## 4 Verification and Evaluation of the Method on Target Particle Statistics Based on Digital Images
### Results Analysis and Method Evaluation  
Three rectangular windows of each digital image were captured at fixed positions. The number of target particles in each window was taken as the value \(N_{2}\), and the observed porosity \(n_{3}^{f}\) was calculated by substituting the calculated value \(N_{2E}\) of the sum of all the window areas into Equation (12). This calculation process was very important, and it was also the basic requirement to meet the consistency of the sample estimation. Figure 8 shows the image processing of working conditions 1-3 of the test experiments, including the original photograph and the image of window 2, the image after greyscale, binarization, and the image after particle statistics and simulation. From Figure 8, it can be seen that there are many black particles obstructed by other particles, such as the black particles shown in the red oval box in the figure, which are clearly obstructed by other particles. These particles were all counted as equal-diameter circular particles. In the statistical analysis process, the method described in Section 4.1 was used to determine the number of sand particles. Table 1 lists the results of four sets of test experiments. Figure 9a shows the relationship between the actual porosity n and \(N_{2}/N_{2E}\). We found that there was an inverse relationship between the two, which is consistent with the inverse relationship shown by Formula (12) derived from the theory. Then, the values of each parameter in Formula (12) are given based on the corresponding physical meanings. \(C_{b}=\overline{\rho}/\rho_{b}=1\) is acceptable, as mentioned above. A mass percentage \(b=14.09\%\) was obtained by testing. Both referring to reference [30] and considering the shape of sand in this test, sphericity \(m=0.85\) is acceptable. For the parameter \(D_{b}\), there is currently no reference, and no experiments have been conducted. If \(D_{b}=2.2\), it can be obtained that \(2m/3C_{b}bD_{b}=1.828\). After analyzing the above parameters, we found that the slope value obtained by Formula (12) is very close to that obtained using the test data in Figure 9. The significance of this value exceeding 2.0 is to indicate that the observable depth is greater than twice the particle size; this aspect is feasible and requires further research. Figure 9b shows the relationship between the actual porosity and the observed porosity; their average could be fitted using a linear relationship with a high degree of correlation, and the value of \(R^{2}\) was 0.92. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Test Group** & **Test** & **Porosity** & **\(N_{2}/N_{2E}\)** & \(n_{3}f\)** & **Test Group** & **Test** & **Porosity** & **\(N_{2}/N_{2E}\)** & \(n_{3}f\)** \\ \hline \multirow{4}{*}{Test 1} & 1 & 0.421 & 0.314 & 0.427 & \multirow{4}{*}{Test 3} & 1 & 0.427 & 0.325 & 0.407 \\ \cline{2-3} \cline{5-8}  & 2 & 0.419 & 0.313 & 0.428 & & 2 & 0.404 & 0.330 & 0.397 \\ \cline{2-3} \cline{5-8}  & 3 & 0.410 & 0.323 & 0.410 & & 3 & 0.396 & 0.339 & 0.380 \\ \cline{2-3} \cline{5-8}  & 4 & 0.404 & 0.324 & 0.408 & & 4 & 0.388 & 0.336 & 0.387 \\ \cline{2-3} \cline{5-8}  & 5 & 0.400 & 0.332 & 0.393 & & 5 & 0.383 & 0.340 & 0.378 \\ \hline \multirow{4}{*}{Test 2} & 1 & 0.410 & 0.321 & 0.412 & & 6 & 0.375 & 0.323 & 0.410 \\ \cline{2-3} \cline{5-8}  & 2 & 0.402 & 0.324 & 0.408 & & 1 & 0.410 & 0.328 & 0.400 \\ \cline{2-3} \cline{5-8}  & 3 & 0.393 & 0.329 & 0.398 & & 2 & 0.402 & 0.335 & 0.387 \\ \cline{2-3} \cline{5-8}  & 4 & 0.385 & 0.337 & 0.385 & & 3 & 0.393 & 0.334 & 0.389 \\ \cline{2-3} \cline{5-8}  & 5 & 0.380 & 0.336 & 0.385 & & 4 & 0.385 & 0.336 & 0.385 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of tests.