# Multilevel Regularization Method for Building Outlines
## 4 Results and Analysis
### Applicability of Analysis for Data of Different Resolutions  
Figure 18 illustrates examples of regularization results for building outlines extracted from images with two different resolutions: 0.075 m and 0.5 m. As shown, for the data with 0.075 m resolution, except for a few complex-shaped buildings, the regularization results from various methods align well with the extracted building outlines, indicating a high level of regularization. However, for the data with 0.5 m resolution, the regularization effects of all methods are reduced to varying degrees. This reduction is primarily characterized by varying degrees of deviation in the main direction of the outlines and a decrease in the overlap ratio between the regularization results and the extracted building outlines.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Time Complexity**} \\ \cline{2-5}  & **First Step** & **Second Step** & **Third Step** & **Sum** \\ \hline A & O(\(n^{2}\)) & O(\(n^{2}\)) & – & O(\(n^{2}\)) \\ \hline B & O(\(n^{2}\)) & O(\(n^{2}\)) & O(\(n\)) & O(\(n^{2}\)) \\ \hline C & O(\(n^{2}\)) & O(\(n^{2}\)) &... & O(\(n^{2}\)) \\ \hline Proposed method & O(\(n^{2}\)) & O(\(n^{2}\)) & O(\(n\)) & O(\(n^{2}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Time complexity of the different methods.  
Figure 18: Regularization results of various methods for data with two different resolutions: (a,c) Results for data with 0.075 m resolution; (b,d) Results for data with 0.5 m resolution.  
Comparing the shapes of the outlines for the two different data resolutions in Figure 18 shows that the image resolution has a significant impact on the accuracy of building outline extraction. To prevent the errors introduced during the extraction process from affecting the evaluation of regularization effects, the extracted building outlines were taken as reference data for evaluation. Table 5 displays the regularization metrics for different methods applied to the data of two different resolutions. For both data resolutions, the proposed method achieves the highest IOU and F1 scores, indicating that it more effectively preserves the shape of building outlines during regularization and achieves greater generality. However, as the data resolution decreases, all methods exhibit decreases in the evaluation metrics. The decreases in P and R are attributable to outline deformations and directional deviations, whereas the decline of F1 suggests the weakening generality of these methods. Notably, the IOU decreases more significantly than the other two metrics, suggesting a substantial reduction in the intersection area between the building outlines before and after regularization as the resolution decreases. This is because lower-resolution data lead to more jagged outlines, resulting in greater changes in shape, area, and position during regularization. Furthermore, when applied to the data with 0.075 m resolution, the proposed method outperforms the other methods across all metrics. When applied to the data with 0.5 m resolution, the proposed method continues to outperform other methods in three metrics, but its superiority is less pronounced.  
In summary, the proposed method performs well in regularizing building outline data extracted from images with different resolutions. Higher resolutions lead to smaller shape variations during regularization, resulting in more regularized outlines. As the resolution of image data grows, the advantages of the proposed method become more pronounced.