# A Fast Hole-Filling Method for Triangular Mesh in Additive Repair
## 6 Experimental Results and Discussion
### Quantitative Evaluation  
Figure 13 shows the hole-filling results on two mechanical models. The results shown in Figure 13c-e are obtained by the algorithms of [PERSON] [6], Polymender [32], and our method respectively. Table 1 gives the statistics of the quantitative evaluation results. Figure 13c demonstrates that [PERSON]'s algorithm has the ability to close the holes of the tube and sphere models, but it fails to recover the geometry shapes of the missing regions. The reason is that the [PERSON]'s algorithm is usually taken as an initial filling technique to roughly close the holes in a simple way. Regarding the Polymender, it falls into the category of voxel-based methods, so the whole models have been modified when performing the hole-filling operation. From the results shown in Figure 13d, it is obvious that the produced results are not satisfactory since many non-hole regions have also been modified. Besides, a large number of degenerated triangles are contained in the resulting models. In contrast, the results obtained by our method (as illustrated in Figure 13e) not only can make the shapes of the filling patches assort with their surrounding meshes but also can ensure that most of the newly added triangles and vertices are both highly regular. Although all three algorithms can produce completed models, the results obtained by our method are obviously better than others. From the view of model shape, out method has the ability to make the filling patch assort with the surrounding mesh well. Before analysing Table 1, Figures 14 and 15, two notations are defined: triangle quality factor \(Q_{t}\) and vertex valence \(V_{p}\). Given a triangle, the ratio of its shortest edge to the radius of its circumscribed \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Incomplete Model**} & \multicolumn{2}{c}{**Hole Number**} & \multicolumn{2}{c}{**Hole Size**} & \multicolumn{2}{c}{**Filling Patch by Ours**} & \multicolumn{2}{c}{**Filling Time (on)**} & \multicolumn{2}{c}{**Triangle Quality Evaluation (\%)**} \\ \cline{2-11}  & \(\mathbf{N_{v}}\) & \(\mathbf{N_{t}}\) & \(\mathbf{N_{u}}\) & \(\mathbf{N_{v}}\) & \(\mathbf{N_{t}}\) & \(\mathbf{N_{v}}\) & **Polymender** & **Liepa** & **Our** & **Polymender** & **Liepa** & **Our** \\ \hline Tube & 5025 & 9992 & 1 & 54 & 345 & 634 & 103 & 28.92 & 4 & / & 46.43 & 91.01 \\ Sphere & 6202 & 12,257 & 1 & 130 & 1573 & 3014 & 158 & 102.73 & 19.00 & / & 48.69 & 86.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of the quantitative evaluation results. Figure 13: Hole filling results on tube and sphere models. (**a**) The incomplete models. (**b**) The results after hole boundary clean. (**c**) The results of [PERSON]’s method. (**d**) The results of [PERSON]’s method. (**e**) The results of our method. circle is denoted as the triangle quality factor \(Q_{t}\). If a triangle is closer to be a regular one, the closer \(Q_{t}\) is to 1.73. In this work, the triangles whose \(Q_{t}\) is greater than 1.39 will be regarded as high-quality ones, and then their percentage in the total number of the filling triangles will be computed. For one vertex, the number of its one-ring adjacent triangles is defined as its vertex valence \(V_{v}\). As one highly regular triangular mesh, the \(V_{v}\) of most vertices is six. From Table 1, it can be observed that the hole-filling times for closing the sphere model by our method is almost one-fifth of that of [PERSON]'s method (19 _ms_ VS 102.73 _ms_), and one-eighth of that of [PERSON]'s method (19 _ms_ VS 158 _ms_). The main reason for the efficient performance of our method is the refined vertex normals can enable the newly added vertices to close the hole in a very fast and proper way. Besides, both the regularity of the inserted triangles and vertex value are guaranteed during the closing stage without needing any further postprocessing steps, which can efficiently reduce the filling times.