# Feature-Preserved Point Cloud Simplification Based on Natural Quadric Shape Models
## 3 Experiments and Discussions
### The Discussion of Self-Adapting Experiment Parameters  
The purpose of this experiment was to discuss the influence of the experimental parameters on the simplification effect, which includes the simplification entropy and the neighborhood range of key points. Firstly, we assumed that the three types of key points could produce a balanced influence. Therefore, we set \(k\) as the number of key points and \(\sigma\) as the neighborhood range of each key point, in order to find the influences of \(k\) and \(\sigma\) on the simplification effect. The experiment used the constant \(c\), where \(c=k\times\sigma\). This experiment used \(c\) = 10,000. We adjusted parameters \(k\) and \(\sigma\) in the FPPS. The reduced point clouds were reconstructed, and the reconstructed surface is shown in Figure 8.  
Figure 8 shows the reconstructed surface. As \(k\) increases, \(\sigma\) will shrink for a constant \(c\), and the difference in the area of the triangle can be observed in Figure 8a-e. In particular, looking at Figure 8d,e, the triangles around the eyes and feet become obviously smaller and more refined, which shows that the reconstructed surface is smoother. A similar change in the triangle can be found around the joining of the bunny's head with its ears. In contrast, the value in the fourth column of Table 3 (Number of Triangles) is bigger from top to bottom, which means that the triangles are refined. However, there are different phenomena in the fourth and fifth rows of Table 3. In the fourth row of Table 3, the values of \"Number of Triangles\" and \"Size\" columns, respectively, are 7145 and 4456, respectively, which are bigger than the values in the fifth row. This phenomenon reveals that the maximum value of \(k\) should be found. The maximum value of \(k\) provides the balance between the satisfied simplification result and the time consumption of FPPS. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data Type** & **Data Set** & **Size** & **Vertices** & **Download** \\ \hline \multirow{3}{*}{Natural data} & Bunny & 1.05 M & 35,947 & [[http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/]([http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/) \\  & Dragon & 14.7 M & 437,645 & [[http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/]([http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/) \\ \multirow{3}{*}{\
\begin{tabular}{c} Mechanical- \\ Components \\ \end{tabular} } & Fandisk & 0.18 M & 6476 & \\  & Turbine & 24.6 M & 882,954 & [[https://www.cc.gatech.edu/projects/large](https://www.cc.gatech.edu/projects/large)]([https://www.cc.gatech.edu/projects/large](https://www.cc.gatech.edu/projects/large))\_models/ \\ \multirow{2}{*}{
\begin{tabular}{c} Man-made \\ \end{tabular} } & Chair & 0.992 M & 15,724 & [[http://www.shapenet.org/](http://www.shapenet.org/)]([http://www.shapenet.org/](http://www.shapenet.org/)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental data set. The simplified results from this experiment are shown in Figure 8. We list the parameters (\(k\), \(\sigma\)) and simplification results (number of triangles, size of \(S\) and simplification ratio) in Table 2, which correspond to Figure 8a-e. In these experiments, the experimental parameter \(c\) was set as 10,000. When \(k\) increases, the number of key points becomes bigger; thus, the overall features of the object are well preserved. However, the process of extracting key points is time consuming. When \(\sigma\) shrinks, the scope of a local area becomes smaller; the local areas which are selected to analyze the shape features become smaller; and the local features of an object, especially the \"corner\" or \"sharp\" features, are well preserved.