An Optimization Method for a Mathematical Model of Industrial Film Characteristic Curves

Abstract

This invention provides an optimization method for the mathematical model of industrial film characteristic curves, belonging to the field of mathematical modeling technology for film characteristic curves. It solves the problems of slow speed, low accuracy, poor repeatability, and difficulty in querying film characteristic parameters in industrial X-ray inspection, which relies on manual table lookup and calculation. This results in long testing cycles and low work efficiency. The method includes the following steps: S1: Collect and organize data corresponding to exposure and density of characteristic curves for different types of industrial films; S2: Analyze the characteristics of the industrial film characteristic curves and adopt... n The mathematical model of the industrial film characteristic curve is obtained by fitting the polynomial regression method; S3: The parameters in the polynomial of the industrial film characteristic curve are adjusted by the least squares method and regression analysis to obtain the optimal mathematical model of the industrial film characteristic curve; This invention is applied to the mathematical modeling of film characteristic curves.

Description

Technical Field

[0001] This invention provides an optimization method for the mathematical model of industrial film characteristic curves, belonging to the field of mathematical modeling technology for film characteristic curves. Background Technology

[0002] Non-destructive testing is a method of conducting highly sensitive and reliable inspections of the properties, state, and structure of an object without damaging or altering its physical and chemical properties, thereby determining its surface and internal integrity, continuity, safety, and other performance indicators.

[0003] X-ray inspection is an important method of non-destructive testing. By transmitting X-rays, a direct image of the defects in the object being tested can be obtained. The qualitative analysis is accurate, and it can even measure the length, width, and height. It is widely used in aviation, aerospace, weaponry, shipbuilding, and special equipment.

[0004] X-rays attenuate to varying degrees when they pass through different materials and are incident on the film. This variation in the degree of darkening of the film due to different exposure levels is called the film characteristic curve.

[0005] The films used in industrial radiography are mainly divided into two types: intensifying films and non-intensifying films. Since intensifying films produce images with lower contrast, non-intensifying films are generally used in industrial radiographic testing.

[0006] In industrial inspection, the proper use of film photosensitivity profiles is crucial for producing high-quality films. Flexible application of film photosensitivity profiles can avoid the blind selection of exposure parameters and improve the quality and sensitivity of X-ray inspection.

[0007] Currently, the use of film characteristic curves mainly relies on manually consulting charts provided by film manufacturers to determine test parameters. However, this manual method can lead to inappropriate parameter selection, resulting in excessive deviations in film density, affecting the sensitivity of X-ray inspection, and failing to guarantee effective detection of defects in the tested object. Studying the mathematical description of the relationship between film density and exposure using film characteristic curves, and calculating test parameters using mathematical models, can improve detection accuracy and efficiency.

[0008] Studies have described the functional relationship between density and exposure in industrial film characteristic curves using mathematical formulas, and the methods generally used are based on the average gradient method. [1][2][3][4] However, if high experimental accuracy is required, this method is considered to introduce errors.

[0009] Currently, in industrial X-ray inspection, characteristic curves are generally drawn manually, and film characteristic parameters are determined by manually looking up tables and calculating. This method has disadvantages such as slow speed, low accuracy, poor repeatability, and difficulty in looking up information. As a result, the testing cycle is long, the work efficiency is low, and the accuracy of the test data is affected. Summary of the Invention

[0010] To address the shortcomings of manually determining film characteristic parameters through table lookups and calculations in industrial X-ray inspection, such as slow speed, low accuracy, poor repeatability, and difficulty in data lookup, which leads to long testing cycles and low work efficiency, this invention proposes an optimization method for the mathematical model of industrial film characteristic curves.

[0011] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: an optimization method for a mathematical model of industrial film characteristic curves, comprising the following steps:

[0012] S1: Collect and organize the exposure and density data corresponding to the characteristic curves of different types of industrial film;

[0013] S2: Analyze the characteristics of industrial film characteristic curves, and use the nth-order polynomial regression method to fit the curves, thereby obtaining a mathematical model of the industrial film characteristic curves.

[0014] S3: By adjusting the parameters in the polynomial of the industrial film characteristic curve using the least squares method, the optimal mathematical model of the industrial film characteristic curve is obtained.

[0015] In step S1, the exposure and density values ​​of the characteristic curves of different types of industrial film are obtained by manually querying and compiling the film charts provided by the manufacturers. The industrial film used is a non-sensitizing film.

[0016] The characteristic curve of industrial film in step S2 is characterized as follows: in the normal exposure area, the characteristic curve of non-intensifying film is a sloping arc with increasing gradient.

[0017] In step S2, an nth-order polynomial regression method is used for fitting, and the mathematical model expression of the industrial film characteristic curve is as follows:

[0018]

[0019] In the above formula: D is the film density, H is the exposure, n is the polynomial order, and a i Here are the polynomial coefficients, N is a natural number, and lg is the polynomial coefficient. i H represents the exposure amount H raised to the power of the logarithm H.

[0020] The optimal mathematical model for the industrial film characteristic curve in step S3 is achieved by repeatedly iterating the polynomial of the industrial film characteristic curve at different orders, requiring the curve curvature to be negative and the difference between it and the measured value to satisfy a threshold range.

[0021] The expression for the optimal mathematical model of the industrial film characteristic curve is as follows:

[0022]

[0023]

[0024] In the above formula: D k The result is the film density fitting calculation. This is the measurement value of film density. Here, D is the film density, H is the exposure, and n and k are the polynomial orders. Here are the polynomial coefficients, and N is a natural number. + R is a non-negative natural number. + H is a non-negative real number. i This represents the calculated exposure rate.

[0025] The advantages of this invention over the prior art are as follows:

[0026] This invention employs a curve fitting and optimization method to solve for the approximate functional relationship of the characteristic curve of industrial film. The approximate formula of the film characteristic curve is input into a computer to solve for the characteristic parameters. The effectiveness of the method is verified through experiments. Within the allowable error range, it greatly improves the calculation speed, calculation accuracy and inspection efficiency, and provides a decision-making basis for the digitization and intelligentization of film systems. Attached Figure Description

[0027] The present invention will be further described below with reference to the accompanying drawings:

[0028] Figure 1 The photosensitivity profile of the intensifying film;

[0029] Figure 2 The photosensitivity profile of non-sensitizing film;

[0030] Figure 3 This is a flowchart of the design scheme of the present invention;

[0031] Figure 4 This is a flowchart of an embodiment of the present invention;

[0032] Figure 5 A fitting curve of the photosensitivity characteristics of AA400 film;

[0033] Figure 6 Fitting curves for the photosensitivity characteristics of Tianjin III;

[0034] Figure 7 The curve is a fitting curve for the photosensitivity characteristics of D7 film. Detailed Implementation

[0035] The following section will explain the relevant concepts of industrial film characteristic curves involved in this invention.

[0036] ①Film density (optical density)

[0037] The blackness of a film refers to its opacity, indicating the degree to which metallic silver makes the film appear black; it is also known as optical density.

[0038] Let the intensity of the light incident on the film be I0, the intensity of the light transmitted through the film be I, and the optical density be D. Then the optical density is defined as... [5] :=lg(I0 / I), such as Figure 1 As shown.

[0039] ②Film characteristic curves

[0040] Film characteristic curves are curves that represent the relationship between relative exposure and film density. [6] In the film characteristic curve, the horizontal axis represents the logarithm of the X-ray exposure, and the vertical axis represents the density value obtained after film development, such as... Figure 1 , Figure 2 As shown. Non-sensitizing films are generally used in industrial X-ray inspection.

[0041] ③ Gradient

[0042] The gradient of a film refers to its inherent ability to display different shades of darkness on a negative at different exposures. It can be represented by the slope of the tangent line at a point on the film's characteristic curve.

[0043] The average gradient of a film is often represented by the slope of the line connecting two points on its characteristic curve. [6] :

[0044]

[0045] ④ Curve fitting

[0046] This refers to using a continuous curve to approximate or analogize the functional relationship between the coordinates of discrete points on a plane. [7] In numerical analysis, curve fitting is the approximation of discrete data using analytical expressions, that is, the formalization of discrete data.

[0047] Based on the above understanding of the concept of industrial film characteristic curves, the following section uses mathematical modeling methods to fit the industrial film characteristic curves, as follows: Figure 3 and Figure 4 As shown, the main steps include:

[0048] 1. System Analysis

[0049] Theoretically, the trend of film curve changes is as follows Figure 1 As shown, however, in practical applications, due to the influence of many factors, the variation pattern of intensifying film curves is not entirely the same as that of non-intensifying film (industrial film) curves. (Comparison) Figure 1 and Figure 2 It can be seen that, in particular, in the normal exposure area, the intensifying film is a slanted straight line, while the non-intensifying film is a slanted arc with an increasing gradient.

[0050] When applying industrial film characteristic curves, the general rule is to use the slope of the straight line connecting the upper and lower limits of net blackness on the curve as the average gradient of the film for calculation. However, since the average gradient and the slope at each point on the curve are not the same, the average gradient is only an approximation. If high precision is required in scientific experiments, this approach will introduce errors.

[0051] If we can find the pattern of the curve's variation and describe it with a formula, it will greatly facilitate subsequent experiments. Moreover, within the identifiable range, by calculating the area outside the curve's specified blackness threshold using the formula, we can expand the thickness tolerance of radiography.

[0052] 2. Curve Fitting

[0053] There are various methods for fitting plane curves, such as univariate linear regression y = a + bx and exponential regression y = ae. bx Geometric regression y = ax b Logarithmic regression y = a1nbx and nth-order polynomial regression Based on the characteristics of the film's characteristic curves, an nth-order polynomial regression method was used for fitting, and the parameters in the polynomial were adjusted using the least squares method to ensure the accuracy and speed of the calculation results.

[0054] Introducing the function descriptor g, the mathematical model for an industrial film curve with a defined model is as follows:

[0055]

[0056] In the above formula: i is the parameter subscript; n is the polynomial order; a i is the polynomial coefficient; D is the film density; H is the exposure.

[0057] 3. Optimization solution

[0058] Through experiments, the least squares method was used to fit the curve. By analyzing the experimental data, the optimal solution was selected within the allowable error range and considering the calculation speed and the gradual change of index values.

[0059] A mathematical model is constructed to find the minimum result where the difference between the measured value and the curve with negative curvature is within a threshold range after multiple iterations of the polynomial of different orders. The optimization model is as follows:

[0060]

[0061]

[0062] In the above formula: i is the parameter subscript; k is the polynomial order; n is the regression analysis number; Here are the polynomial coefficients; D is the film density; H is the exposure. This refers to the film's density and exposure measurements; D k The result is the film density fitting calculation, where N is a natural number. + R is a non-negative natural number. + H is a non-negative real number. i This represents the calculated exposure rate.

[0063] The film curve fitted by the method of the present invention will be calculated and verified below.

[0064] Take, for example, a certain company's AA400 industrial film.

[0065] 1. Data Collection

[0066] The characteristic curve data of AA400 industrial film are summarized in Table 1:

[0067]

[0068] Table 1. Curves of AA400 industrial film.

[0069] 2. Fitting Experiment

[0070] According to the method of the present invention, the curve of AA400 industrial film was fitted, and the fitting results are shown in Table 2 below. Figure 5 As shown, Figure 5 The curves in the table represent the fitting results of the data in Table 2, and the "+" signs correspond to the data in Table 1.

[0071] Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > 0 0.6505 3 -13.9883 6 0.9593 1 -3.3976 4 10.5040 7 -0.0857 2 10.0461 5 -4.3680

[0072] Table 2. AA400 film order i and corresponding coefficient a i .

[0073] 3. Regression Analysis

[0074] The results of the regression parameter error analysis after multiple fitting experiments are shown in Table 3.

[0075] Order n MSE RMSE MAE MAPE R2 4 0.0011 0.0329 0.0229 0.0253 0.9995 5 0.0009 0.0300 0.0207 0.0259 0.9996 6 0.0008 0.0275 0.0190 0.0242 0.9997 7 0.0007 0.0269 0.0177 0.0170 0.0997 8 0.0007 0.0266 0.0177 0.0191 0.9997 9 0.0007 0.0258 0.0168 0.0206 0.9997 10 0.0006 0.0237 0.0156 0.0233 0.9998

[0076] Table 3. Regression error of AA400 film curve parameters.

[0077] The analysis of the experimental data in Table 3 shows that an order of n=7 is considered reasonable.

[0078] 4. Experimental verification

[0079] To verify the validity of the above conclusions, the following text was used. [2] The data in Table 1 were used for verification; since the X-ray machine current i used in the original experiment was fixed, the exposure time s can be used to represent the exposure amount H. The approximate formula in the original text is as follows:

[0080] H a =bD+c;

[0081] Where H is the amount of exposure received by the film; a, b, c are constants; and D is the film density.

[0082] The original data (x = exposure time, y = blackness test value) were fitted using the method of this invention. The regression parameter error analysis results are shown in Table 4.

[0083] 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

[0084] Table 4 shows the regression error of film curve parameters.

[0085] Too high or too low an order can lead to overfitting or underfitting. Analyzing the experimental data in Table 4, an order of n=5 is considered reasonable. The corresponding coefficients are shown in Table 5 below.

[0086] 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

[0087] Table 5 shows the order i and its corresponding coefficient a. i .

[0088] The polynomial coefficients were tested using the data in Table 5 and compared with the results in the original paper. The comparison results are shown in Table 6.

[0089]

[0090] Table 6 Comparative experiments.

[0091] Analysis of the data in Table 6 shows that:

[0092] 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 <0.1;

[0093] ②The original text actually takes three sets of relevant data in the characteristic curve with the same multiple relationship for the exposure threshold. The purpose is to extract a common factor so that the subsequent equation system can be solved. In essence, it is still the average gradient method of two points determining a line.

[0094] 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 comparative analysis of relative errors, it can be seen that the approximate solution method used in the present invention is superior to the original method.

[0095] The method of the present invention will be further described below with reference to specific embodiments.

[0096] The characteristic curves of Tianjin III and D7 type films were calculated using the method of this invention.

[0097] 1. The calculation steps for the Tianjin Type III film curve are as follows.

[0098] 1.1 Data Acquisition

[0099] Exposure and density data of Tianjin Type III film were collected and summarized in Table 7 below:

[0100]

[0101] Table 7. Curves of Tianjin Type III film.

[0102] 1.2 Fitting Experiment

[0103] The results of the regression parameter error analysis after multiple fitting experiments are shown in Table 8.

[0104] Order n MSE RMSE MAE MAPE R2 4 0.0022 0.0258 0.0212 0.0359 0.9997 5 0.0015 0.0239 0.0208 0.0253 0.9997 6 0.0007 0.0179 0.0148 0.0207 0.9999 7 0.0006 0.0178 0.0150 0.0198 0.0999 8 0.0002 0.0110 0.0089 0.0078 0.9999 9 0.0002 0.0109 0.0090 0.0081 0.9999 10 0.0002 0.0107 0.0092 0.0104 0.9999

[0105] Table 8 Regression error of Tianjin Type III film curve parameters.

[0106] 1.3 Model Determination

[0107] According to Table 8, the Tianjin III film grade i and the corresponding coefficient a are obtained. i The relationship is shown in Table 9 below. Figure 6 As shown, Figure 6 The curves in the figure represent the fitting results for the data in Table 9;

[0108] Order i <![CDATA[Coefficient α i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > 0 -12.0761 3 213.2304 6 -19.0114 1 71.4285 4 -158.3015 7 2.7755 2 -168.7345 5 71.1638 8 -0.1705

[0109] Table 9 Tianjin III film grade i and corresponding coefficient a i .

[0110] 2. Calculation of D7 film, the steps are as follows.

[0111] 2.1 Data Acquisition

[0112] Exposure and density data of D7 film were collected and summarized in Table 10 below:

[0113]

[0114] Table 10. D7 film curves.

[0115] 2.2 Fitting Experiment

[0116] The results of the regression parameter error analysis after multiple fitting experiments are shown in Table 11:

[0117] Order n MSE RMSE MAE MAPE R2 4 0.0264 0.0901 0.0754 0.1345 0.9903 5 0.0038 0.0384 0.0329 0.0580 0.9982 6 0.0007 0.0181 0.0152 0.0256 0.9996 7 0.0002 0.0103 0.0081 0.0151 0.0999 8 0 0.0067 0.0046 0.0057 0.9999 9 0 0.0063 0.0049 0.0086 1.0000 10 0 0.0039 0.0032 0.0073 1.0000

[0118] Table 11 Regression error of D7 film curve parameters.

[0119] 2.3 Model Determination

[0120] According to Table 11, the D7 film order i and the corresponding coefficient a are obtained. i The relationships are shown in Table 12 below. Figure 7 As shown, Figure 7 The curves in the table represent the fitting results for the data in Table 12.

[0121] Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > Order i <![CDATA[Coefficient a i > 0 2.4084 3 -138.3202 6 40.1685 1 -20.0765 4 152.6966 7 -8.7320 2 72.9033 5 -101.5108 8 0.8044

[0122] Table 12 D7 film order i and corresponding coefficient a i .

[0123] This invention uses a computer to calculate film characteristic parameters. By analyzing the original measurement data of different types of film and performing extensive data calculations, a corresponding film characteristic curve model is constructed. The model formula calculation replaces manual methods, which can greatly improve the accuracy of characteristic parameters, reduce human error, and improve detection efficiency.

[0124] The references cited in this invention are as follows:

[0125] [1] Zhang Jianhe, Chen Cunzhu. Comparative determination method and application of characteristic curves of industrial X-ray film [J]. Nondestructive Testing. 2002, 24(9):404-406;

[0126] [2] Zhou Wei. Approximate algorithm for analytical expression of film characteristic curve [J]. Nondestructive Testing, 2000, 22(10):456-457;

[0127] [3] Feng Changchun, Zhou Wei. Practical analytical expression of film characteristic curves [J]. Nondestructive Testing, 1997, (1): 42-44;

[0128] [4] Zheng Shicai (ed.). X-ray Detection [M]. Machinery Industry Press, 2004;

[0129] [5] Qiang Tianpeng (ed.). X-ray Detection [M]. China Labor and Social Security Press, 2006;

[0130] [6] Song Tianmin (ed.). X-ray Detection [M]. China Petrochemical Press, 2011;

[0131] [7] Ye Qixiao, Jiang Qiyuan et al. Mathematical Modeling [M]. Machinery Industry Press, 2009.

[0132] Regarding the specific structure of this invention, it should be noted that the connection relationships between the various component modules used in this invention are definite and achievable. Except as specifically described in the embodiments, their specific connection relationships can bring about corresponding technical effects and solve the technical problems proposed by this invention without relying on the execution of corresponding software programs. The models of the components, modules, and specific components appearing in this invention, the connection methods between them, and the conventional usage methods and expected technical effects brought about by the above technical features, unless specifically described, are all publicly disclosed content in patents, journal articles, technical manuals, technical dictionaries, and textbooks that can be obtained by those skilled in the art before the application date, or belong to conventional technology, common knowledge, and other existing technologies in this field. There is no need to elaborate, which makes the technical solution provided in this case clear, complete, and achievable, and can reproduce or obtain corresponding physical products based on this technical means.

[0133] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions 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 method of optimizing a mathematical model of an industrial film characteristic curve, characterized in that: The method comprises the following steps: S1: collecting data corresponding to exposure and blackness of characteristic curves of different types of industrial films, and collating; S2: Analyzing the characteristics of the industrial film characteristic curve, using n a polynomial regression method to fit, get the mathematical model of the industrial film characteristic curve, the expression of the mathematical model of the industrial film characteristic curve is as follows: ； In the above formulae: D H is the blackness of the negative, H E is the exposure, n P is the polynomial order, a i C is the polynomial coefficient, N N is a natural number, E represents the exposure H logarithmic value of i power; S3: adjusting parameters in the polynomial of the characteristic curve of the industrial film by least square method and regression analysis to obtain an optimized mathematical model of the characteristic curve of the industrial film.

2. The method of claim 1, wherein: The exposure and blackness values of the characteristic curves of different types of industrial films in the step S1 are obtained by manually querying and collating film charts provided by manufacturers, and the industrial film is a non-sensitized film.

3. The method of claim 2, wherein: The characteristic curve of the industrial film in the step S2 is a gradient-increasing inclined arc in a normal exposure area.

4. The method of claim 1, wherein: The optimized mathematical model of the characteristic curve of the industrial film in the step S3 is obtained by repeatedly iterating different orders of the polynomial of the characteristic curve of the industrial film and requiring the curve curvature to be negative and the difference between the curve and the measured value to satisfy a threshold range.

5. The optimization method for a mathematical model of industrial film characteristic curves according to claim 4, characterized in that: The expression of the optimized mathematical model of the characteristic curve of the industrial film is as follows: ； In the above formulae: D k is a calculated result of the film blackness fitting, is a measured value of the film blackness, is a measured value of the exposure amount, D is the film blackness, H is the exposure amount, n, k is the polynomial order, 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 denotes the exposure rate calculated value.

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