and continuing iteration calculation until an iteration termination condition is met. The method effectively improves fitting precision of continuous distribution in the imaging area, simplifies complexity of an imaging model, and has important application value in the field of laser absorption spectrum imaging.

Description

TECHNICAL FIELD

[0001] The application relates to a two-dimensional temperature and concentration reconstruction method based on mixed radial basis iterative solution, and belongs to the field of laser absorption spectrum imaging. A mixed radial basis function matrix is formed by using a base function vector of a discrete grid point in an imaging area to approximate spectral line absorption density values of each point, mixed radial basis function fitting coefficients are iteratively solved, a calculation result is corrected and a radial basis function weight factor is updated in an iteration process, and two-dimensional distributions of temperature and concentration in the imaging area are calculated according to a double-line measurement method. BACKGROUND

[0002] Flame temperature distribution and component concentration distribution of a combustion field are core parameters of combustion field measurement, and can directly reflect fuel combustion conditions and combustion efficiency. Laser absorption spectrum technology combined with tomography technology can be used for tomographic imaging measurement of flame combustion parameters, and has been used for monitoring of combustion field parameter distribution. However, due to the limitation of measurement space and mechanical structure of a sensor, only limited-angle laser absorption spectrum can be detected, which brings a severe challenge to high-quality reconstruction of flame combustion parameters.

[0003] In 2017, Xia et al. proposed a two-step algebraic reconstruction algorithm in the article titled 'Two-step tomographic reconstructions of temperature and species concentration in a flame based on laser absorption measurements with a rotation platform' published in Optics and Lasers in Engineering, vol. 90, pp. 10-18. This method involves obtaining the temperature distribution using an algebraic reconstruction algorithm based on measured values, then substituting the temperature values into the original equation, and solving a linear equation system with concentration values as unknowns using an algebraic reconstruction algorithm again. However, this method has the problems of strong ill-conditioned equations and a large number of unknowns to be solved. In 2021, Bao et al. published an article titled 'Relative entropy regularized TDLAS tomography for robust temperature imaging' in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-9. To alleviate the problem of large reconstruction error and many artifacts caused by underdetermined equations, they used the ratio of two absorption lines as a smoothing constraint and added it to the objective function to solve the minimum value of the objective function. However, this method still has many reconstruction artifacts for complex distributions. In 2022, Si et al. published an article titled 'A quality-hierarchical temperature imaging network for TDLAS tomography' in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-10. They proposed a quality-hierarchical imaging network using the training advantages and learning induction capabilities of deep learning technology. The imaging network outputs two types of reconstruction images: one with poor imaging quality but high time resolution, and the other with high imaging quality but time-consuming calculation, to achieve a balance between reconstruction accuracy and efficiency.In the same year, Gao et al. published Sparse Zernike fitting for dynamic LAS tomographic images of temperature and water vapor concentration in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-14, which utilized finite-term Zernike polynomials to fit the absorption rate distribution of the absorption spectral line in the imaging region, and solved the fitting coefficients of the Zernike polynomials sparsely according to the measured values, but the Zernike polynomials had a scope of the entire imaging region, had good global fitting characteristics, but had a large fitting error for complex distribution.

[0004] Therefore, it is necessary to select a base function with stronger approximation ability to improve the local fitting ability, both to meet the overall fitting accuracy requirements and to meet the local area detail fitting accuracy, so as to improve the reconstruction accuracy of the combustion parameter distribution. The radial basis function value is only related to the distance from the sample to the data center. Taking advantage of the advantages of radial basis function in function expression and local data fitting, it is applied to the tomographic imaging inversion solving model. In 2017, Jia et al. published an article entitled "Two-dimensional temperature field distribution reconstruction based on least square method and radial basis function approximation" in Mathematical Problems in Engineering, Vol. 1-7, 2017. An improved reconstruction algorithm based on least square method and radial basis function approximation is proposed. The radial basis function is used to approximate the temperature distribution in the imaging area. The least square method is used to solve the coefficients of the base function according to the measured value, and finally the temperature distribution of the measured area is obtained. In 2021, Zhang et al. published an article entitled "Acoustic tomography of two dimensional velocity field by using meshless radial basis function and modified Tikhonov regularization method" in Measurement, Vol. 175. A method of combining improved Tikhonov regularization with meshless radial basis function (RBF) to reconstruct velocity field is proposed. The continuity of the reconstructed parameters is used as prior information. The radial basis function is used to fit the continuous distribution. The improved Tikhonov regularization method is used to solve the coefficients, and finally the measured flow field velocity distribution is obtained. The above methods prove the feasibility of radial basis function in solving practical problems. However, the selection of a single type of radial basis function has limited fitting accuracy, making it difficult to reconstruct a high-precision parameter distribution.

[0005] Based on the above background, the application provides a two-dimensional temperature concentration reconstruction method based on mixed radial basis iterative solution. A mixed radial basis function matrix is constructed by using different radial basis function vectors at discrete grid points in an imaging area, the spectral line absorption rate density value of the discrete points is expressed as a linear combination of the mixed radial basis function, and the mixed radial basis function matrix coefficient is iteratively solved, which effectively simplifies the complexity of the imaging solution model and improves the two-dimensional temperature concentration reconstruction accuracy. SUMMARY

[0006] The application aims to provide a two-dimensional temperature concentration reconstruction method based on mixed radial basis iterative solution, mixed radial basis function vectors are calculated according to the determined radial basis function type and weight factor, a mixed radial basis function matrix is constructed by using the mixed radial basis function vectors of the discrete points in the imaging area, the spectral line integral absorption rate density distribution in the imaging area is fitted, the radial basis function coefficient vector is iteratively solved, and the two-dimensional distribution of temperature and concentration in the imaging area is calculated by using two spectral lines with different central wavenumbers. The proposed reconstruction method has a simple solution model and high calculation accuracy, and is an effective laser absorption spectrum imaging method.

[0007] The technical scheme of the application is as follows:

[0008] Step one, obtaining laser absorption spectrum integral absorption rate through multiple angles and multiple laser paths in the imaging area. A plurality of lasers are arranged around the imaging area, and laser with varying frequency is output. When the laser passes through the to-be-measured area, the light intensity is attenuated, the transmitted light intensity of the laser passing through the imaging area is received by the detector, M measurement values are obtained, the imaging area is uniformly discretized into N grids, and the pressure, temperature and gas molar fraction in each grid are uniformly distributed. According to the Beer-Lambert absorption law, the integral absorption rate A v,i of the i-th laser with central wavenumber v can be expressed as:

[0009]

[0010] Wherein, i (i=1, 2, …, M) and j (j=1, 2, …, N) are the serial numbers of the laser beam and the discrete grid, I t,i (v) and I o,i (v) are the transmitted light intensity and incident light intensity of the i-th laser, L i is the path length of the laser passing through the imaging area, P j , T j [K] and X j respectively represent the pressure, temperature and gas molar fraction in the jth grid, S(v, T j ) is the spectral line intensity corresponding to the temperature, and φ(v) is a linear function, which satisfies the normalization condition, that is, The line integral absorption in formula (1) can be expressed as:

[0011]

[0012] where L i,j represents the length of the i-th laser passing through the j-th grid, a ν,j is the integral absorption density of the absorption line with central wavenumber v in the j-th grid.

[0013] Step two, the mixed radial basis function matrix is used to approximate the integral absorption density values of the discrete points in the imaging area. The radial basis function is a real-valued function only related to the distance from the determined point. After the type and center point of the radial basis function are determined, the radial basis function distribution of the center point can be calculated. The center coordinates of the j-th grid point are denoted as (x j ,y j ), and the mixed radial basis function value of the point (x,y) is:

[0014]

[0015] where f and f represent the basis function values of the basis function type t1 and t2 of the point (x,y), respectively, and w t1 and w t2 represent the weight values of the basis function types t1 and t2, respectively.

[0016] The integral absorption density value of the j-th grid point (x j ,y j ) in the imaging area can be expressed as a linear combination of the radial basis functions,

[0017]

[0018] where Q is the total number of radial basis function center points participating in the fitting, satisfying Q≤N, a q is the mixed radial basis function coefficient of the q-th grid point as the basis function center, and φ q (x j ,y j ) represents the mixed radial basis function value of the point (x j ,y j ) with the q-th grid point as the basis function center. Therefore, the N×Q mixed radial basis function matrix φ of the N discrete points in the imaging area can be obtained,

[0019]

[0020] By combining formula (2) and (4), we can obtain

[0021]

[0022] where The matrix form of which can be written as

[0023] W = Lφ (7)

[0024] where the M x Q matrix W represents the integral of the mixed radial basis functions along the M laser paths, and the M x N matrix L is the sensitivity matrix, whose values are determined by the optical path geometry; then equation (6) can be expressed as

[0025]

[0026] For the integral absorption rate vector A obtained by M optical path detection v , the linear solution equation can be written as

[0027] Wα v = A v (9)

[0028] where the Q x 1 vector α v is the fitting coefficient corresponding to the Q radial basis function center points;

[0029] Step three: Iterative solution of mixed radial basis coefficient vector and correction of integral absorption rate density distribution; The number of laser paths that can be arranged in the finite measurement space and the number of measurement values obtained are very limited, and the radial basis function coefficient vector solving problem is an underdetermined equation solving problem. The Simultaneous Algebraic Reconstruction Technique (SART) is used to solve the coefficient vector, which has obvious advantages in reconstruction speed and anti-interference performance. The iterative solution formula of equation (9) is

[0030]

[0031] where and are the basis function coefficient vectors of the kth iteration and the k+1th iteration, respectively, and λ is the relaxation factor, whose value affects the iteration calculation accuracy and speed; according to equation (4), the spectral line integral absorption rate density distribution of the center frequency v [cm -1 ] at the k+1th iteration is calculated

[0032] In order to reduce or eliminate the noise and singular values appearing in the image reconstruction results, the OLS (One Step Late) algorithm combined with median filtering is used to correct the iterative calculation results, that is,

[0033]

[0034] where and respectively represent the integrated absorption density distribution of the kth iteration and the k+1th iteration after filtering correction by OLS algorithm; represents the integrated absorption density distribution of the k+1th iteration without filtering correction by OLS algorithm; is the median filtering operation on The weight factor τ is used to adjust the weight of the correction value;

[0035] the integrated absorption density distribution of the k+1th iteration after filtering correction As the final result of the k+1th iteration, the values less than 0 in the reconstructed integrated absorption density distribution result are all set to 0 by using the prior information of non-negative constraint;

[0036] Step four: judge whether the iteration termination condition is met; judge whether the current iteration number exceeds the set maximum iteration number K or the current error precision is less than the set error precision η, m

[0037]

[0038] If the termination condition is not met, the weight factor of the radial basis function of the k+1th iteration is updated according to the current result,

[0039]

[0040] and returns to step two to update the hybrid radial basis function matrix according to formula (3), calculate the current measurement value deviation, and start the next iteration;

[0041] If the iteration termination condition is met, the iteration is stopped, and the absorption spectral line integrated absorption density distribution calculated in the current iteration is output;

[0042] Step five: look up the table to obtain the flame temperature and concentration distribution; use the ratio of the integrated absorption density of two absorption spectral lines with different central frequencies to look up the table to obtain the flame temperature and component concentration distribution. BRIEF DESCRIPTION OF DRAWINGS

[0043] Figure 1 is the specific implementation diagram of the method.

[0044] Figure 2 is the original temperature distribution and water vapor concentration distribution.

[0045] Figure 3 is the temperature distribution and water vapor concentration distribution obtained by using the reconstruction method. DETAILED DESCRIPTION

[0046] The specific implementation diagram and the attached Figure 1 are combined below. Figure 2 ​The specific implementation model shown further illustrates the present application.

[0047] Step one: 5 lasers are arranged at equal angle intervals around the imaging area, and the output frequency of the tunable laser is controlled to change. When the laser with central wave number v [cm -1 ] passes through the imaging area, the laser intensity will be attenuated to a certain extent, and the transmitted laser intensity is received by the detector array. The 5-angle laser beams collectively obtain 120 measurement values. The imaging area is uniformly dispersed into 2500 grids, and the pressure, temperature and gas molar fraction in each grid are uniformly distributed. Then, according to the Beer-Lambert absorption law, the integral absorption A v,i of the i-th laser passing through the imaging area can be expressed as:

[0048]

[0049] where i (i = 1, 2, …, 120) and j (j = 1, 2, …, 2500) represent the serial numbers of the laser beams and the dispersed grids, respectively, I t,i (v) and I o,i (v) are the transmitted intensity and incident intensity of the i-th laser, respectively, L i is the path length of the laser passing through the imaging area, P j , T j [K] and X j are the pressure, temperature and gas molar fraction in the j-th grid, respectively, S(v, T j ) is the spectral line intensity at central wave number v, and φ(v) is a linear function that satisfies the normalization condition, i.e. Then, the integral absorption A in equation (14) can be expressed as:

[0050]

[0051] where L i,j represents the length of the i-th laser passing through the j-th grid, a v,j is the integral absorption density in the j-th grid.

[0052] Step two: fitting the spectral line integral absorption density distribution of the imaging area with a mixed radial basis function matrix. The radial basis function is a real-valued function only related to the distance between the center point of the basis function and the selected Gaussian radial basis function (GRBF) and inverse multiquadric radial basis function (IMRBF) constitute a mixed radial basis function. The center coordinates of the j-th grid point are denoted as (x j , y j), and as the radial basis function center point, the Gaussian radial basis function value and the inverse multi-quadratic radial basis function value at the point (x, y) are and The mixed radial basis function value of the point is:

[0053]

[0054] In the formula, w GRBF and w IMRBF respectively represent the weight values of the Gaussian radial basis function and the inverse multi-quadratic radial basis function, and the initial weight values are both set to 0.5;

[0055] The integral absorption density value of the jth grid point (x j , y j ) can be expressed as a linear combination of the mixed radial basis function vector,

[0056]

[0057] In the formula, Q is the total number of mixed radial basis function center points participating in fitting, satisfying Q≤N, and a q is the basis function coefficient when the basis function center is at the qth grid point, and φ q (x j , y j ) represents the mixed radial basis function value at the point (x j , y j ) with the qth grid point as the basis function center; here, 1250 discrete points are selected as basis function center points at an interval of 1 grid between adjacent center points, and then a 2500×1250 mixed radial basis function matrix φ can be obtained for the 2500 discrete points in the imaging area,

[0058]

[0059] By combining equations (16) and (17), we can obtain:

[0060]

[0061] Let Then the matrix form of equation (19) can be written as:

[0062] W=Lφ (20)

[0063] Where the 120×1250 matrix W represents the integral of the mixed radial basis function along the 120 laser paths, and the 120×2500 matrix L is the sensitivity matrix, the value of which is determined by the optical path geometry; then equation (19) can be expressed as:

[0064]

[0065] The integral absorption rate vector A obtained by 120 light paths v The linear solution equation can be written as:

[0066] Wα v = A v (22)

[0067] In the formula, the 1250 × 1 vector α v is the fitting coefficient corresponding to the 1250 radial basis function center points;

[0068] Step three: iteratively solve the mixed radial basis coefficient vector and correct the integral absorption rate density distribution; the number of laser light paths that can be arranged in the limited measurement space and the number of measurement values obtained are very limited, and the radial basis function coefficient vector solving problem is an underdetermined equation solving problem, the Simultaneous Algebraic Reconstruction Technique (SART) is used to solve the coefficient vector, and the iterative solution formula of formula (22) is

[0069]

[0070] In the formula, and are the basis function coefficient vectors of the kth iteration and the (k+1)th iteration, respectively, λ is a relaxation factor, and its value affects the iteration calculation accuracy and speed; according to formula (17), the spectral line integral absorption rate density distribution of the center frequency v [cm -1 ] at the (k+1)th iteration is calculated

[0071] In order to reduce or eliminate the noise and singular values appearing in the image reconstruction result, the OSL (One Step Late) algorithm combined with median filtering is used to correct the iterative calculation result, that is,

[0072]

[0073] In the formula, and respectively represent the integral absorption rate density distribution of the kth iteration and the (k+1)th iteration after filtering correction by using the OSL algorithm; represents the (k+1)th iteration integral absorption rate density distribution without filtering correction by using the OSL algorithm; is the median filtering operation on , and the weight factor τ is used to adjust the weight of the correction value;

[0074] The integral absorption rate density distribution after filtering correction of the (k+1)th iteration As the final result of the k+1th iteration, and using the prior information of non-negative constraint, all the values less than 0 in the reconstructed integral absorption density distribution are set to 0;

[0075] Step four: judging whether the iteration termination condition is met; judging whether the current iteration number exceeds the set maximum iteration number K m or the current error precision is less than the set error precision η,

[0076]

[0077] If the termination condition is not met, the weight factor of the radial basis function of the k+1th iteration is updated according to the current result,

[0078]

[0079] and returns to step two to update the hybrid radial basis function matrix according to formula (16), calculate the current measurement value deviation, and start the next iteration; if the iteration termination condition is met, stop the iteration, and output the absorption spectral line integral absorption density distribution calculated in the current iteration;

[0080] Step five: look up the table to obtain the flame temperature and concentration distribution; using the ratio of the integral absorption of two absorption spectral lines with different central frequencies, look up the table to obtain the flame temperature and component concentration distribution.

Claims

1. A two-dimensional temperature-concentration reconstruction method based on hybrid radial basis iteration, characterized in that: The type of radial basis function (RBF) is determined, and discrete points within the imaging region are used as RBF centers. RBF vectors of different types are calculated and combined to obtain a mixed RBF vector. The mixed RBF vectors of different discrete grid points within the imaging region are used to construct a mixed RBF matrix, which approximates the integral absorbance density values ​​of the absorption lines at the discrete grid points. Using absorption spectral data acquired by a multi-angle detector array, the coefficient vector of the mixed RBF is iteratively solved. In each iteration, the integral absorbance density distribution of the spectral lines is corrected using smoothing and non-negativity prior information, and the weighting factors of different types of RBFs in the mixed RBF matrix are updated until the iteration termination condition is met. Based on the mixed RBF matrix and coefficient vector, the integral absorbance density distribution of the absorption lines is obtained. Using the ratio of the integral absorbance densities of two absorption lines with different center wavenumbers, a table is consulted to obtain the two-dimensional temperature and concentration distribution within the imaging region.

2. The two-dimensional temperature-concentration reconstruction method based on hybrid radial basis iteration solution according to claim 1, characterized in that... The integral absorbance density of discrete grid points in the imaging region is approximated using a mixed radial basis function matrix, and the coefficient vector of the mixed radial basis function is iteratively solved. This process includes the following steps: Step 1: Obtain the integral absorptivity of the laser absorption spectrum from multiple laser paths passing through the imaging region at multiple angles. Multiple lasers are arranged around the imaging region, outputting lasers with varying frequencies. As the laser passes through the test area, its intensity is attenuated to some extent due to absorption by the components. The transmitted laser intensity through the imaging region is received by M detectors. The imaging region is uniformly discretized into N grids, with pressure, temperature, and gas mole fraction uniformly distributed within each grid. Then, according to the Beer-Lambert absorption law, the integral absorptivity A of the i-th laser with center wavenumber v... v,i It can be represented as: Where i (i = 1, 2, ..., M) and j (j = 1, 2, ..., N) are the indices of the laser beam and the discrete grid, respectively. t,i (v) and I o,i (v) represent the transmitted light intensity and incident light intensity of the i-th laser beam, respectively. i P is the path length of the laser through the imaging region. j T j [K] and X j Let S(v,T) represent the pressure, temperature, and gas mole fraction in the j-th grid, respectively. j ) represents the spectral line intensity at the corresponding temperature, and φ(v) is a line type function that satisfies the normalization condition, i.e. Then the integral absorption rate in equation (1) can be expressed as: Among them, L i,j Let a represent the length of the i-th laser beam passing through the j-th grid. v,j The integral absorbance density of the absorption spectral line with center wavenumber v in the j-th grid. Step 2: Approximate the integral absorptivity density values ​​of discrete points within the imaging region using a hybrid radial basis function matrix. Radial basis functions are real-valued functions that depend only on the distance to a given point. Once the type of radial basis function and its center point are determined, the hybrid radial basis function for that center point can be calculated based on the weighting factors. The center coordinates of the q-th grid point are denoted as (x...). q ,y q If ), then the mixed radial basis function value of the point (x,y) is: In the formula, and Let w represent the basis function values ​​of basis function types t1 and t2 for the point (x, y), respectively. t1 and w t2 These represent the weight values ​​for basis function type t1 and basis function type t2, respectively. The j-th grid point (x) within the imaging region j ,y j The integral absorbance density value of ) can be expressed as a linear combination of mixed radial basis functions. In the formula, Q represents the total number of radial basis function center points participating in the fitting, satisfying Q≤N, α q The coefficients of the mixed radial basis functions are the basis function centers at the q-th grid point, φ. q (x j ,y j ) indicates that the q-th grid point is the center of the basis function, and the point (x) j ,y j The mixed radial basis function value at point () is obtained; therefore, an N×Q mixed radial basis function matrix φ can be obtained from N discrete points within the imaging region. Combining equations (2) and (4) yields make The matrix form of the above equation can then be written as: W=Lφ (7) In the formula, the M×Q matrix W represents the integral of the mixed radial basis function along the M laser paths, and the M×N matrix L is the sensitivity matrix, the value of which is determined by the optical path geometry; then equation (6) can be expressed as: For the integral absorptivity vector A obtained from M optical paths... v The linear solution to the equation can be written as: Wα v =A v (9) In the formula, Q×1 vector α v These are the fitting coefficients corresponding to the center points of the Q radial basis functions; Step 3: Iteratively solve the hybrid radial basis coefficient vector and correct the integral absorptivity density distribution; the number of laser optical paths that can be arranged and the number of measured values ​​that can be obtained in the finite measurement space are very limited. The problem of solving the radial basis function coefficient vector is an underdetermined equation problem. The Simultaneous Algebraic Reconstruction Technique (SART), which has obvious advantages in reconstruction speed and anti-interference performance, is used to solve the coefficient vector. The iterative solution formula of equation (9) is as follows: In the formula, and These are the basis function coefficient vectors for the k-th and (k+1)-th iterations, respectively. λ is the relaxation factor, whose value affects the accuracy and speed of the iteration calculation. According to equation (4), the center frequency v[cm] for the (k+1)-th iteration is calculated. -1 ] spectral line integral absorbance density distribution To reduce or eliminate noise and outliers in the image reconstruction results, an OLS (One-Step Late) algorithm combined with median filtering is used to correct the iterative calculation results. In the formula, and These represent the integral absorption rate density distributions of the k-th and (k+1)-th iterations after filtering correction using the OLS algorithm, respectively. This represents the integral absorption rate density distribution of the (k+1)th iteration without OLS algorithm filtering correction; Yes Median filtering is performed, and the weight factor τ is used to adjust the weights of the correction values. The integral absorbance density distribution after filtering correction in the (k+1)th iteration As the final result of the (k+1)th iteration, and using the prior information of the non-negative constraint, all values ​​less than 0 in the reconstructed integral absorption rate density distribution result are set to 0. Step 4: Determine if the iteration termination condition is met; determine if the current iteration count exceeds the set maximum iteration count K. m Or is the current error precision less than the set error precision η? If the termination condition is not met, then update the weight factor of the radial basis function in the (k+1)th iteration based on the current result. Then return to step two, update the mixed radial basis function matrix according to equation (3), calculate the current measurement deviation, and start the next iteration; If the iteration termination condition is met, the iteration stops and the integral absorbance density distribution of the absorption spectrum calculated in the current iteration is output. Step 5: Obtain the flame temperature and concentration distribution by referring to the table; use the ratio of the integrated absorbance density of two absorption spectral lines with different center frequencies to look up the flame temperature and component concentration distribution in the table.

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