A mathematical modeling method for DT exposure curve

Abstract

The present invention provides a mathematical modeling method for a D-T exposure curve, which belongs to the technical field of mathematical modeling of a D-T exposure curve; E ‑ T The curve is drawn manually, which requires manual query of tables and manual calculation to determine the parameters of the curve, resulting in long test cycles and low work efficiency. The method includes the following steps: S1: Using curve fitting combined with optimization method to construct a non-sensitized industrial film D ‑ H Characteristic curve mathematical model, obtain the characteristic parameters corresponding to the characteristic curves of different types of industrial films; S2: According to the characteristics of industrial films D ‑ H Characteristic curve mathematical model construction D ‑ T Exposure curve mathematical model: by proving the linear relationship, establish the nonlinear D ‑ H Mapping of film characteristic curves, construction D ‑ T Exposure curve mathematical model; the present invention is applied to quantify D-T exposure curve.

Description

Technical Field

[0001] The invention provides a mathematical modeling method for a DT exposure curve, belonging to the technical field of mathematical modeling of DT exposure curves. Background Art

[0002] Nondestructive testing is a highly sensitive inspection and high-reliability test of the properties, state, and structure of the object being tested without destroying or changing its physical and chemical properties, and then judging its surface and internal integrity, continuity, safety, and other performance indicators.

[0003] X-ray flaw detection is an important method of non-destructive testing. Through X-ray penetration, a direct image of the defects of the object being tested can be obtained. It is qualitatively accurate and can even measure the length, width, and height. It is widely used in aviation, aerospace, weapons, ships, special equipment and other fields.

[0004] In industrial X-ray nondestructive testing, the selection of process parameters (such as tube voltage V, exposure E, focal length F, etc.) is generally determined by querying the exposure curve based on the material and thickness T of the workpiece.

[0005] ET exposure curves are unique. Different X-ray machines, or even the same X-ray machine at different stages of its life, have different radiation quality and irradiation rates, and thus different corresponding exposure curves. This requires the exposure curve to be recreated or revised based on actual usage.

[0006] Query the DT exposure curve and determine the corresponding ET exposure curve based on the exposure parameters of the penetration thickness corresponding to the benchmark blackness. For the DT curve, it is generally considered to be the three points {(T i , D i )|i=1,2,3} [1] However, there is no research on the mathematical description of this curve, that is, the use of mathematical formulas to describe the DT function relationship. [2][3][4] The above-mentioned references are given in the detailed description.

[0007] Currently, in industrial X-ray inspection, the selection of process parameters (ray energy, exposure, focal length, etc.) is usually determined by querying the ET exposure curve, which is produced based on the curve drawn by the DT test.

[0008] Due to the uniqueness of the ET exposure curve, the characteristic curve is generally drawn manually, and the parameters are determined by manual query of tables and manual calculation. This has the disadvantages of slow speed, low accuracy, poor repeatability, and difficulty in query. This results in a long detection test cycle, low work efficiency, and affects the determination of the accuracy of the test data.

[0009] Therefore, the present invention proposes a mathematical modeling method for the DT exposure curve, constructs a DT exposure curve model by establishing a relationship with the DH characteristic curve model (film characteristic curve model), and verifies the effectiveness and practicality of this method through experiments. Summary of the Invention

[0010] In order to solve the problem that manual drawing of ET curve requires manual table query and manual calculation to determine the parameters of the curve, resulting in long test cycle and low work efficiency, the present invention proposes a mathematical modeling method for DT exposure curve.

[0011] In order to solve the above technical problems, the technical solution adopted by the present invention is: a mathematical modeling method of the DT exposure curve, comprising the following steps:

[0012] S1: Use curve fitting combined with optimization method to construct the mathematical model of the DH characteristic curve of non-sensitized industrial film, and obtain the characteristic parameters corresponding to the characteristic curves of different types of industrial films;

[0013] S2: Construct a mathematical model of the DT exposure curve based on the mathematical model of the DH characteristic curve of industrial films: By proving the linear relationship between H′ and T, establish a mapping with the nonlinear DH film characteristic curve and construct a mathematical model of the DT exposure curve;

[0014] Where D represents the darkness of the film, H represents the exposure, T represents the transillumination thickness, and H' represents a temporary variable.

[0015] The steps for constructing the mathematical model of the DH characteristic curve of the industrial film in step S1 are as follows:

[0016] The n-order polynomial regression method is used to fit and obtain the mathematical model of the industrial film characteristic curve;

[0017] The parameters of the industrial film characteristic curve polynomial are adjusted by the least square method to obtain the optimal mathematical model of the industrial film characteristic curve.

[0018] The expression of the mathematical model of the DH characteristic curve of the industrial film is as follows:

[0019]

[0020] The expression of the optimized mathematical model of the DH characteristic curve of industrial film is as follows:

[0021]

[0022]

[0023] In the above formula: lg i H represents the i-th power of the logarithm of the exposure value H, Dk is the result of the film blackness fitting calculation, is the negative film blackness measurement value, is the exposure measurement value, D is the film blackness, H is the exposure, n and k are the polynomial orders, is the polynomial coefficient, N is a natural number, N + is a non-negative natural number, R + is a non-negative real number, H i Represents the calculated value of exposure.

[0024] The expression of H' in step S2 is as follows:

[0025] H′=h(T)=lg E0-μT+lg C′;

[0026] In the above formula: E0 represents the exposure amount. The exposure amount is generally considered to be represented by H in the film system and E in the transillumination test. Here, the subscript 0 of E represents the exposure amount in the initial state. T represents the thickness of the test block, μ = μ'lge, μ' represents the linear absorption coefficient of the material, and C' = αZV 2 / F 2 , α represents the proportional coefficient, i represents the tube current, Z represents the atomic number of the target material, V represents the tube voltage, F represents the focal length, and C' is a constant for a given transillumination system.

[0027] In the expression of H', for a given transillumination system, C' is a constant, and C = lg E0 + lg C' is assumed. If E0 is constant, then H' and T are in a linear relationship, denoted as h, and D = 8(H'). Then D is in a mapping relationship with H' after the linear transformation of T by the operator h.

[0028] For the mapping D = g(H′), H′ = h(T), D = g(h(T)), that is, DT has the functional relationship: gh:D→T;

[0029] Therefore, the mathematical model of the DT curve can be transformed according to the mathematical model of the DH curve.

[0030] According to the optimized mathematical model of the industrial film DH characteristic curve, the given exposure E0 and tube voltage kV are obtained. i The mathematical expression of the DT model is as follows:

[0031]

[0032]

[0033] In the above formula: kV represents kilovolts, and i represents the i-th power of the logarithm of the exposure H.

[0034] The beneficial effects of the present invention compared to the prior art are:

[0035] This method demonstrates the linear relationship between H′ and T, establishes a mapping with the nonlinear DH characteristic curve, and constructs a DT exposure curve model. The effectiveness and practicality of this model are verified through experiments and application examples. Within the allowable error range, it significantly improves calculation speed, accuracy, and inspection efficiency, providing a decision-making basis for the digitalization and intelligentization of radiographic systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The present invention will be further described below with reference to the accompanying drawings:

[0037] Figure 1 A flow chart of the design scheme of the present invention;

[0038] Figure 2 A flow chart of an embodiment of the present invention;

[0039] Figure 3 This is a schematic diagram of the step test block transillumination;

[0040] Figure 4 Schematic diagram of projection transformation from DH to DT;

[0041] Figure 5 It is the DH characteristic curve;

[0042] Figure 6 It is the DT exposure curve;

[0043] Figure 7 DT curve (E = 5kV.min) diagram;

[0044] Figure 8 This is the DT curve (E=200kV.min) diagram. DETAILED DESCRIPTION

[0045] The following first describes the relevant concepts of the ET exposure curve involved in the present invention.

[0046] ①E-T exposure curve

[0047] The ET exposure curve describes the functional relationship between the radiation energy I, exposure E, focal length F and penetration thickness T under specified conditions. Theoretically, it is proved that the curve with penetration voltage as parameter is approximately a first-order linear relationship. [5] , as shown below:

[0048] lg E=μT+C (1).

[0049] ②D-T exposure curve

[0050] The DT exposure curve is a curve that describes the relationship between the exposure thickness T and the corresponding blackness D when the same exposure E is selected and different tube voltages V are used to illuminate the step test block under the premise of certain camera model, film, intensifying screen and focal length. [6] , referred to as DT curve.

[0051] ③Film blackness (optical density)

[0052] The blackness of a film refers to the degree of opacity of the film, which is used to indicate the degree to which metallic silver makes the film darker, and is also called optical density.

[0053] Assume that the light intensity incident on the film is I0, the light intensity passing through the film is I, and the optical density is D. Then the optical density is defined as [7] : =lg(I0 / I)

[0054] ④D-H film characteristic curve

[0055] The DH film characteristic curve is a curve that shows the relationship between relative exposure and film blackness. [6] In the film characteristic curve, the horizontal axis represents the logarithmic value of the X-ray exposure (H), and the vertical axis represents the black value (optical density D) obtained after the film is developed. Non-sensitized film is generally used in industrial X-ray detection.

[0056] ⑤ Gradient (G)

[0057] The gradient of a film refers to the inherent ability of the film to display different black levels on the negative film at different exposure levels. It can be expressed by the slope of the tangent line at a certain point on the film characteristic curve.

[0058] The slope of the line connecting two points on the characteristic curve is often used to represent the average gradient of the film. [7] :

[0059]

[0060] ⑥ Curve fitting

[0061] Refers to the use of continuous curves to approximate or compare the functional relationship between the coordinates represented by discrete points on a plane [8] ,In numerical analysis, curve fitting is the use of analytical expressions to approximate discrete ,data, that is, the formulation of discrete data.

[0062] ⑦ Radiation field

[0063] Use a directional radiation X-ray machine for transillumination, specify the focal length F0 = 100 cm, and the radiation angle α = 16-26° to form the radiation space as the radiation field [7] .

[0064] ⑧Mathematical model

[0065] Transform abstract practical problems into mathematical problems, grasp the main factors, and express and calculate them using mathematical formulas that are easy to understand and calculate [8] .

[0066] Based on the above understanding of the concept of ET exposure curve, the ET exposure curve is fitted according to the mathematical modeling method, which mainly includes the following steps:

[0067] 1. System Analysis

[0068] The determination of the parameters of the X-ray transillumination test needs to be based on the ET exposure curve. The drawing of the ET exposure curve needs to rely on the DT exposure curve of each correction test. Since the DT exposure curve is affected by multiple factors and its variability directly affects the transillumination test results, it is necessary to study the DT model to facilitate the determination of the test parameters.

[0069] After overall analysis, the design and implementation plans are determined as follows: Figure 1 and Figure 2 shown.

[0070] 2. Film DH characteristic curve modeling

[0071] By analyzing the changing law of the characteristic curve of non-sensitized film, it is found that in the normal exposure area, the characteristic curve of non-sensitized film is an inclined arc with increasing gradient.

[0072] According to the characteristics of the film characteristic curve, the n-order polynomial regression method is used for fitting, and the parameters in the polynomial are adjusted by the least square method to ensure the accuracy and speed of the calculation results.

[0073] Introducing the function descriptor g, the mathematical model of the industrial film curve with a certain model is as follows:

[0074]

[0075] In the above formula: i is the parameter subscript; n is the polynomial order; a i are the polynomial coefficients; D is the film blackness; H is the exposure, and lg′H represents the i-th power of the logarithm of the exposure H.

[0076] Through experiments, the least square method is used to perform curve fitting. By analyzing the experimental data, the optimal solution is selected within the allowable error range and taking into account the calculation speed and the smooth transformation of indicator values.

[0077] A mathematical model was constructed to find the minimum result under multiple iterations of polynomials of different orders, requiring the difference between the curvature of the curve and the measured value to meet the threshold range. The optimization model is as follows:

[0078]

[0079]

[0080] In the above formula: i is the parameter subscript; k is the polynomial order; is the polynomial coefficient; D is the film blackness; H is the exposure; D is the film blackness and exposure measurement value; k is the result of the film blackness fitting calculation, N is a natural number, N + is a non-negative natural number, R + is a non-negative real number.

[0081] In order to verify the validity of the above conclusions, we use [1] The data in Table 1 are used for verification; because the X-ray machine current i used in the original experiment is fixed, the exposure time s can be used to represent the exposure H. The original approximate formula is as follows:

[0082] H a =bD+c;

[0083] Among them, H is the exposure amount received by the film; a, b, c are constants; D is the film blackness.

[0084] The original data (x = exposure time, y = blackness test value) was fitted using the method of the present invention. The results of the regression parameter error analysis are shown in Table 1:

[0085] Order n MSE RMSE MAE MAPE R2 2 0.0036 0.0256 0.0211 0.0146 0.9994 3 0.0009 0.0157 0.0117 0.0079 0.9998 4 0.0003 0.0103 0.0079 0.0050 0.9999 5 0.0001 0.0075 0.0053 0.0028 0.9999 6 0.0001 0.0066 0.0052 0.0032 1.0000 7 0.0001 0.0056 0.0041 0.0019 1.0000

[0086] Table 1. Regression errors of film curve parameters.

[0087] Too high or too low an order will result in overfitting or underfitting. After analyzing the test data in Table 1, it is believed that the order n=5 is more reasonable. The corresponding coefficients are shown in Table 2 below:

[0088] Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > 0 -18.7017 2 -64.5275 4 -10.0937 1 55.8214 3 36.6225 5 1.1008

[0089] Table 2 Order i and corresponding coefficient a i .

[0090] The data in Table 2 were used to test the polynomial coefficients and compared with the original results. The comparison results are shown in Table 3:

[0091]

[0092] Table 3 Comparative experiments.

[0093] From the analysis of the data in Table 3, we can see that:

[0094] ① The original solution method requires that the selected sample data must be the solution of the calculated approximate formula (such as {42,126,360} in the table), that is, the experimental value is equal to the calculated value, so the relative error will be less than 0.1;

[0095] ② The original text actually takes three sets of related data with the same multiple relationship of exposure threshold in the characteristic curve, with the purpose of extracting a common factor to facilitate the solution of the subsequent equation system. In essence, it is still the average gradient method of determining a line from two points.

[0096] ③ Using multiple parameters of the present invention to adjust the approximate calculation value is beneficial to improving the calculation accuracy and reducing the error. Through the comparative analysis of relative errors, it can be seen that the approximate solution method adopted by the present invention is superior to the original method.

[0097] 3. Model solution

[0098] The ray intensity at any point in the radiation field space is [5] :

[0099] I0=αiZV 2 / F 2 (2);

[0100] In the above formula: I0—ray intensity; α—proportional coefficient; i—tube current; Z—atomic number of target material; V—tube voltage; F—focal length.

[0101] For a given transillumination system, assume that the parameters α, Z, V, and F remain constant, and let C′ = αZV 2 / F 2 , we have I0=C′i, C′ is a constant, it can be seen that the intensity of the rays is proportional to the tube current.

[0102] The test process of the transillumination step block is as follows Figure 3 As shown, since F>>ΔT, it can be considered that the radiation energy I0 (exposure E0) irradiated to the workpiece surface is equal, and the attenuation law of the transmitted radiation is:

[0103] I d =I0e -μ′T (3);

[0104] In the above formula: I0, I d - the intensity of the ray before and after the ray enters the test block; μ′- the material linear absorption coefficient; T is the thickness of the test block; ΔT is the thickness difference of the test block.

[0105] Taking the logarithm of both sides of formula (3), we have lg I0=μ′T lg e+lg I d Assume that the exposure time is t, add lgt on both sides of the above equation, and we have lg I0t=μ′T lg e+lg I dt; Since I0=C′i, then lg C′it=μ′T lg e+lg I d t, let E0 = it, μ = μ′lge, and rearrange it to get:

[0106] lg E0=μT+lg I d t-lg C′ (4);

[0107] In the above formula: is the received exposure. If the transillumination blackness D is specified, then is a constant, let C0=lgI d t-lg C′, lgE0=μT+C0 [7] , is the general approximate expression of ET exposure curve; let H=I d t, we have lg E0=μT+lgH-lg C′ and further lg E0=μT+g -1 (D) -lg C′ is sorted to get g -1 (D) = lg E0 - μT + lg C′. Performing g operation on both sides yields:

[0108] D = g(IgE0-μT+IgC′) (5).

[0109] Assume temporary variable H′=h(T)=lgE0-μT+lgC′, let C=lgE0+lgC′, stipulate that E0 is constant, then parameter C is a constant, so H′ and T are in linear relationship, denoted as h; Formula (5) can be simplified to D=g(H′); From this, it can be concluded that D and H′ after the linear transformation of T by operator h are in mapping relationship g; since lgH=g- 1 (D) = h(T) = -μT + lgC, it can be seen that lgH ~ T is a linear relationship. In summary, it can be considered that the change trend of the DT curve is the same as the change trend of the DH characteristic curve, such as Figure 3 Medium H i The law of change.

[0110] For the mapping D = g(H′), H′ = h(T), D = g(h(T)), that is, DT has the functional relationship: Within the threshold range of D and T, since ① every value of T has a unique D value corresponding to it; ② all values ​​in D can find a unique T value corresponding to it, so It is a one-to-one mapping; because h is a linear function, it is a scaling transformation of the T value with a coefficient of -μ and a vertical translation transformation with a distance of C, so the DT curve can be obtained by transforming the DH curve.

[0111] Figure 4 The projection transformation process from the DH curve model to the DT curve model is explained from a visualization perspective.

[0112] Combined with the DH curve model, the DT curve model under a given exposure E0 and tube voltage V is obtained as follows:

[0113]

[0114]

[0115] For a given DH characteristic curve model, given g, in the mapping h, there are only two parameters, μ and C, and the operator can be solved using two binary coordinates (T, D)

[0116] 4. Comparative experiment

[0117] To verify the above model, we use

[10] The original paper used AGFAC7 industrial film. The model is: polynomial order parameter k = 7, other parameters and visual effects are shown in Table 4. Figure 5 As shown, Figure 5 The curve in is the fitting result of the measured data, and “+” corresponds to the measured data;

[0118] Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > 0 -2.20e+2 3 1.94e+3 6 -8.45e+1 1 9.91e+2 4 -1.19e+3 7 7.05e+0 2 -1.88e+3 5 4.28e+2

[0119] Table 4 AGFAC7 film order i and corresponding coefficient a i .

[0120] According to the above DH curve model,

[10] Taking the test data in Tables 2 and 3 (V = 250kV, E = 2.0min; V = 275kV, E = 4.0min) as an example, the blackness D value shown in Table 5 is obtained:

[0121]

[0122] Table 5 Blackness D value.

[0123] Use the test data in Table 5 to determine the DT function relationship under different conditions The conclusions are shown in Table 6. Figure 6 The relative errors between the test value and the calculated value are calculated using the obtained functional relationship as shown in Table 7. Figure 6 The curves in the figure correspond to the function models under different conditions. “+” represents the experimental data in Table 5. It can be seen that the original experimental data fluctuates around the curve, and the experimental data is consistent with the DT curve model.

[0124]

[0125] Table 6 Order i and corresponding coefficient a i .

[0126] Use original text

[10] The DT curve model under the corresponding conditions is solved using the test data in Tables 1 to 5, and the transillumination thickness T when D = 2.5 is calculated. Compared with the test data in Table 6 of the original text, the calculated relative error is shown in Figure 7:

[0127]

[0128] Table 7 Relative error of transillumination thickness T (mm) (D=2.5).

[0129] It can be seen from Tables 5 and 7 that the test values ​​are very close to the calculated values. Taking the test values ​​as the benchmark, the absolute value of the relative error between the calculated values ​​and the test values ​​is very small (blackness D≤2%, transillumination thickness T≤2%), which verifies that model (6) is reliable.

[0130] Through theoretical deduction and comparative tests (error ≤ 2%), it is proved that the variation law of the DT curve model comes from the DH characteristic curve model, that is, the essence of the DT model is the DH model.

[0131] When the DH characteristic curve model is known, the function under given conditions (tube voltage V, exposure E) can be determined by more than two sets of (T, D) binary data in the DT test.

[0132] By inputting the DT model into the computer, other exposure parameters can be easily calculated, such as the thickness tolerance under a given tube voltage V and exposure E, such as Figure 6 shown.

[0133] The method of the present invention is further described below with reference to specific embodiments.

[0134] Calculate the DT curve of AA400 film under given test conditions.

[0135] 1. Data Collection

[0136] The DT test process parameters are shown in Table 8:

[0137] X-ray machine: 300EGM2 Intensifying screen: Pb0.1mm×2 focal length: 1000mm film: AA400 Darkroom processing: PRO@NDT80 film processor Film blackness: 2.0

[0138] Table 8 DT test conditions.

[0139] Under the process parameter conditions shown in Table 8, the DT test measurement parameters are shown in Table 9:

[0140]

[0141] Table 9 DT test data.

[0142] 2. Model calculation

[0143] The given DH model parameters are shown in Table 10:

[0144] Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i <!-- 9 -->]]> 0 0.6505 3 -13.9883 6 0.9593 1 -3.3976 4 10.5040 7 -0.0857 2 10.0461 5 -4.3680

[0145] Table 10 AA400 film order i and corresponding coefficient a i .

[0146] Calculate the H′-T mapping relationship, as shown in Table 11 and Table 12:

[0147] <![CDATA[a0]]> 37.9096 43.5355 50.7099 54.5371 59.7925 60.8208 66.8165 72.6165 74.7668 <![CDATA[α1]]> -12.4391 -14.2196 -16.4571 -17.5517 -19.0397 -19.2110 -21.1508 -22.8698 -23.4020

[0148] Table 11 H′-T linear relationship (E=5 kV.min).

[0149] <![CDATA[a0]]> 77.8711 87.6329 111.0844 99.2227 93.6528 94.1241 121.8700 123.1783 130.6836 <![CDATA[a1]]> -19.5887 -21.9891 -30.1541 -24.5051 -20.9636 -19.9530 -29.8679 -30.6464 -32.4749

[0150] Table 12 H′-T linear relationship (E=200 kV.min).

[0151] 3. Experimental verification

[0152] According to the mapping relationships given in Tables 11 and 12, the calculated values ​​are substituted into model (6) and compared with the original measured values ​​in Table 9. The comparison results are shown in Table 13 below:

[0153]

[0154] Table 13D-T comparison test.

[0155] The film order i and the corresponding coefficient a in Table 1, Table 3 and Table 7 in the above embodiments of the present invention i Calculated through the mathematical model of industrial film curve.

[0156] The references used in the present invention are as follows:

[0157] [1] Du Chuanguo. Method for making X-ray exposure curve [J]. Journal of Shandong Electric Power College, 2001, 4(1), 26-29;

[0158] [2] Zhou Wei. Approximate algorithm for analytical expression of film characteristic curve[J]. Nondestructive Testing, 2000, 22(10):456-457;

[0159] [3] Zhang Jianhe, Chen Cunzhu. Comparative determination method and application of industrial X-ray film characteristic curve [J]. Nondestructive Testing, 2002, 24(9): 404-406;

[0160] [4] Hu Lianwei. New method and application of exposure curve preparation[J]. Nondestructive Testing, 2001, (2): 11-15;

[0161] [5] Zheng Shicai, ed. Radiographic Detection[M]. Machinery Industry Press, 2004;

[0162] [6] Song Tianmin (ed.). Radiographic Detection[M]. China Petrochemical Press, 2011;

[0163] [7] Qiang Tianpeng (ed.). Radiographic Detection[M]. China Labor and Social Security Press, 2006.

[0164] [8] Ye Qixiao, Jiang Qiyuan et al. Mathematical Modeling[M]. Machinery Industry Press, 2009;

[0165] [9] ISO 7004-2002. Industrial radiographic films, etc. [S], International Organization for Standardization, 2002;

[0166]

[10] Feng Jian. Preparation of XXHA-3505 exposure curve[J]. Value Engineering, 2015(3):74-75.

[0167] Regarding the specific structure of the present invention, it should be noted that the connection relationship between the various component modules adopted in the present invention is definite and feasible. Except for those specifically described in the embodiments, the specific connection relationship can bring about corresponding technical effects and solve the technical problems raised by the present invention without relying on the execution of corresponding software programs. The components, modules, models of specific components appearing in the present invention, the connection methods between each other, and the conventional usage methods and expected technical effects brought about by the above-mentioned technical features, except for those specifically described, all belong to the disclosed contents in patents, journal articles, technical manuals, technical dictionaries, and textbooks that can be obtained by technical personnel in this field before the application date, or belong to the existing technologies such as conventional technology and common knowledge in this field. There is no need to elaborate, so that the technical solution provided in this case is clear, complete, and feasible, and the corresponding physical products can be reproduced or obtained based on this technical means.

[0168] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A mathematical modeling method for a DT exposure curve, characterized by: The steps include: S1: Use curve fitting combined with optimization method to construct the mathematical model of the DH characteristic curve of non-sensitized industrial film, and obtain the characteristic parameters corresponding to the characteristic curves of different types of industrial films; The steps for constructing the mathematical model of the DH characteristic curve of the industrial film in step S1 are as follows: The n-order polynomial regression method is used to fit and obtain the mathematical model of the industrial film characteristic curve; By adjusting the parameters of the polynomial of the industrial film characteristic curve through the least square method, the optimal mathematical model of the industrial film characteristic curve is obtained; S2: Construct a mathematical model of the DT exposure curve based on the mathematical model of the DH characteristic curve of industrial films: By proving the linear relationship between H′ and T, establish a mapping with the nonlinear DH film characteristic curve and construct a mathematical model of the DT exposure curve; Where D represents the darkness of the film, H represents the exposure, T represents the transillumination thickness, and H′ represents a temporary variable.

2. The mathematical modeling method for a DT exposure curve according to claim 1, wherein: The expression of the mathematical model of the DH characteristic curve of the industrial film is as follows: The expression of the optimized mathematical model of the DH characteristic curve of industrial film is as follows: In the above formula: lg i H represents the i-th power of the logarithm of the exposure value H, D k is the result of the film blackness fitting calculation, is the negative film blackness measurement value, is the exposure measurement value, D is the film blackness, H is the exposure, n and k are the polynomial orders, is the polynomial coefficient, N is a natural number, N + is a non-negative natural number, R + is a non-negative real number, H i Represents the calculated value of exposure.

3. The mathematical modeling method for a DT exposure curve according to claim 2, wherein: The expression of H' in step S2 is as follows: H′=h(T)=lgE0-μT+lgC′; In the above formula: E0 represents the exposure amount, T represents the thickness of the test block, μ = μ'lge, μ' represents the material linear absorption coefficient, C' = αZV 2 / F 2 , α represents the proportional coefficient, Z represents the atomic number of the target material, V represents the tube voltage, F represents the focal length, and C' is a constant for a given transillumination system.

4. The mathematical modeling method for a DT exposure curve according to claim 3, wherein: In the expression of H′, for a given transillumination system, C′ is a constant, and C=lgE0+lgC′ is assumed. If E0 is constant, then H′ and T are in a linear relationship, denoted as h, and D=g(H′). Then D is in a mapping relationship with H′ after the linear transformation of T by the operator h. For the mapping D = g(H′), H′ = h(T), D = g(h(T)), that is, DT has the functional relationship: gοh:D→T; Therefore, the mathematical model of the DT curve can be transformed according to the mathematical model of the DH curve.

5. The mathematical modeling method for a DT exposure curve according to claim 4, characterized in that: According to the optimized mathematical model of the industrial film DH characteristic curve, the given exposure E0 and tube voltage kV are obtained. i The mathematical expression of the DT model is as follows: gοh:D→T In the above formula: kV represents kilovolts, lg i H represents the i-th power of the logarithm of the exposure H.

Images