Method for testing martensite critical cooling speed of steel
Through a new steel testing method, including heating, cooling, and image processing steps, the rotate cooler cooling speed of steel martensite is accurately determined, and the problems that are not accurate and cannot simulate steel rail scabies in existing technologies. Implement the efficient and accurate test results.
Patent Information
- Application Number
- CN202411511425.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-28
- Publication Date
- 2025-01-28
AI Technical Summary
The prior art is difficult to accurately determine the martensite critical cooling speed of steel, and it is impossible to obtain the upper and lower critical cooling speeds at the same time. The test results are relatively broad, and are greatly affected by human factors, and it is impossible to simulate the rail abrasion conditions.
Adopt a steel -generating critical cooling speed test method of a steel, including preparing steel samples, heating to the preservation point above the phase change critical point to form austenite tissue, use different cooling speed cooling, obtain gold for retracting, calculate the image, calculate Matrheous tissue volume score, fit cooling speed-martensite volume score curve, obtain the upper and lower critical cooling speed.
The martensite critical cooling speed that can obtain steel accurately and quantitatively, including the upper and lower critical cooling speeds, the test method is scientific, simple, and efficient. It can simulate the tissue of the white layer of the steel rail.
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Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of heat treatment of metal materials, and in particular relates to a method for testing the critical cooling rate of martensite of steel. Background Art
[0002] The critical cooling rate of martensite in steel plays an important role in obtaining the desired structure. The critical cooling rate can be divided into the upper critical cooling rate and the lower critical cooling rate. The upper critical cooling rate refers to the minimum cooling rate that ensures that austenite does not decompose and is completely supercooled to the martensite zone during the continuous cooling process. The lower critical cooling rate refers to the maximum cooling rate that ensures that austenite is completely decomposed and does not undergo martensitic transformation during the continuous cooling process. When the cooling rate used in the cooling stage of the heat treatment process is greater than the upper critical cooling rate, a full martensitic structure can be obtained; when the cooling rate used is less than the upper critical cooling rate but greater than the lower critical cooling rate, the obtained structure is pearlite or bainite and martensite, in which the percentage of martensite is related to the cooling rate; when the cooling rate used is less than the lower critical cooling rate, the obtained structure is all pearlite or bainite. In practical engineering applications, the upper and lower critical cooling rates have special significance. For example, when simulating the martensitic white layer produced by rail abrasion, it is necessary to obtain a complete martensitic structure, so it is necessary to test the upper critical cooling rate of the rail steel; during the rail welding process, martensitic structure cannot appear, so it is necessary to test the lower critical cooling rate of the rail steel. Therefore, it is very necessary to measure accurate critical cooling rate data.
[0003] In the field of heat treatment technology of steel, different cooling rates are usually used to approximate the critical cooling rate of steel. For the test method, see YB / T 5128-2018 "Determination of the expansion method of the continuous cooling transformation curve of steel". Taking the above critical cooling rate test method as an example, v1 is first selected as the cooling rate based on experience. When the cooling rate v1 is used, the obtained structure is a small amount of martensite and most of pearlite, then the cooling rate is tentatively increased to v2; when the cooling rate v2 is used, the structure is mostly martensite and a small amount of pearlite, then the cooling rate is tentatively increased to v3; when the cooling rate v3 is used, the structure is mostly martensite and a very small amount of pearlite, then the cooling rate is tentatively increased slightly to v4; when the cooling rate v4 is used, the structure is 100% martensite, then the upper critical cooling rate of the steel is considered to be v3~v4. The lower critical cooling rate test method is similar, and it is also necessary to conduct a series of attempts in combination with the metallographic structure to finally find the lower critical cooling rate range for obtaining 100% pearlite.
[0004] However, this method has the following drawbacks:
[0005] First, this traditional test method can only obtain a critical cooling rate range, and cannot simultaneously obtain the specific values of the upper and lower critical cooling rates. The test results are relatively broad, inaccurate, not quantitative, and not very applicable. In addition, the critical point of the cooling rate needs to be determined through multiple attempts based on experience, which is labor-intensive and greatly affected by human factors.
[0006] Second, the traditional testing method is to fully austenitize the steel, with slow heating speed and long insulation time. The martensitic structure formed is very thick, which cannot simulate the actual rail steel scratch working conditions. Rail scratch is one of the common damages to rails. It is caused by abnormal sliding between wheels and rails. The heat generated by sliding friction causes the surface structure of the rail to austenitize quickly. After the wheel passes, the surface of the rail cools quickly, causing the austenite to transform into a martensitic white layer structure. The whole process only takes a few seconds, the austenitization is not sufficient, and the martensitic white layer formed is very thin, at the micron level. Short-term insulation is very important for the formation of the abrasion martensitic white layer.
[0007] Therefore, a better solution is urgently needed in the prior art to solve this problem. Summary of the invention
[0008] In view of the defects of the prior art, the present invention provides a method for testing the critical cooling rate of martensite of steel. The method of the present invention can obtain accurate and quantitative critical cooling rate of martensite of steel, and can obtain upper and lower critical cooling rates at the same time. At the same time, the method of the present invention can obtain martensite under insufficient austenitization conditions, and can simulate the white layer structure of rail abrasion in the field.
[0009] The technical solution of the present invention is as follows.
[0010] The present invention provides a method for testing the critical cooling rate of martensite of steel, the method comprising:
[0011] 1) Prepare steel samples;
[0012] 2) heating the steel sample prepared in step 1) to above the critical point of phase transformation, and keeping the temperature to form an austenite structure;
[0013] 3) cooling the austenite structure obtained in step 2) at two or more different cooling rates to obtain a cooled austenite structure;
[0014] 4) using the austenite structure cooled in step 3) to obtain metallographic samples at the different cooling rates, and taking metallographic photographs under an optical microscope;
[0015] 5) using the Otsu algorithm to process the metallographic photograph obtained in step 4) to calculate the area fraction of the martensite structure, thereby obtaining the volume fraction of the martensite structure (in this field, generally speaking, the area fraction of the martensite structure is used to approximate the volume fraction of the martensite structure);
[0016] 6) fitting and drawing a cooling rate-martensite volume fraction curve based on the cooling rate and the martensite volume fraction as coordinate axes;
[0017] 7) Obtain the upper and lower critical cooling rates based on the curve in step 6).
[0018] Preferably, in step 1), the steel sample may be an austenitized steel sample, such as a U71MnG rail.
[0019] Preferably, in step 2), the heating rate is 80-120° C. / s.
[0020] Preferably, in step 2), the steel sample prepared in step 1) is heated to 900-1100°C.
[0021] Preferably, in step 2), the insulation time is 1 to 5 seconds.
[0022] Preferably, in step 3), the two or more different cooling rates are independently 4-10° C. / s.
[0023] Preferably, in step 3), the austenite structure obtained in step 2) is cooled at three or more (for example, three) different cooling rates to obtain a cooled austenite structure.
[0024] Preferably, in step 5), the Otsu algorithm is as follows:
[0025] i) converting the metallographic photograph obtained in step 4) into a grayscale photograph in 8-bit format, that is, converting the original photograph with 256-bit color resolution into a grayscale photograph with only 8-bit color resolution;
[0026] ii) counting the number of pixels at the grayscale level of each grayscale histogram of the grayscale photo obtained in step i), and calculating the pixel ratio corresponding to the grayscale level of each grayscale histogram, that is, the area percentage;
[0027] iii) finding the minimum gray value according to the area percentage in step ii), looping from the minimum gray value, calculating the average gray value of the foreground and background when each gray value is used as the threshold, and the pixel ratio of the foreground and background when each gray value is used as the threshold, so as to determine the inter-class variance;
[0028] iv) According to the inter-class variance determined in step iii), the gray value with the largest inter-class variance is selected as the maximum threshold, and the final foreground and background are divided by the maximum threshold to form a two-dimensional image to calculate the area fraction of the martensitic structure, thereby obtaining the volume fraction of the martensitic structure.
[0029] Definition of the maximum threshold: The algorithm traverses all possible thresholds of each grayscale value from 0 to 255. For each grayscale value threshold, the image is divided into two parts (foreground and background) according to this threshold, the inter-class variance of the two parts is calculated, and the grayscale value with the largest inter-class variance is selected as the maximum threshold.
[0030] The process of forming a two-dimensional image by the maximum threshold: According to the maximum threshold, the image is segmented into a binary image, that is, all pixels less than this maximum threshold are set to one color (such as black), and all pixels greater than or equal to this maximum threshold are set to another color (such as white).
[0031] Preferably, in step iii), the inter-class variance is calculated by the following formula:
[0032]
[0033] in, is the inter-class variance corresponding to each gray value t; ω0(t) and ω1(t) respectively represent the pixel proportions of the background and foreground when each gray value is used as the threshold; μ0(t) and μ1(t) respectively represent the average gray value of the background and foreground when each gray value is used as the threshold.
[0034] The Otsu algorithm is an algorithm that automatically sets the image threshold. It determines the maximum threshold through the inter-class variance, thereby converting the grayscale image into a two-dimensional image to calculate the grayscale area percentage (that is, it can be considered as the volume fraction of the martensitic structure).
[0035] In a specific embodiment, the Otsu algorithm of the present invention is as follows: first, the metallographic photograph is converted into a grayscale photograph; then, the histogram of the grayscale image is calculated, the number of pixels of the grayscale level of each grayscale histogram is counted, and the pixel ratio corresponding to the grayscale level of each grayscale histogram is calculated; then, starting from the minimum grayscale value, the average grayscale value of the foreground and background when each grayscale value is used as the threshold, and the pixel ratio of the foreground and background when each grayscale value is used as the threshold are calculated to determine the inter-class variance; then, according to the inter-class variance, the grayscale level corresponding to the maximum value is selected as the maximum threshold, and all pixels in the image with grayscale values greater than the maximum threshold are marked as foreground, and the remaining pixels are marked as background, thereby forming a two-dimensional image to calculate the area fraction of the martensitic structure, thereby obtaining the volume fraction of the martensitic structure.
[0036] Preferably, after determining the upper and lower critical cooling rates, the accuracy of the upper and lower critical cooling rates may be further verified to ultimately determine the upper and lower critical cooling rates.
[0037] Compared with the prior art, the present invention has at least the following beneficial technical effects:
[0038] The invention can obtain accurate and quantitative critical cooling speed of martensite of steel, and can obtain upper and lower critical cooling speeds simultaneously.
[0039] The testing method of the present invention is scientific, simple and efficient, and the testing results are accurate and quantifiable.
[0040] In view of the problems in the prior art that the critical cooling rate is tested in a wide range, is greatly affected by human factors, and has low applicability, the present invention, based on experiments, adopts a method of data fitting and then substituting it into a solution to obtain accurate critical cooling rate data.
[0041] In view of the problem that the existing technology for testing the upper and lower critical cooling speeds requires repeated "heating-insulation-cooling-metallographic structure observation" experiments, which is a large workload, the present invention adopts a method of data fitting and then substituting the data into a solution, which only requires at least two experiments. BRIEF DESCRIPTION OF THE DRAWINGS
[0042] Figure 1 It is a flow chart of the method for testing the critical cooling rate of martensite of steel proposed by the present invention;
[0043] Figure 2 Metallographic photographs of the sample of Example 1 before and after image processing;
[0044] Figure 3 It is the fitting curve diagram of the sample of Example 1;
[0045] Figure 4 Metallographic photographs of the sample of Example 2 before and after image processing;
[0046] Figure 5 It is the fitting curve diagram of the sample of Example 2;
[0047] Figure 6 Metallographic photographs of the sample of Example 3 before and after image processing;
[0048] Figure 7 It is the fitting curve diagram of the sample of Example 3;
[0049] Figure 8 This is a metallographic photograph of the sample of Comparative Example 1. DETAILED DESCRIPTION
[0050] The present invention is further described below in conjunction with specific examples and accompanying drawings, but the embodiments do not limit the present invention in any form. Unless otherwise specified, the reagents, methods and equipment used in the present invention are conventional reagents, methods and equipment in the art. Unless otherwise specified, the reagents and materials used in the present invention are commercially available.
[0051] Embodiment 1: Martensite critical cooling rate testing method of the present invention
[0052] like Figure 1 FIG. 1 is a flow chart of a method for testing the critical cooling rate of martensite according to the present invention. The method comprises:
[0053] 1) Prepare U71MnG rail samples;
[0054] 2) The U71MnG rail sample prepared in step 1) was heated to above the phase transformation critical point using a Formastor-FII dilatometer, and then kept warm to form an austenite structure. The specific process parameters are shown in Table 1;
[0055] 3) cooling the austenite structure obtained in step 2) at three different cooling rates to obtain a cooled austenite structure. The specific cooling rates are shown in Table 1;
[0056] 4) using the austenite structure cooled in step 3) to obtain metallographic samples at different cooling rates, and taking metallographic photographs under an optical microscope;
[0057] 5) The metallographic photographs obtained in step 4) are processed using the Otsu algorithm to calculate the volume fraction of the martensite structure; the Otsu algorithm is an algorithm that automatically sets the image threshold, and determines the optimal grayscale threshold by maximizing the inter-class variance, thereby converting the grayscale image into a two-dimensional image to calculate the grayscale area percentage, which is the volume percentage. See the photos before and after image processing. Figure 2 The obtained martensite volume fractions are 11.70%, 34.40% and 89.51% respectively.
[0058] 6) With the cooling rate and the volume fraction of martensite structure as the coordinate axes, a cooling rate-martensite volume fraction curve was fitted and drawn; that is, with the cooling rate as the abscissa and the martensite volume fraction as the ordinate, three groups of data points (4.89, 11.70), (5.87, 34.40), and (8.80, 89.51) were linearly fitted to obtain a fitting curve y=19.65x-82.91. The fitting curve is shown in FIG. Figure 3 .
[0059] 7) Obtain the upper and lower critical cooling rates according to the curve of step 6). Specifically, intersect the fitting curve with the curve y=100 to obtain the intersection coordinates (9.31, 100), then 9.31°C / s is the upper critical cooling rate. Intersect the fitting curve with the curve y=0 to obtain the intersection coordinates (4.22, 0), then 4.22°C / s is the lower critical cooling rate.
[0060] Table 1 Heat treatment process parameters of U71MnG rail material
[0061]
[0062] 8) Other samples of U71MnG rail, such as sample 4, were cooled at a cooling rate of 9.31° C. / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% martensite.
[0063] Other samples of U71MnG rail, such as sample 5, were cooled at a cooling rate of 4.22°C / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% pearlite.
[0064] It is thus determined that the upper critical cooling rate of the tested U71MnG rail is 9.31°C / s, and the lower critical cooling rate is 4.22°C / s. Verification has proven that these values are accurate and reliable.
[0065] Embodiment 2: Martensite critical cooling rate testing method of the present invention
[0066] like Figure 1 FIG. 1 is a flow chart of a method for testing the critical cooling rate of martensite according to the present invention. The method comprises:
[0067] 1) Prepare U71Mn rail specimens;
[0068] 2) The U71Mn rail sample prepared in step 1) was heated to above the phase transformation critical point using a Formastor-FII dilatometer, and then kept warm to form an austenite structure. The specific process parameters are shown in Table 2;
[0069] 3) cooling the austenite structure obtained in step 2) at three different cooling rates to obtain a cooled austenite structure. The specific cooling rates are shown in Table 2;
[0070] 4) using the austenite structure cooled in step 3) to obtain metallographic samples at different cooling rates, and taking metallographic photographs under an optical microscope;
[0071] 5) The metallographic photographs obtained in step 4) are processed using the Otsu algorithm to calculate the volume fraction of the martensite structure; the Otsu algorithm is an algorithm that automatically sets the image threshold, and determines the optimal grayscale threshold by maximizing the inter-class variance, thereby converting the grayscale image into a two-dimensional image to calculate the grayscale area percentage, which is the volume percentage. See the photos before and after image processing. Figure 4 As shown in the figure, the volume fractions of martensite obtained are 27.96%, 47.74% and 82.34% respectively.
[0072] 6) With the cooling rate and the volume fraction of martensite structure as the coordinate axes, a cooling rate-martensite volume fraction curve was fitted and drawn; that is, with the cooling rate as the abscissa and the martensite volume fraction as the ordinate, three groups of data points (5, 27.96), (6, 47.74), and (7, 82.34) were linearly fitted to obtain a fitting curve y=27.19x-110.46. The fitting curve is shown in FIG. Figure 5 .
[0073] 7) Obtain the upper and lower critical cooling rates according to the curve of step 6). Specifically, intersect the fitting curve with the curve y=100 to obtain the intersection point coordinates (7.74, 100), then 7.74°C / s is the upper critical cooling rate. Intersect the fitting curve with the curve y=0 to obtain the intersection point coordinates (4.06, 0), then 4.06°C / s is the lower critical cooling rate.
[0074] Table 2 Heat treatment process parameters of U71Mn rail material
[0075]
[0076] 8) Other samples of U71Mn rail, such as sample 4, were cooled at a cooling rate of 7.74° C. / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% martensite.
[0077] Other samples of U71Mn rail, such as sample 5, were cooled at a cooling rate of 4.06°C / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% pearlite.
[0078] It is thus determined that the upper critical cooling rate of the tested U71Mn rail is 7.74℃ / s, and the lower critical cooling rate is 4.06℃ / s. Verification has proven that these values are accurate and reliable.
[0079] Embodiment 3: Martensite critical cooling rate testing method of the present invention
[0080] like Figure 1FIG. 1 is a flow chart of a method for testing the critical cooling rate of martensite according to the present invention. The method comprises:
[0081] 1) Prepare U71Mn rail specimens;
[0082] 2) The U71Mn rail sample prepared in step 1) was heated to above the phase transformation critical point using a Formastor-FII dilatometer, and then kept warm to form an austenite structure. The specific process parameters are shown in Table 3;
[0083] 3) cooling the austenite structure obtained in step 2) at three different cooling rates to obtain a cooled austenite structure. The specific cooling rates are shown in Table 3;
[0084] 4) using the austenite structure cooled in step 3) to obtain metallographic samples at different cooling rates, and taking metallographic photographs under an optical microscope;
[0085] 5) The metallographic photographs obtained in step 4) are processed using the Otsu algorithm to calculate the volume fraction of the martensite structure; the Otsu algorithm is an algorithm that automatically sets the image threshold, and determines the optimal grayscale threshold by maximizing the inter-class variance, thereby converting the grayscale image into a two-dimensional image to calculate the grayscale area percentage, which is the volume percentage. See the photos before and after image processing. Figure 6 As shown in the figure, the obtained martensite volume fractions are 21.54%, 56.61% and 91.68% respectively.
[0086] 6) With the cooling rate and the volume fraction of martensite as coordinate axes, a cooling rate-martensite volume fraction curve was fitted and drawn; that is, with the cooling rate as the abscissa and the martensite volume fraction as the ordinate, three groups of data points (6, 21.54), (7, 56.61), and (10, 91.68) were linearly fitted to obtain a fitting curve y=16.19x-67.48. The fitting curve is shown in FIG. Figure 7 .
[0087] 7) Obtain the upper and lower critical cooling rates according to the curve of step 6). Specifically, intersect the fitting curve with the curve y=100 to obtain the intersection coordinates (10.34, 100), and 10.34°C / s is the upper critical cooling rate. Intersect the fitting curve with the curve y=0 to obtain the intersection coordinates (4.17, 0), and 4.17°C / s is the lower critical cooling rate.
[0088] Table 3 Heat treatment process parameters of U71Mn rail material
[0089]
[0090] 8) Other samples of U71Mn rail, such as sample 4, were cooled at a cooling rate of 10.34° C. / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% martensite.
[0091] Other samples of U71Mn rail, such as sample 5, were cooled at a cooling rate of 4.17°C / s, and then the samples were taken out to prepare metallographic samples. Metallographic photographs were taken under an optical microscope, and the structure was 100% pearlite.
[0092] It is thus determined that the upper critical cooling rate of the tested U71Mn rail is 10.34°C / s, and the lower critical cooling rate is 4.17°C / s. Verification has proven that these values are accurate and reliable.
[0093] Comparative Example 1: Test method for critical cooling rate of martensite
[0094] in accordance with Figure 1 The test is performed by the martensite critical cooling rate test method shown in the figure. The method specifically comprises:
[0095] 1) Prepare U71Mn rail specimens;
[0096] 2) The U71Mn rail sample prepared in step 1) was heated to above the phase transformation critical point using a Formastor-FII dilatometer, and then kept warm to form an austenite structure. The specific process parameters are shown in Table 4;
[0097] 3) cooling the austenite structure obtained in step 2) at three different cooling rates to obtain a cooled austenite structure. The specific cooling rates are shown in Table 3;
[0098] 4) Using the austenite structure cooled in step 3), metallographic samples at different cooling rates are obtained, and metallographic photographs are taken under an optical microscope. Figure 8 As shown;
[0099] according to Figure 8 It can be clearly seen that the metallographic photographs at the three cooling rates are all white martensite, that is, the volume fraction of martensite is 100%.
[0100] Table 4 Heat treatment process parameters of U71Mn rail material
[0101]
[0102]
[0103] In summary, using this heat treatment parameter, a mixture of pearlite and martensite cannot be obtained, and the critical cooling rate of martensite cannot be obtained.
Claims
1. A method for testing the critical cooling rate of martensite of steel, the method comprising: 1) Prepare steel samples; 2) heating the steel sample prepared in step 1) to above the critical point of phase transformation, and keeping the temperature to form an austenite structure; 3) cooling the austenite structure obtained in step 2) at two or more different cooling rates to obtain a cooled austenite structure; 4) using the austenite structure cooled in step 3) to obtain metallographic samples at the different cooling rates, and taking metallographic photographs under an optical microscope; 5) using the Otsu algorithm to process the metallographic photograph obtained in step 4) to calculate the volume fraction of the martensite structure, thereby obtaining the volume fraction of martensite; 6) fitting and drawing a cooling rate-martensite volume fraction curve based on the cooling rate and the martensite volume fraction as coordinate axes; 7) Obtain the upper and lower critical cooling rates based on the curve in step 6).
2. The method according to claim 1, wherein: In step 1), the steel sample may be an austenitized steel sample, such as a U71MnG rail.
3. The method according to claim 1 or 2, wherein: In step 2), the heating speed is 80-120° C. / s.
4. The method according to any one of claims 1 to 3, wherein: In step 2), the steel sample prepared in step 1) is heated to 900-1100°C.
5. The method according to any one of claims 1 to 4, wherein: In step 2), the insulation time is 1 to 5 seconds.
6. The method according to any one of claims 1 to 5, wherein: In step 3), the two or more different cooling rates are independently 4-10° C. / s.
7. The method according to any one of claims 1 to 6, wherein: In step 3), the austenite structure obtained in step 2) is cooled at three or more different cooling rates to obtain a cooled austenite structure.
8. The method according to any one of claims 1 to 7, wherein: In step 5), the Otsu algorithm is as follows: i) converting the metallographic photograph obtained in step 4) into a grayscale photograph in 8-bit format, that is, converting the original photograph with 256-bit color resolution into a grayscale photograph with only 8-bit color resolution; ii) counting the number of pixels at the grayscale level of each grayscale histogram of the grayscale photo obtained in step i), and calculating the pixel ratio corresponding to the grayscale level of each grayscale histogram, that is, the area percentage; iii) finding the minimum gray value according to the area percentage in step ii), looping from the minimum gray value, calculating the average gray value of the foreground and background when each gray value is used as the threshold, and the pixel ratio of the foreground and background when each gray value is used as the threshold, so as to determine the inter-class variance; iv) According to the inter-class variance determined in step iii), the gray value with the largest inter-class variance is selected as the maximum threshold, and the final foreground and background are divided by the maximum threshold to form a two-dimensional image to calculate the area fraction of the martensitic structure, thereby obtaining the volume fraction of the martensitic structure.
9. The method according to any one of claims 1 to 8, wherein: In step iii), the inter-class variance is calculated by the following formula: in, is the inter-class variance under each gray value threshold t; ω0(t) and ω1(t) respectively represent the pixel proportions of the background and foreground when each gray value is used as the threshold; μ0(t) and μ1(t) respectively represent the average gray value of the background and foreground when each gray value is used as the threshold.