Predicting and Analyzing Rock Mechanical Properties Using Image Processing Techniques

Figure 1 shows the technical roadmap of this research. This paper first studies the segmentation of rock microscopic images for predicting mechanical properties. The minimum threshold method, commonly used for image segmentation, is based on selecting a threshold value to divide the image pixels into target and background, which is simple and requires low computational effort. Due to the complexity of rock microstructures, the minimum threshold method can serve as a preliminary segmentation means to quickly delineate general areas, providing a basis for subsequent finer analysis. The maximum interclass variance method (also known as the Otsu method) is an adaptive image segmentation method that automatically calculates the optimal threshold by a statistical method to maximize the variance between the two categories (target and background) after segmentation. In the processing of rock microscopic images, the maximum interclass variance method can precisely separate different components of the rock, especially when the boundaries of rock particles are not very clear.

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Figure 1. Technical roadmap of this research

2.1 Basic structure segmentation of rocks

For the implementation of the minimum threshold method in rock microscopic image segmentation, the first step is to convert the rock microscopic images to grayscale, extract the grayscale histogram, and analyze its distribution. Then, using the minimum threshold method to process the grayscale histogram, and calculate the optimal threshold. This value should maximally differentiate the rock microstructure from the non-structural background. To enhance the segmentation effect of microstructures, the Laplacian histogram method can be further applied to intensify details. This method uses the Laplace operator to highlight the edges of the image, making the micro-fine structures within the rock, such as cracks and pores, more apparent. Assume the threshold is $\varphi$, the average gray level distribution of the object is represented by $\alpha$, the background average gray level is represented by x, and the standard deviation of the normal distribution probability density w(c) is represented by e. Calculate α and x, then calculate e. Compare e with the threshold $\varphi$, and if $\varphi$ ≤ e, then increase the value. Repeat the operation until $\varphi$ > e. The probability distribution functions are represented by O(e) and o(e), with a schematic diagram shown in Figure 2, and the derivation formula is as follows:

$o(e)=\frac{d o(e)}{d e}$    (1)

$O(e)=\int_0^e o(e) d e$       (2)

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Figure 2. Schematic diagram of the probability density function

The Laplacian histogram method enhances the parts with sharp grayscale changes through the second derivative of the image, which usually corresponds to the edges or cracks in the rock structure. The result of the second derivative is to reduce the grayscale values in smoother areas, while producing high values in high-frequency details, such as edges and textures. After taking the absolute value, the constructed grayscale histogram exhibits a distinct bimodal characteristic in edge areas, making the possibly inconspicuous details of rock structures prominent. Combined with the minimum threshold method, effective segmentation of rock edge features can be achieved. By analyzing the bimodal distribution of the Laplacian histogram and calculating the optimal threshold, the minimum threshold method can automatically differentiate the basic structure of the rock from the background. When the image contrast is high, traditional global threshold methods may not effectively segment all key features, whereas the combination of local feature enhancement with the Laplacian histogram and global analysis with the minimum threshold method can achieve more accurate and robust segmentation results.

In digital image processing, an image is essentially a discrete data structure composed of a finite set of pixel points, rather than a continuous entity. Each pixel point has one or more corresponding numerical values that represent different colors or grayscale levels. Therefore, unlike traditional continuous functions, digital images cannot directly apply calculus operations from the continuous domain, such as differentiation and integration. When segmenting the basic structure of rocks, the Laplacian histogram method involves the second derivative of the image, which is clearly defined in continuous mathematics. However, in discrete image processing, it is necessary to introduce discrete forms of differential operators to approximate the effects of continuous differentiation.

Specifically, let the image grayscale levels be 1 to l, and the number of pixels with grayscale value e be v, the total number of pixels can be calculated by the following formula:

$V=\sum_{e=1}^l v_e$     (3)

The following formula calculates the probability of each grayscale level:

$o_e=\frac{v_e}{V}$       (4)

Then, the image is divided into foreground and background, corresponding to $X_1=\{1 \sim j\}$ and $X_2=\{j+1 \sim l\}$. Assuming the entire image's average grayscale value is represented by $i=\sum_{e=1}^l O_e$, the following formulas calculate the probabilities of the two parts:

The Probability of $X_1 \quad X_1 q_1=\sum_{e=1}^j o_e=q(j)$         (5)

The Probability of $X_2 \quad q_2=\sum_{e=j+1}^l o_e=1-q(j)$        (6)

The Average Value of $X_1 \quad \beta=\sum_{e=1}^j \frac{e o_e}{q_1}=\frac{i(j)}{q(j)}$     (7)

The Average Value of $X_2 \quad \alpha=\sum_{e=j+1}^l \frac{e o_e}{q_2}=\frac{i-i(j)}{1-q(j)}$     (8)

The average of the sums of the two groups of average grayscale values can be calculated by the following formula:

$\varepsilon=\frac{x+\alpha}{2}$     (9)

If $e \leq \varepsilon 1$, then continue the above calculations until $e>\varepsilon$, at which point $e$ equals the sought threshold $\varphi$.

2.2 Microstructure segmentation within rocks

The maximum interclass variance method (also known as the Otsu method) is used in image processing for automatic threshold selection. Its main advantage is the effective differentiation between the foreground and background in an image, making it particularly suitable for distinguishing microstructures within rocks. This method determines the optimal threshold by maximizing the variance between the foreground and background, thereby achieving precise segmentation of rock structures. Specifically, first, the color rock image is converted to a grayscale image for easier processing. Then, all possible thresholds are calculated for the entire image and the interclass variance is computed for each threshold, and the threshold that maximizes the interclass variance is selected as the final segmentation threshold. Finally, apply this threshold to binarize the grayscale image, thus obtaining the segmented microstructure image of the rock.

Firstly, to measure the distinction between foreground and background under different thresholds, it is necessary to calculate the average grayscale value for each area in the image. In rock image processing, this paper divides the areas into foreground (the microstructures of the rock) and background. For each possible threshold, the image is divided into two parts, and the sum of the pixel values for each part is divided by the number of pixels to obtain the average grayscale values for the foreground and background under that threshold. Assume the number of pixels with grayscale value u is represented by v, the range of grayscale levels by $D=(0,1,2, \ldots, M-1)$, and the total number of pixels by $V=\sum^{M-1}{ }_{u=0} v_u$, with the grayscale threshold represented by S, and the probability of each grayscale occurring represented by $O_u$, where $O_u=v_u / V$. The image is divided into two regions, X and Y, with the probabilities of occurrence being $O_X=\sum^S{ }_{u=0} O_u, O_Y=\sum^{M-1}{ }_{u=S+1} O_u$. The following formulas calculate the average grayscale values $\gamma_X$ and $\gamma_Y$ of regions $X$ and $Y$:

$\gamma_X=\sum_{u=0}^S \frac{uo_U}{O_X}$         (10)

$\gamma_Y=\sum_{\mathrm{u}=\mathrm{S}+1}^{M-1} \frac{\mathrm{u} o_{\mathrm{u}}}{\mathrm{o}_{\mathrm{Y}}}$     (11)

Then, to provide a measure of the dispersion of pixel grayscale values within each area, further understanding how different thresholds will affect the separation effect of rock microstructure from the background, variance for each area is calculated, that is, the average of the squares of the differences between each pixel's grayscale value and the average grayscale value of that area. The following formulas calculate the variances $\delta^2{ }_X$ and $\delta^2{ }_Y$ of regions $X$ and $Y$:

$\delta_X^2=\sum_{u=0}^S \frac{\left(u-\gamma_X\right)^2 o_u}{o_X}$       (12)

$\delta_Y^2=\sum_{u=0}^S \frac{\left(u-\gamma_Y\right)^2 o_u}{o_Y}$          (13)

The calculation of the overall average grayscale involves calculating the average grayscale value for the entire image. This is done by summing all pixel grayscale values and dividing by the total number of pixels. The following formula gives the calculation for the overall average grayscale: 

$\gamma=\sum_{X=0}^{M-1} u O_u=\gamma_X O_X+\gamma_Y O_Y$         (14)

Interclass variance refers to the size of the difference between the average grayscale values of the foreground and background. For each possible threshold, calculate the average grayscale values of the foreground and background, then square the difference between these two averages. The larger this value, the more distinct the separation between foreground and background. Thus, the threshold point with the maximum interclass variance is considered the ideal segmentation point. The interclass variance for regions X and Y can be calculated by the following formula: 

$\delta_j^2=O_X\left(\gamma_X-\gamma\right)^2-O_Y\left(\gamma_Y-\gamma\right)^2$        (15)

Intraclass variance refers to the average of the squares of the differences between each pixel's grayscale value and the average grayscale value of that area within the same region. The calculation of intraclass variance indicates the similarity among pixels within the same area (foreground or background). The smaller the intraclass variance, the more consistent the pixel grayscale values within the same area, indicating higher homogeneity of that area. The intraclass variance for regions X and Y can be calculated by the following formula:

$\begin{aligned} & \delta_q^2=O_X \delta_X^2+O_Y \delta_Y^2 \\ & =\sum_{u=0}^S \frac{\left(u-\gamma_X\right)^2 o_u}{O_X}+\sum_{u=S+1}^{M-1} \frac{\left(u-\gamma_Y\right)^2 o_u}{O_Y}\end{aligned}$         (16)

After calculating the intraclass and interclass variances for all possible thresholds, the total variance of the image can be computed, which measures the dispersion of all pixel grayscale values across the entire image. The total variance is a fixed value, not changing with the threshold. In the maximum interclass variance method, the interclass variance is maximized while keeping the total variance constant. Therefore, when the interclass variance is maximized, it means the differentiation between the foreground and background is maximized, and the consistency within each area is also ensured. The interclass variance coefficient is represented by $\sigma \delta_j^2$, with the corresponding optimal value represented by S_MAX, the following formula gives the calculation for the total variance:

$\sigma_T^2=\sigma_k^2+\sigma_w^2$      (17)

Once the basic structure and microstructure of the rock are successfully segmented using the maximum interclass variance method, these segmentation results are further used to predict the mechanical properties of the rock. This usually involves quantifying the segmentation results and extracting key features, such as porosity, crack density, particle size distribution, and shape parameters. These features are closely related to the mechanical behavior of the rock, thus establishing the relationship between microstructural features and macroscopic mechanical properties.

From Tables 1, 2, and 3, we can observe the performance of various rock micro-image segmentation methods under different standard deviations. The standard deviation represents the level of noise in the image, simulating different degrees of image quality variations encountered in actual situations. Across all three tables, the segmentation method proposed in this paper demonstrates significant superiority in three metrics measuring segmentation quality (MCR, PSNR, EPM). A lower MCR (Misclassification Rate) indicates a smaller proportion of misclassified pixels, a higher PSNR (Peak Signal-to-Noise Ratio) indicates better image quality, and a lower EPM (Error Pixel Mapping) indicates a smaller proportion of incorrectly segmented pixels. The values for MCR and EPM with the proposed method are significantly lower than those for CycleGAN and GrabCut, and its PSNR values are substantially higher than the other two methods. This indicates that the proposed method provides more accurate segmentation, maintaining robust performance even at higher noise levels. Analyzing the above experimental results, we can conclude that the rock micro-image segmentation method proposed in this paper significantly outperforms the CycleGAN and GrabCut methods in terms of noise resistance and segmentation accuracy. By combining the minimum threshold method and the Laplacian histogram method to preliminarily process the basic structure of the rock, and then using the maximum interclass variance method to refine the segmentation of the microstructure, the proposed method effectively enhances segmentation accuracy. Even under gradually increasing noise levels, the proposed method still maintains a lower misclassification rate and error pixel mapping, as well as a higher peak signal-to-noise ratio, emphasizing its effectiveness and practicality in rock micro-image processing.

Table 1. Comparison of MCR among different rock micro-image segmentation methods

Table 2. Comparison of PSNR among different rock micro-image segmentation methods

Table 3. Comparison of EPM among different rock micro-image segmentation methods

Standard Deviation	10	11	12	13	14	15
CycleGAN	0.6235	0.6147	0.6258	0.6125	0.6235	0.5895
GrabCut	0.9147	0.8795	0.8326	0.8145	0.7458	0.7156
Proposed Segmentation Method	37.9854	36.1245	35.6985	33.1245	32.5689	31.2547

Table 4. Data comparison of porosity calculations for rock samples by different methods

Rock Sample Number	Morphological Analysis Method	The Proposed Method	Rock Sample Number	Morphological Analysis Method	The Proposed Method
1	1.85	6.63	18	2.15	11.23
2	1.69	7.12	19	2.23	12.15
3	1.73	6.89	20	2.35	6.78
4	1.82	6.89	21	4.56	8.23
5	1.63	6.87	22	4.62	8.21
6	1.67	6.78	23	1.57	7.78
7	1.59	6.89	24	4.88	9.21
8	1.71	6.45	25	2.12	8.12
9	1.82	6.89	26	1.39	8.62
10	1.71	6.32	27	2.14	6.78
11	1.12	4.89	28	2.23	6.69
12	0.94	4.32	29	1.65	8.69
13	0.82	3.79	30	2.18	8.89
14	0.81	3.76
15	1.12	4.23
16	1.23	5.23
17	1.69	6.35

Table 4 presents a comparison of porosity data for rock samples calculated by two different methods, covering 35 rock samples with different numbers. Comparing the porosity data obtained by the morphological analysis method and the proposed method, it is evident that the porosity calculated by the proposed method is generally higher than that obtained by the morphological analysis method. The porosity data calculated by the proposed method show larger values, meaning that the proposed method is more sensitive and precise in identifying and calculating pore spaces. Especially in samples with high porosity, the difference between the proposed method and the morphological analysis method is more pronounced, indicating that the proposed method has better resolution in rock samples with higher porosity.

From the comprehensive analysis of these data, we can conclude that the porosity calculation method proposed in this paper is more effective in extracting microstructural parameters, especially when dealing with samples with complex pore structures. The proposed method employs more advanced image analysis technology or introduces more detailed parameter considerations, thus providing more comprehensive and accurate porosity measurements.

Table 5. Analysis results of rock porosity, maximum particle area, and particle size variance regression analysis

	df	SS	MS	F	Significance F
Regression Analysis	3	221.235	71.23458	31.15489	7.25468E-06
Residuals	12	27.65415	2.356248
Total	14	235.2356
	Coeffcients	Standard Deviation	tStat	P-value	Lower95%
Intercept	6.78546245	3.526345	1.887951	0.081457	-1.123257485
Maximum Void Diameter	-0.0087956	0.01526	-0.51264	0.612358	-0.044512568
Particle Area	0.03624589	0.00658	5.523154	0.000124	0.021458579
Particle Size	-0.001236	0.000514	-2.64259	0.017895	-0.002315487

Table 6. Regression statistical results of rock porosity with maximum particle area and particle size

Regression Statistics
Multiple R	0.92358795
R Square	0.87451263
Adjusted R Square	0.84521685
Standard Error	1.53264859
Observations	15

From the variance analysis results in Table 5, it can be seen that the significance F value of the regression model is 7.25468E-06, far less than the commonly used significance level of 0.05, indicating that the model's regression effect is significant, meaning at least one predictive variable in the model is effective for predicting rock porosity. Specifically, the P value for the maximum particle area is 0.000124, far less than 0.05, indicating a significant positive impact on porosity, while the P value for particle size is 0.017895, also less than 0.05, indicating a significant negative impact on porosity. The P value for the maximum void diameter is higher, indicating its impact on porosity is not significant. Table 6 further reveals the regression statistical results, where the R Square value is 0.8745, indicating that about 87.45% of the variation in porosity can be explained by the predictive variables in the model, showing strong explanatory power of the model.

Integrating these analysis results, we can conclude that the rock microstructural parameter extraction method and porosity prediction model proposed in this paper are effective. The maximum particle area, as a positive influencing factor, has a significant correlation with rock porosity, and particle size, as a negative influencing factor, also has a significant correlation with porosity. These findings validate the complex relationship between microstructural parameters and rock mechanical properties, and emphasize the importance of these parameters in predicting and understanding rock porosity.

Further, this paper analyzes the fractal characteristics of porosity using core samples and explores the relationship between the pore fractal dimension and rock mechanical properties. Comparing rock samples under core classification with their fractal dimensions in Figure 3, a consistency between them is found, indicating that fractal dimension can serve as an indicator to identify and differentiate the pore structure characteristics of different rock samples. However, the content of rock samples and fractal dimension do not show a simple positive correlation, indicating that rock sample content is not the sole factor controlling pore distribution. The development of pores in rocks is affected by multiple factors, such as mineral composition, rock genesis, stress conditions, etc. Moreover, when ignoring the precondition of rock sample classification, the relationship between rock sample content and fractal dimension has a low degree of linear fit, indicating no apparent correlation without considering specific rock sample conditions. From these experimental results, it can be concluded that the complexity of rock microstructure leads to the relationship between porosity and rock mechanical properties not being fully described by linearity or a single variable. Fractal dimension, as a quantitative indicator of pore structure complexity, shows a certain degree of correlation with rock mechanical properties under the context of rock samples, but this correlation is influenced by the combined effects of rock type, composition, and other factors affecting pore development.

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Figure 3. Relationship between different rock samples and fractal dimension

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Figure 4. Relationship between rock sample porosity and elastic modulus

In the experimental research of this paper, by meticulously extracting rock microstructural parameters such as particle size, particle size distribution, and total area, a discussion on the relationship between rock mechanical properties and microstructure was conducted. The experimental data in Figure 4 shows that there is a certain relationship between total porosity and the rock's elastic modulus. Specifically, before the experiment, the correlation coefficient R2 between total porosity and elastic modulus was only 0.0047, indicating almost no correlation. However, after the experiment, the correlation coefficient significantly increased to R2=0.455, suggesting the experimental process affected the rock microstructure, thereby altering its elastic modulus. This lower but significant correlation coefficient hints that although there is a correlation between porosity and elastic modulus, the total porosity of the rock is not the sole factor affecting the elastic modulus, and the rock's macroscopic stratification structure and other characteristics also contribute.

Further analysis shows that rock porosity has a negative correlation with its elastic modulus: as porosity increases, the elastic modulus gradually decreases, which is reflected in the linear relationship in Figure 4-2. This negative correlation reveals an important mechanical behavior, that rocks with higher porosity exhibit more plastic characteristics under stress, reflecting a decrease in the rock's elastic capabilities. This finding is significant for predicting and analyzing the mechanical response of rock materials in engineering applications, emphasizing the need to comprehensively consider microstructural characteristics, especially porosity, when assessing the structural integrity and long-term stability of rock materials.

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Figure 5. Relationship between rock sample porosity and Poisson's ratio

In this study, an in-depth analysis of rock microstructure, including particle size, total area, and dimensions, was performed, further exploring the relationship between these microstructural parameters and rock mechanical properties, particularly Poisson's ratio. By comparing changes in Poisson's ratio before and after the experiment, a significant impact of total porosity on Poisson's ratio was observed. The correlation coefficient for Poisson's ratio increased from R2=0.1079 to R2=0.2811 before and after the experiment, showing a clear upward trend. This indicates that, although the correlation coefficient is not high, the impact of total porosity on Poisson's ratio is significant during the experimental process. Importantly, it is recognized that porosity is just one of many influencing factors; rock's strain characteristics are not only affected by the internal pore structure but also closely related to the content and distribution of mineral components.

Continuing to analyze the relationship between Poisson's ratio and porosity, experimental results revealed a key indicator of rock plastic characteristics: as porosity increases, Poisson's ratio also tends to rise, clearly demonstrated in Figure 5-2. A higher Poisson's ratio means the rock's lateral expansion capability under vertical compression is enhanced, indicating stronger plasticity of the rock. This trend is crucial for understanding the deformation behavior of rocks under stress. Therefore, it can be concluded that an increase in porosity leads to a rise in Poisson's ratio, thereby affecting the rock's plastic response. In predicting and analyzing rock mechanical properties, considering microstructural parameters like porosity is indispensable for accurately predicting the behavior of rock in engineering applications.

This paper has made significant progress in exploring the relationship between rock microstructure analysis and rock mechanical properties. Firstly, the research successfully preliminarily segmented the basic structure of rocks using the minimum threshold method and Laplacian histogram method, followed by a more refined subdivision of the rock's internal microstructure using the maximum interclass variance method, greatly improving the accuracy of rock micro-image segmentation. The effective combination of these methods not only enhanced the identification of rock microstructure but also laid a solid foundation for subsequent structural parameter extraction.