# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 3 Experiments and Discussions
### Comparison of the Simplification Effect with Other Algorithms
#### 3.3.2 Point Cloud Simplification for the Machine Component Data Set  
This data set has an explicit and regular shape feature, which can be easily identified and matched by simplification rules. Therefore, theoretically it is suitable to use the FPPS algorithm for this type of data. The data set, experiment parameter, and simplification results are described in Table 4, and the responding reconstructed surfaces are shown in Figures 15 and 16. In this experiment, the data sets, called \"Fandisk\" and \"Turbine Blade\", were used. We set experiment parameter \(c\) based on the results of the first experiment, as shown in the second column of Table 4. For FPPS, the simplification ratios were 90.7% and 40.7%, respectively. Using the FPPS's simplification ratio, we executed the uniform-based, grid-based, and curvature-based algorithms. The grid-based algorithm adjusted the simplification ratio based on the grid resolution. For the \"Fandisk\", the grid resolution is set as 0.086. According to the grid resolution, the simplification ratio is configured as 89.9%; for the \"Turbine Blade\", the grid resolution is set as 1.384, and the simplification ratio is 41.8%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Data Set**} & \multirow{2}{*}{\(c\)} & **Simplification** & **Simplification** & **Deviation** & **Standard** \\  & & **Ratio** & **Method** & **Range** & **Deviation** \\ \hline \multirow{4}{*}{Fandisk} & \multirow{4}{*}{6000} & 90.7\% & Uniform & \([-0.884,0.374]\) & 0.015 \\  & & 89.9\% & Grid & \([-0.885,0.85]\) & 0.019 \\  & & 90.7\% & Curvature & \([-0.885,0.088]\) & 0.018 \\  & & 90.7\% & FPPS & \([-0.885,0.182]\) & 0.01 \\  & & 40.7\% & Uniform & \([-2.985,\) & 2.262 \\  & & & & \([-29.883,\) & 0.775 \\  & & & & \([-27.420,\) & 0.603 \\  & & 40.7\% & FPPS & \([-29.539,\) & 0.751 \\  & & & & \([-29.539,\) & 0.751 \\  & & & & \([16.794]\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The experiment parameters and simplification results for the Fandisk and Turbine Blade data sets. Figure 15: Simplification effect comparison of four simplification algorithms using the Fandisk data set: (**a**) uniform-based simplification algorithm; (**b**) grid-based simplification algorithm; (**c**) curvature-based simplification algorithm; (**d**) FPPS. In Table 4, the deviation range and standard deviation are used to reflect the simplification quality. The simplification quality can also be observed in Figures 15 and 16. The deviation range and standard deviation lists shown in Table 4 were less for the FPPS and curvature-based algorithms. Similar results can be seen in Figures 15 and 16. From Figure 14, we can see that the FPPS had a better effect on the surface's corner and boundary. However, for the flat and continuous area, FPPS had simplified the data with a bigger ratio, leading to greater deviation--these are the experimental results that we expected. In order to compare the effects of simplification, we added the red box and red circle in Figure 15. The red box is the boundary of \"Fandisk\"; Figure 15b shows an unregular boundary line, and the boundary surface in Figure 15a,c,d was reconstructed completely. When comparing with the red circle areas in Figure 15a-d, sub-figures c and d show the least deviation, which means that the curvature-based and FPPS algorithms preserve the surface features well. Furthermore, with the aim of analyzing the simplification effect, we list the sub-figure to show the details of the Turbine Blade data set in Figure 16a-d. The sub-figure is captured at another viewport, and the trademark can be seen in the sub-figure. Figure 16c,d are clearer than the other sub-figures; therefore, the curvature-based and FPPS algorithms produce less deviation from the original surface. Figure 16: Simplification effect comparison of four simplification algorithms using the Turbine Blade data set: (**a**) uniform-based simplification algorithm; (**b**) grid-based simplification algorithm; (**c**) curvature-based simplification algorithm; (**d**) FPPS.