# A Quantitative Evaluation of UAV Flight Parameters for SfM-Based 3D Reconstruction of Buildings
## 4 Validation
### Single Shooting Pattern
#### 4.1.6 Pareto Optimization of Single Shooting Patterns  
Based on previous analyses, it is common for various performance goals pursued in UAV-SfM-based 3D modeling tasks, such as reducing computation time, minimizing reprojection error, maximizing the number of points, and maximizing reconstruction completeness, to be in conflict with each other (trade-offs). In this section, a Pareto optimization analysis was performed by pairing the main performance objectives (e.g., computational time-reprojection error, computational time-number of points, reprojection error-number of points, computational time-reconstruction completeness). Pareto optimization is a multi-objective optimization methodology that aims to identify a set of \"non-dominated\" solutions, where improving one specific objective necessarily results in a loss in one or more other objectives, thereby forming the Pareto Frontier. Among the entire set of shooting configurations (data points) analyzed for each goal combination, different configurations were identified as Pareto optimal solutions, and Table 9 summarizes the analysis results for each goal combination. Visualizing the Pareto frontier for each goal combination clearly revealed the trade-off characteristics between goals (Figure 7). The frontier for the 'Error_vs_Points' combination exhibited a concave shape toward the upper left, indicating low error and high point count simultaneously. This suggests that there is an efficient range where the number of points can be significantly increased by slightly allowing (increasing) the error, while there is a \begin{table}
\begin{tabular}{c c c} \hline
**Objective Pair Analyzed** & **N Points on Frontier** & **Key Frontier Characteristics and Implication** \\ \hline Time vs. Error & 4 & Weakly concave shape (bottom-left); diminishing returns for error reduction over time \\ Time vs. Points & 5 & Concave rising shape; diminishing returns for point increase over time \\ Error vs. Points & 4 & Concave shape (top-left); clear trade-off exists, optimal balance selection needed \\ Time vs. Completeness & 4 & Nearly horizontal shape (top-left); diminishing returns for completeness improvement over time \\ \hline \end{tabular}
\end{table} 
Table 9: Summary of Pareto Optimization Analysis for Key Objective Pairs. Figure 6: Surface pattern RSM visualization. (**a**) Forward overlap and Side overlap. (**b**) Forward overlap and Tilt angle. (**c**) Side overlap and Tilt angle. range where the number of points must be significantly sacrificed to reduce the error below a certain level. On the other hand, the frontier for the 'Time-vs-Completeness' combination is in the upper left corner, indicating short computation time and high reconstruction completeness, and exhibits a nearly horizontal shape. This indicates a 'diminishing returns' phenomenon, meaning that even with significantly more time invested beyond a certain point, the improvement in completeness is minimal, while it is possible to achieve a fairly high level of reconstruction completeness (0.985 or higher) within a very short computation time (e.g., 3-8 min). The frontier of the 'Time_vs_Error' combination is located in the lower left (short time, low error) and exhibits a slightly convex shape, indicating that investing a little more time initially (e.g., 3-6 min) can significantly reduce errors, but the error reduction effect gradually diminishes with further time investment. Finally, the frontier of the 'Time_vs_Points' combination shows a convex upward trend from the lower left to the upper right, indicating that initial time investment (e.g., 3-14 min) is highly effective in increasing the number of points, but additional time investment beyond that point contributes significantly less to further increases in the number of points.