## 3 Results
### Prediction by Multiple Regression Analysis  
Figure 4 shows the correlation matrix of 18 soil surface image features. A number of features showed a very strong relationship, such as Value and Gray, Dissimilarity and Contrast, and H0 and H1. Hue, Sat, Value, Gray, and Skewness, which are color features, showed high correlation with H0 and H1 among shape features. Texture features showed low correlation with other features and relatively high correlation between texture features.  
Figure 3: Relationship between soil properties and features from soil surface images: (**a**) coefficient of correlation between water content and features; (**b**) coefficient of correlation between bulk density and features.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Source** & **Feature** & **Min** & **Mean** & **Max** & **Range** \\ \hline \multirow{4}{*}{Color features} & \multirow{4}{*}{Color image} & Hue & \(12\) & \(14.6\) & \(24.6\) & \(12.6\) \\  & & Sat & \(37.6\) & \(108.3\) & \(168.9\) & \(131.3\) \\  & & Value & \(57.1\) & \(126.9\) & \(183.7\) & \(126.7\) \\ \cline{2-7}  & \multirow{4}{*}{Grayscale image} & Gray & \(41.5\) & \(105.4\) & \(172\) & \(130.5\) \\  & & Skewness & \(-1.19\) & \(-0.27\) & \(0.82\) & \(2.01\) \\  & & Kurtosis & \(-0.64\) & \(0.7\) & \(5.89\) & \(6.53\) \\ \hline \multirow{4}{*}{Texture features} & \multirow{4}{*}{Gray level co-occurrence matrix} & Contrast & \(201.4\) & \(1185.2\) & \(2666.2\) & \(2464.8\) \\  & & Dissimilarity & \(9.9\) & \(24.6\) & \(39.3\) & \(29.4\) \\ \cline{1-1}  & & Homogeneity & \(0.028\) & \(0.059\) & \(0.158\) & \(0.13\) \\ \cline{1-1}  & & Energy & \(0.007\) & \(0.012\) & \(0.027\) & \(0.019\) \\ \cline{1-1}  & & Correlation & \(0.1\) & \(0.47\) & \(0.8\) & \(0.7\) \\ \hline \multirow{4}{*}{Shape features  
Table 4 summarizes the results of calculating the VIFs of 18 soil surface image features. Since the VIF of 15 features exceeded 10, it was judged that multicollinearity would appear if these features were used in multiple regression analysis. The process of removing the feature with the highest VIF and calculating the VIF again was repeated until the VIF of all features was less than 10.  
Table 5 summarizes the VIF calculation results of the selected features after removing features that may affect multicollinearity. A total of seven features were selected, and all VIFs were below 10. Multiple regression analysis was performed with these as independent variables.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline
**Color features** & **Hue** & **Sat** & **Value** & **Gray** & **Skewness** & **Kurtosis** \\ \hline
**VIF** & 194.4 & 529.1 & 8052.3 & 6644.8 & 35.2 & 15.9 \\ \hline
**Texture features** & Contrast & Dissimilarity & Homogeneity & Energy & Correlation & \\ \hline
**VIF** & 875.6 & 3551.4 & 649.3 & 747.1 & 109.8 & \\ \hline
**Shape features** & H0 & H1 & H2 & H3 & H4 & H5 & H6 \\ \hline
**VIF** & 4926.6 & 7755.0 & 85.2 & 97.6 & 1.4 & 1.7 & 1.0 \\ \hline \end{tabular}
\end{table}
Table 4: Variance inflation factor of color, texture, and shape features extracted from soil surface images.  
Figure 4: Correlation matrix of color, texture, shape features extracted from soil surface images.