A two-dimensional temperature and concentration field imaging method based on Zernike polynomial fitting

Abstract

The present invention relates to a two-dimensional temperature field and concentration field imaging method based on Zernike polynomial fitting, which mainly includes the following steps: for a multi-angle laser absorption spectrum imaging sensor, a certain number of detectors are arranged at equal intervals at each angle, the laser emitted by the laser passes through the area to be measured and is received by the photoelectric detector, and M laser absorption spectrum data are obtained; the area to be measured is discretized into N grids, and the local integral absorption rate distribution of the absorption spectrum line in the discrete grid is fitted with K-term Zernike polynomials; the integral absorption rate obtained by detection is used to solve the Zernike polynomial fitting coefficient, and the local integral absorption rate distribution of the absorption spectrum line is obtained by combining the Zernike polynomial matrix, and the temperature field and concentration field information in the area to be measured are obtained by using the double-line method measurement principle. The method converts the quantity to be solved from the local integral absorption rate in the discrete grid to the Zernike polynomial coefficient, reduces the number of unknowns, effectively reduces the morbidity of the equation, and has important application value in the field of combustion dynamic monitoring.

Description

Technical Field

[0001] The present invention relates to a two-dimensional temperature field and concentration field imaging method based on Zernike polynomial fitting, which belongs to the field of laser absorption spectrum imaging. The prior information of flame temperature continuous distribution is used to fit the local integral absorption rate distribution of different center frequency spectral lines in the test area through finite-term Zernike polynomial fitting, and then the temperature field and concentration field in the test area are calculated according to the double-line method measurement principle. Background Art

[0002] The temperature and component concentration distribution of combustion gases are two very important parameters in the combustion process. Their parameters can reflect the fuel combustion efficiency and evaluate the burner performance. Laser absorption spectroscopy, as a non-invasive, fast-response, and anti-interference optical detection technology, has been widely used in the detection of combustion gas temperature and component concentration. The direct absorption method can directly obtain the entire absorption line shape and is the most commonly used measurement method of laser absorption spectroscopy. However, during the gas combustion process, due to airflow disturbances, uneven distribution of combustibles, and other reasons, the gas parameter distribution in the area to be measured is usually non-uniform. In order to obtain the spatial distribution information of the combustion gas parameters, two-dimensional measurement is required. A suitable two-dimensional imaging algorithm is the key to obtaining spatial distribution information.

[0003] In 2008, Cai et al. published an article entitled “Application of simulated annealing for multispectral tomography” in Computer Physics Communications, Vol. 179, No. 4, pp. 250-255. They transformed the inversion problem into a minimum optimization problem and used the simulated annealing algorithm to solve the minimum problem. This method made full use of prior information to improve the ill-posed nature of the inversion problem. This method solves the inverse problem by searching for an optimization algorithm that approximates the global optimal solution, and it often takes a long time to search for the optimal solution. In 2019, Liu et al. published an article titled "Toward Customized Spatial Resolution in TDLAS Tomography" in IEEE Sensors Journal, Volume 19, Issue 5, Pages 1748-1755. Based on the optimization of beam arrangement, the Tikhonov regularization method was used to compare the effects of different angles and number of detectors on spatial resolution. However, this method has limitations on the number of reconstructed grids, that is, it can only solve a certain degree of pathological problems. In 2020, Zhao et al. published an article titled "A WMS Based TDLAS Tomographic Systemfor Distribution Retrievals of Both Gas Concentration and Temperature inDynamic Flames" in IEEE Sensors Journal, Volume 20, Issue 8, Pages 4179-4188. They used the joint algebraic reconstruction algorithm (SART) method to monitor the gas concentration and temperature distribution of dynamic flames. This method can obtain better results when the number of reconstruction grids is small, but the spatial resolution is low at this time. When the number of reconstruction grids increases, artifacts appear in the reconstructed image.In the same year, Huang et al. published an article entitled “Frequency-Division Multiplexing and Main Peak Scanning WMS Method for TDLAS Tomography in Flame Monitoring” in IEEE Transactions on instrumentation and measurement, Vol. 69, No. 11, pp. 9087-9096. In the article, they used the Landweber method to monitor the temperature and concentration fields of a dynamic Bunsen burner in real time. The imaging results obtained by this method had a low spatial resolution within an acceptable error range.

[0004] Zernike orthogonal polynomials are orthogonally complete in the unit circle domain. Therefore, when characterizing and analyzing free-form surfaces, Zernike circle domain polynomials can be used as basis functions to obtain analytical orthogonal polynomials of the surface. In 2007, Xin et al. published an article entitled "Accurate Computation of Zernike Moments in Polar Coordinates" in IEEE Transactions on Image Processing, Vol. 69, No. 11, pp. 581-587. They proposed an algorithm for high-precision numerical calculation of Zernike moments. By introducing the polar coordinate calculation scheme, the geometric error and numerical integration error caused by the traditional Cartesian coordinate calculation were avoided. In 2010, Chen et al. published an article entitled "A Zernike Moment Phase-Based Descriptor for Local Image Representation and Matching" in IEEE Transactions on Image Processing, Vol. 19, No. 1, pp. 205-219, evaluating the matching ability of Zernike moments and evaluating their description performance through a set of local images. In 2006, Ares et al. published an article entitled "Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction" in Applied Optics, Vol. 45, No. 27, pp. 6954-6964, applying cubic B-spline and Zernike techniques to wavefront fitting and comparing the fitting effects of the two methods. Both methods can obtain good fitting results for simple wavefront distributions.

[0005] Considering that the temperature distribution of combustion gas changes continuously and evenly, Zernike polynomials have good performance in free surface characterization and analysis. Based on the above background, the present invention proposes a two-dimensional temperature field and concentration field imaging method based on Zernike polynomial fitting. Zernike polynomials are used to characterize the local absorption rate distribution of gas absorption lines, and then the temperature field and concentration field information of the combustion gas in the measured area are obtained, which effectively reduces the number of unknown quantities to be solved, and the reconstruction speed is fast and the reconstruction accuracy is high. Summary of the invention

[0006] The purpose of the present invention is to propose a two-dimensional temperature field and concentration field imaging method based on Zernike polynomial fitting. The characteristic of continuous temperature distribution is utilized to fit the continuously changing distribution in the imaging area with a finite-term Zernike polynomial, and the quantity to be solved is converted from the grayscale value in the discrete grid to the polynomial coefficient. The proposed method has high reconstruction accuracy and can effectively reconstruct the temperature field and concentration field in the measured area. It is an effective and novel laser absorption spectroscopy imaging method.

[0007] The technical solution of the present invention is:

[0008] Step 1: Obtain the integrated absorption rate of the laser absorption spectrum imaging sensor at multiple angles and multiple paths. N tunable lasers are arranged at equal angles around the area to be measured, and a certain number of photodetectors are arranged at equal intervals at each angle to form a detector array. Control the tunable laser to scan within a certain frequency range. When a beam with a central frequency of v[cm -1 When a laser of length L passes through the area to be tested, part of the laser light intensity is absorbed by the gas to be tested in the area to be tested, and the transmitted light intensity I t (v) is detected by the detector array. According to the Beer-Lambert absorption law, the absorption rate α v It can be expressed as:

[0009]

[0010] Where I0(ν) is the incident laser intensity, I t (v) is the transmitted light intensity, P(x), T(x), X abs (x) and S[T(x)] are the pressure, temperature, gas concentration and line intensity at x, respectively;

[0011] Since the linear function φ satisfies the normalization condition, that is, The integrated absorption rate can be expressed as:

[0012]

[0013] In laser absorption spectral tomography, the region of interest is discretized into N grids, and the pressure, temperature and gas concentration in each grid are considered to be constants. The transmitted light intensity of the lasers emitted by the lasers arranged at equal angles after passing through the region to be measured is detected by the detection array, and a total of M integrated absorption rates can be calculated; the integrated absorption rate A obtained by the i-th laser passing through the region to be measured is v,i It can be expressed as follows:

[0014]

[0015] Where, Lij is the length of the i-th light ray passing through the j-th grid, and the local integrated absorption rate of the center frequency v in the j-th grid is recorded as a v,j , formula (3) can be written in matrix form as:

[0016] La v =A v (4)

[0017] Among them, the M×N matrix L is the sensitivity matrix, which represents the length of M laser beams passing through N grids respectively, which is determined by the structure of the sensor and the optical path layout, and the N×1 column vector a v represents the N local integrated absorption rates to be determined in the area to be measured, and the M×1 column vector A v represents the M integrated absorption rates obtained on the detector array.

[0018] Step 2: Use Zernike polynomials to fit the local integrated absorptivity distribution in the area to be measured. The temperature distribution in the area of ​​interest changes continuously and evenly, and the local integrated absorptivity corresponding to the absorption spectrum line with the center frequency ν in the area also changes continuously, which can be fitted by a finite-term Zernike polynomial. The coordinates of the fitting point are expressed in polar coordinate form as (ρ, θ). Then, the k-th Zernike polynomial Zk(ρ, θ) at the fitting point (ρ, θ) can be calculated by the following formula:

[0019]

[0020] Where p is the order of the polynomial, which represents the highest power of the radial variable ρ; q is the azimuth frequency, which represents the coefficient of the azimuth variable θ. The relationship between p and q is as follows: p-|q|≥0 and the difference is an even number; the radial function is a polynomial in the radial variable ρ:

[0021]

[0022] Therefore, the local integrated absorption rate in the jth grid can be obtained by fitting the K-term Zernike polynomial:

[0023] a v,j (ρ,θ)=b1Z j,1 (ρ,θ)+…+b K Z j,K (ρ,θ) (7)

[0024] Then the local integral absorption rate in N discrete grids in the area to be measured can obtain N fitting equations in total,

[0025]

[0026] Formula (8) can be written in matrix form as:

[0027] ZB=a v (9)

[0028] Among them, the N×K matrix Z is a Zernike polynomial matrix, which is determined by the fitting range and the total number of Zernike polynomials, and the K×1 column vector B represents the K polynomial coefficients. Therefore, the local integrated absorption rate distribution corresponding to the spectral line center frequency v can be obtained by fitting according to formula (9).

[0029] Step 3: Combine equations (4) and (9), and the integrated absorption rate can be expressed as follows by the sensitivity matrix L and the Zernike polynomial matrix Z:

[0030] LZB=A v (10)

[0031] Formula (10) transforms the N local integral absorption rates to be determined in the original formula (4) into K Zernike polynomial coefficients. The number of polynomial terms K is much smaller than the number of discrete grids N, which reduces the ill-conditioned nature of the equation.

[0032] Step 4: Calculate the integrated absorption rate A based on the detector detection signal v Vector, the regularization solution method is used to solve the Zernike polynomial coefficient B in equation (10), and equation (10) is written as the following l1 norm regularization form:

[0033]

[0034] in, is the l1 norm of the Zernike coefficient vector B, and the regularization parameter λ>0; Lagrange duality is used to simplify the solution model of equation (11) by introducing the dual variable δ i and the equality constraint γ i =(LZB-A ν ) i The Lagrange function of formula (11) can be obtained:

[0035] L(B,γ,δ)=γ T γ+λ||B||1+δ T (LZB-A v -γ) (12)

[0036] Then the dual objective function of formula (12) is:

[0037]

[0038] According to the weak duality of the dual problem, any dual feasible point δ can obtain the optimal value c of the original problem (11) * The lower bound of G(δ)≤c* ; Usually, the suboptimal bound on B can be calculated based on the dual feasible point;

[0039]

[0040] Step 5: The original problem is an l1 regularized least squares problem, which is transformed into a convex quadratic problem. The polynomial coefficient b i The range of variation is expressed by the inequality -w i ≤ b i ≤w i , then the original problem can be equivalent to the problem of finding the minimum value of the following formula:

[0041]

[0042] In the formula, τ>0 is a parameter representing the approximate accuracy. Its value determines the solution speed and solution accuracy of the above formula. In each iteration, the value of τ is updated in the form of a geometric progression to achieve faster convergence.

[0043] τ=r t τ0 (16)

[0044] In the formula, r is the common ratio of the geometric sequence, t is the update frequency of the parameter τ, and τ0 is the initial value;

[0045] The Newton iteration method is used to solve equation (15). The search direction for each iterative update can be calculated by the following formula:

[0046]

[0047] In the formula, the Hessian matrix is the gradient of the current iteration point (B,w);

[0048] After calculating the search direction, select an appropriate search step size to update the iteration point (B, w) + s (ΔB, Δw). Here, a method for calculating the approximate exact search step size is used. After determining the search direction, the coefficient b i The value range of the search step length can be calculated to obtain the upper limit s max , within the range of the search step length, around the rotation point (s r ,Φ r ) Rotate the curve corresponding to the search step and the objective function by an angle θ so that the axis of symmetry of the curve is perpendicular to the s axis. The coordinates of the point (s, Φ) after rotating by an angle θ are (s′, Φ′).

[0049]

[0050] The curve equation of the point (s′, Φ′) is expressed as follows:

[0051] Φ′=a1s′2 +a2s′+a3 (19)

[0052] Where a1, a2 and a3 are the coefficients of the curve equation; the four unknowns including the rotation angle θ are uniquely solved by the four points on the curve after the rotation angle, and the search step value s after the rotation angle θ can be calculated by the following formula: *′ ,

[0053]

[0054] According to formula (18), the optimal search step value s is obtained * And update the current point (B,w)+s * (ΔB, Δw).

[0055] Step 6: Determine whether the current iteration termination condition is met.

[0056]

[0057] Where ε is the relative tolerance; if the iteration termination condition is met, the iteration is stopped and the calculated Zernike polynomial coefficients are recorded as Otherwise, return to step 5 and continue to update the search direction and current point (B, w);

[0058] Select two absorption lines with different low-level energies E″. The low-level energies of the two lines are recorded as E″1 and E″2, respectively. The center frequencies of the lines are v1 and v2, respectively. The integral absorption rates calculated for the two lines are and According to formula (9), the local integrated absorption rate distribution of the spectral line can be fitted by Zernike polynomials respectively, and the Zernike fitting coefficients of the local integrated absorption rate distribution of the two spectral lines can be calculated by formula (11): and Local integrated absorptivity of two spectral lines and According to formula (22), the temperature T in the jth grid can be calculated: j ,

[0059]

[0060] Where h is Planck's constant, c is the speed of light in vacuum, T0 is the reference temperature, and k is the Boltzmann constant. and are the local integrated absorption rates of the corresponding spectral lines in the jth grid;

[0061] When the temperature T in the jth grid is obtained j After that, the concentration of the gas to be measured in the grid can be calculated by the following formula:

[0062] BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 It is a specific implementation diagram of this method.

[0064] Figure 2 are the original temperature distribution and water vapor concentration distribution.

[0065] Figure 3 are the temperature distribution and water vapor concentration distribution obtained using this reconstruction algorithm. DETAILED DESCRIPTION

[0066] The following is combined with Figure 1 The specific implementation diagram and attached diagram of this method shown in Figure 2 The specific implementation model shown is used to further illustrate the present invention.

[0067] Step 1: Arrange 5 lasers at equal angles in the area to be tested. At the same time, 12 photodetectors are evenly spaced at each angle to form a detector array. Control the tunable laser to scan within a certain frequency range. When the center frequency of a beam is v[cm -1 ], when the laser with light intensity I0(ν) passes through the area to be measured, part of the light intensity is absorbed by the gas to be measured in the area to be measured, and the photoelectric detector obtains the laser transmission light intensity I t (ν), according to the Beer-Lambert absorption law, the absorption rate α ν It can be expressed as:

[0068]

[0069] Among them, the pressure at x is P(x), the temperature is T(x), and the gas concentration is X abs (x), the line intensity is S[T(x)].

[0070] The circular region of interest is discretized into 1876 grids, and the pressure, temperature and gas concentration in each grid are considered to be constant. The fan-shaped laser beam arranged at each angle passes through the area to be measured and is detected by 12 photodetectors arranged at equal intervals. Then, a total of 60 integral absorption rates can be calculated for the detector array at 5 angles. Among them, the integral absorption rate obtained by the i-th (1≤i≤60) laser path is:

[0071]

[0072] Where, L ij is the length of the i-th ray passing through the j-th grid, a v,j The local integrated absorption rate of the spectrum line with the center frequency v in the jth grid is expressed in matrix form in equation (25):

[0073] La v =A v (26)

[0074] The matrix L represents the length of 60 beams passing through 1876 grids respectively, which depends on the laser size and optical path layout. The 1876×1 column vector a v Represents 1876 local integral absorption rates to be determined, 60×1 column vector A v It represents the integrated absorption rate calculated from 60 detection values.

[0075] Step 2: Zernike polynomial fitting of the local integrated absorption rate distribution in the area to be measured. The local integrated absorption rate of the spectrum line with the center frequency v in the area of ​​interest changes continuously and smoothly, which can be obtained by fitting with a finite-term Zernike polynomial. The polar coordinates of the fitting point are (ρ, θ), and the k-th Zernike polynomial Zk(ρ, θ) at the fitting point is:

[0076]

[0077] Where p is the order of the polynomial, representing the highest power of the radial variable ρ; q is the azimuth frequency and the coefficient representing the azimuth variable θ, p and q satisfy p-|q|≥0 and are even numbers; the radial function is a polynomial in the radial variable ρ,

[0078]

[0079] 300 Zernike polynomials are used to fit the local integrated absorption rate distribution in the measured area, and the local integrated absorption rate in the jth grid can be obtained by the following formula:

[0080] a v,j (ρ,θ)=b1Z j,1 (ρ,θ)+…+b 300 Z j,300 (ρ,θ) (29)

[0081] Then the local integrated absorption rate in 1876 discretized grids can obtain 1876 fitting equations,

[0082]

[0083] Formula (30) can be written in matrix form as:

[0084] ZB=a v (31)

[0085] Among them, the 1876×300 matrix Z represents the Zernike polynomial matrix, which is determined by the fitting range and the number of fitting items, and the 300×1 column vector B represents 300 polynomial coefficients.

[0086] Step 3: Use 60 integrated absorption rates to solve the Zernike polynomial coefficients. By combining equations (26) and (31), the integrated absorption rate can be expressed by the sensitivity matrix L and the Zernike polynomial matrix Z as follows:

[0087] LZB=A v (32)

[0088] In this way, the original 1876 local integral absorption rates to be determined are converted into 300 Zernike polynomial coefficients, which reduces the number of unknowns and effectively reduces the ill-conditioning of the equation.

[0089] Step 4: Use the regularization method to solve equation (32). The objective function is:

[0090]

[0091] In the formula, is the l1 norm of the coefficient vector B, and the regularization parameter λ satisfies λ>0; the Langrange duality is used to simplify the equation (33) to solve the model, and the dual variable δ is introduced i and the equality constraint γ i =(LZB-A ν ) i , then we can get the Langrange form of the original problem,

[0092] L(B,γ,δ)=γ T γ+λ||B||1+δ T (LZB-A v -γ) (34)

[0093] Then the dual objective function of the above problem is:

[0094]

[0095] According to the weak duality of the dual problem, any dual feasible point δ can obtain the optimal value c of the original problem (33) * The lower bound of G(δ)≤c * ; The dual feasible point can be calculated by the following formula:

[0096]

[0097] Step 5: Polynomial coefficient b i Using linear inequality-w i ≤ b i≤w i By constraining its value range, the l1 regularized least squares problem can be transformed into a convex quadratic problem. Then the original problem is equivalent to solving the minimum value of the following problem:

[0098]

[0099] Among them, the parameter τ that controls the approximate accuracy satisfies τ>0, and its value determines the solution speed and solution accuracy. In each iteration process, the value of the parameter τ is updated in a geometric sequence to converge faster.

[0100] τ=r t τ0 (38)

[0101] In the formula, r is the common ratio, t is the update frequency of parameter τ, and τ0 is the initial value;

[0102] The optimization method is used to solve the minimum value of equation (37), and the search direction of each iteration is calculated by Newton's method.

[0103]

[0104] In the formula, is the Hessian matrix, is the gradient of the current iteration point (B,w);

[0105] After calculating the search direction, select the appropriate search step size s and update the point (B, w) according to (B, w) + s (ΔB, Δw); after determining the search direction, according to the coefficient b i The upper limit of the search step length s can be obtained from the range of max , within the range of the search step size, select 4 points on the curve corresponding to the search step size and the objective function, and use the point (s1, Φ1) as the rotation point, rotate the curve around the rotation point by an angle θ so that the symmetry axis of the curve is perpendicular to the s axis, and the point (s, Φ) is rotated to obtain (s′, Φ′),

[0106]

[0107] The curve expression corresponding to the search step size and the objective function is:

[0108] Φ′=a1s′ 2 +a2s′+a3 (41)

[0109] Where a1, a2 and a3 are the coefficients of the curve equation; by solving the four unknowns including the rotation angle θ through the four points on the selected curve, the optimal search step is the point s after the rotation angle θ *′ for,

[0110]

[0111] According to formula (40), the optimal search step length s before the rotation angle is obtained * And update the current point (B,w).

[0112] Step 6: Determine whether the current coefficient vector B meets the iteration termination condition.

[0113]

[0114] In the formula, ε is the relative tolerance; if the iteration termination condition is met, the iteration is stopped and the calculated Zernike polynomial coefficients are recorded as Otherwise, return to step 5 and continue updating the search direction.

[0115] The temperature field and concentration field of the area to be measured are calculated according to the measurement principle of the two-line method. Two absorption spectra with different low-level energies E″ are selected, and the low-level energies of the two spectra are represented by E″1 and E″2 respectively. The center frequencies of the spectra are v1 and v2 respectively. The integral absorption rates calculated for the two spectra are and According to formula (31), the local integrated absorption rate distribution of the spectral line can be obtained by fitting the Zernike polynomial respectively. According to formula (33), the Zernike fitting coefficients of the local integrated absorption rate distribution of the two spectral lines are calculated: and Local integrated absorption rate and According to formula (44), the temperature T in the jth grid can be obtained: j ,

[0116]

[0117] In the formula, h and c are Planck's constant and the speed of light in vacuum, respectively, T0 is the reference temperature, k is the Boltzmann constant, and are the local integrated absorption rates of the two spectral lines in the jth grid respectively;

[0118] When the temperature T in the jth grid is obtained j After that, the concentration of the gas to be measured in the jth grid can be obtained according to the following formula:

[0119]

Claims

1. A two-dimensional temperature field and concentration field imaging method based on Zernike polynomial fitting, characterized in that: Laser absorption spectroscopy is used to realize two-dimensional distribution imaging of temperature field and concentration field in the area to be measured. By arranging multiple lasers outside the area to be measured and obtaining spectral absorption data on the photodetector array, the prior information of continuous distribution of the temperature field is used to fit the local integrated absorption rate distribution corresponding to the spectrum lines of different central wave numbers in the area to be measured with a finite-term Zernike polynomial; and according to the integrated absorption rate vector of the corresponding spectrum line detected by the photodetector, the Zernike polynomial coefficient vector is solved to obtain the local integrated absorption rate distribution of the spectrum lines of different central wave numbers in the area to be measured. According to the Zernike polynomial coefficient vector and the Zernike polynomial matrix, the local integrated absorption rate distribution of the spectrum lines of different central wave numbers in the area to be measured is obtained. According to the measurement principle of the two-line method, the two-dimensional distribution reconstruction of the temperature field and concentration field in the area to be measured is realized; the specific implementation is carried out according to the following steps: Step 1: Obtain the multi-angle and multi-path integrated absorption rate of the laser absorption spectrum imaging sensor; arrange multiple tunable lasers with equal angle distribution and photoelectric detectors with equal spacing at each angle in the test area, control the tunable laser to scan within a certain frequency range, and when the laser beam with a center frequency of v passes through the test area, a part of the light intensity is absorbed by the gas to be measured in the test area, and the transmitted light intensity I after the laser passes through the test area with a length of L is t (v) It is detected and received by the photodetector. According to the Beer-Lambert absorption law, the absorption rate α v It is expressed as: Among them, I0(v) and I t (v) are the incident laser intensity and the transmitted laser intensity, P(x), T(x), X abs (x) and S[T(x)] are the pressure, temperature, gas concentration and line intensity at x, respectively; Since the linear function φ satisfies the normalization condition, that is, The integrated absorption rate is expressed as: The region of interest is discretized into N grids. The pressure, temperature and gas concentration in each grid are considered to be constant. The transmitted light intensity of the fan-shaped laser beams arranged at equal angles after passing through the area to be measured is detected by the photodetector. Then, M integral absorption rates are calculated by the total detection, where the integral absorption rate A obtained by the i-th laser is v，i It is expressed as the following formula: Where j represents the number of discrete grids, L ij is the length of the i-th laser passing through the j-th grid, a v，j represents the local integrated absorption rate of the spectrum line with center frequency v in the jth grid; Equation (3) is expressed in matrix form as: The v =A v (4) Where, the M×N matrix L represents the length of M light beams passing through N grids. The specific value of the matrix is ​​determined by the sensor size and light path layout. The N×1 column vector a v represents the local integrated absorption rate in N discrete grids, and the M×1 column vector A v represents the integrated absorption rate obtained from M detections; Step 2: Get the local integral absorption rate distribution in the test area by fitting a finite-term Zernike polynomial. The distribution of the test material in the region of interest changes continuously and smoothly, and Zernike polynomials are often used to describe the geometric shape information of continuous distribution because of their orthogonality within the unit circle. Therefore, Zernike polynomials are used to fit the local integral absorption rate distribution of the absorption spectrum line with the center frequency v. The polar coordinates of the fitting point are (ρ, θ). At the fitting point (ρ, θ), the kth Zernike polynomial Z k (ρ, θ) is expressed as: Where p is the order of the polynomial, which represents the highest power of the radial variable ρ; q is the azimuth frequency, which represents the coefficient of the azimuth variable θ. p and q satisfy p-|q|≥0 and the difference is an even number. The radial function is a polynomial in the unit circle domain with respect to the radial variable ρ, Therefore, the local integrated absorption rate in the jth grid is fitted by a K-term Zernike polynomial, a v，j (ρ,θ)=b1Z j，1 (p, i)+…+b K Z j，K (p, i) (7) Where b k is the fitting coefficient of the kth Zernike polynomial; Then the local integrated absorption rate in N discrete grids can obtain N fitting equations, Formula (8) can be written in matrix form as: ZB=a v (9) Among them, the N×K matrix Z represents the Zernike polynomial matrix, which is determined by the fitting range and the number of fitting items, the K×1 column vector B represents the K polynomial coefficients, and the N×1 column vector a v is the local integrated absorption rate within N discrete grids; Step 3: Use the Zernike polynomial matrix Z to represent the integrated absorption rate; by combining equations (4) and (9), the integrated absorption rate vector is represented by the matrix L and the Zernike polynomial matrix Z as follows: LZB=A v (10) In this way, the original N local integral absorption rates to be calculated are transformed into K Zernike polynomial coefficients. The number of polynomial terms K is much smaller than the number of discretized grids N, which reduces the ill-conditioning of the equation. Step 4: Calculate the integrated absorption rate vector according to the signal detected by the photodetector, and solve equation (10) in regularized form. Then equation (10) can be written as follows in l1-norm regularized form: in, represents the l1 norm of the coefficient vector B, λ is the regularization parameter, and satisfies λ>0; Lagrange duality is used to simplify the solution model of formula (11), and the dual variable δ is introduced i and the equality constraint γ i =(LZB-A v ) i , then the original problem is transformed into the extreme value problem of solving the Lagrange dual function of equation (11), that is, solving the extreme value problem of the following equation: L(B,γ,δ)=γ T c+λ||B||1+d T (LZB-A v -c) (12) Then the dual objective function of formula (12) is: According to the weak duality of the dual problem, any dual feasible point δ gives the optimal value c of the original problem (11) * The lower bound of G(δ)≤c * ; The dual feasible point will obtain an easily computable suboptimal bound for any B, Step 5: The original problem (11) is an l1 regularized least squares problem, which is transformed into a convex quadratic problem. The fitting coefficient b of the Zernike polynomial is i It can be expressed as -w using a linear inequality i ≤ b i ≤w i , so the original problem is equivalent to finding the minimum value of the following problem, Among them, τ>0 is a parameter that determines the approximate accuracy. Its value affects the calculation speed and result accuracy of the minimum solution of equation (15). Here, the parameter τ is updated in each iteration to converge faster. τ=r t τ0 (16) Among them, r represents the common ratio, t is the update frequency of parameter τ, and τ0 is the initial value; Newton's method is used to solve the minimum value of equation (15), and the search direction is calculated according to the following formula: in, is the Hessian matrix, is the gradient of the current iteration point (B, w); After the search direction is obtained by equation (17), a suitable search step size s is selected and the point (B, w) is updated according to (B, w) + s (ΔB, Δw). Here, the search step size adopts an approximate and precise search step size calculation method. When the search direction is determined, the fitting coefficient b of the Zernike polynomial is used. i The range of the search step size is obtained by max , within the search step range, the curve corresponding to the search step and the objective function is rotated around the point (s r , Φ r ) is rotated by an angle θ to make the axis of symmetry of the curve perpendicular to the s axis. The point (s, Φ) is rotated to obtain (s′, Φ′). The equation of the curve where the point (s′, Φ′) lies is, Φ′=a1s′ 2 +a2s′+a3 (19) Among them, a1, a2 and a3 are the coefficients of the equation; select 4 points on the curve to solve 4 unknowns including the rotation angle θ, then the optimal search step is the point s after the rotation angle θ * ' is calculated by the following formula: According to formula (18), the optimal search step length s is obtained * , update the current point (B, w) + s * (ΔB, Δw); Step 6: Determine whether the iteration termination condition is met according to the following formula: In the formula, ε is the relative tolerance; if the iteration termination condition is met, the iteration is stopped and the calculated Zernike polynomial coefficients are recorded as Otherwise, return to step 5 and continue to update the parameter τ and the search direction; Step 7, respectively solve the Zernike polynomial coefficients of the two absorption lines, and use the double-line measurement principle to obtain the temperature field and concentration field information of the measured area; For two absorption lines with different low-level energies E″, the center frequencies of the absorption lines are denoted as v1 and v2, respectively. The integrated absorption rates of the two lines are calculated based on the measured values ​​of the photodetector and denoted as and The Zernike polynomial coefficients corresponding to the local integrated absorption rate distribution of these two spectral lines are denoted as and According to the Zernike polynomial matrix and equation (9), the local integrated absorption rates of the two spectral lines are calculated respectively: and According to formula (22), the temperature T in the jth grid is obtained j , In the formula, E ′ 1 ′ and E ′ 2 ′ are the low-energy state energies of the two spectral lines, h and c are Planck constant and the speed of light in vacuum, T0 is the reference temperature, k is the Boltzmann constant, and are the local integrated absorption rates of the two spectral lines in the jth grid respectively; When the temperature T in the jth grid is obtained j Then, calculate according to the following formula: Get the gas concentration X to be measured j .

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