# Point Cloud Repair Method via Convex Set Theory
## 4 Conclusions  
This paper proposed a point-cloud repair method via a convex set for an incomplete point cloud. First, we constructed a cube unit \(S\) centered on several adjacent hole boundary points and discretized \(S\) into \(K(K=9)\) subunits. Second, pointless subunits that are roughly screened by the difference of normal vectors are called unstable unit sets. Then, the objective function was calculated from the joint distribution function of the attribute information of the unstable unit set and the state distribution function of MRF, and the MAP was solved by using variational inference to determine subunit \(s^{*}\) that needed to be filled. Lastly, \(s^{*}\) and central unit \(s_{1}\) formed a specific subunit \(s\). Combined with the weakened convex set, new 3D points were added to the hole area, so that subunit \(s\) became a convex  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **SpareNet [31]** & **SnowflakeNet [30]** & **PF-Net [11]** & **SCCR [32]** & **Ours** \\ \hline Airplane & 3.28 & 2.52 & 2.79 & 1.85 & **0.23** \\ Chair & 4.06 & 3.38 & 4.10 & 1.58 & **1.04** \\ Table & 18.56 & 13.39 & 19.31 & 4.25 & **0.51** \\ Car & 6.33 & 10.30 & 12.39 & 2.96 & **1.13** \\ Monster & 811.98 & 5567.67 & 6092.97 & 240.32 & **107.44** \\ Monkey & 290.11 & 529.26 & 603.14 & 2.77 & **1.21** \\ Sphere & 148.25 & 182.33 & 223.45 & 26.01 & **2.47** \\ Mean & 183.22 & 901.26 & 994.02 & 39.96 & **16.29** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Quantitative comparison of NHSD (\(\times 10^{-2}\)).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **SpareNet [31]** & **SnowflakeNet [30]** & **PF-Net [11]** & **SCCR [32]** & **Ours** \\ \hline Airplane & 18.54 & 19.31 & 18.01 & 40.38 & **62.41** \\ Chair & 15.15 & 14.48 & 15.11 & 32.05 & **37.71** \\ Table & 19.22 & 19.91 & 13.38 & 41.52 & **58.95** \\ Car & 15.48 & 14.53 & 13.51 & 40.20 & **44.23** \\ Monster & 4.81 & 5.01 & 2.33 & 32.41 & **51.72** \\ Monkey & \(-2.52\) & 3.01 & \(-3.74\) & 37.36 & **50.83** \\ Sphere & \(-7.17\) & \(-9.31\) & \(-8.63\) & 10.17 & **33.45** \\ Mean & 11.78 & 9.56 & 7.14 & 33.44 & **48.47** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Quantitative comparison of GPSNR (DB).  
set. Our method could accurately infer the distribution of new filling points to ensure that the repaired shape was as consistent with the real surface shape as possible. The proposed method was not limited by the category of point clouds and could repair various categories of point clouds while maintaining the position of points in non-hole areas unchanged. The experimental results show that our method is superior to the compared methods in terms of GPSNR and NHSD. However, the method in this paper was limited by the size of unit \(S\). Large or small units \(S\) reduce the final repair effect. Therefore, we will consider using more prior information to repair incomplete point clouds in the future.  
Conceptualization, T.D.; methodology, T.D. and Y.Z.; software, Y.Z.; formal analysis, Y.Z.; data curation, Y.B.; writing--original draft preparation, Y.Z.; writing--review and editing, M.L.; visualization, Y.B. and M.L.; funding acquisition, T.D. All authors have read and agreed to the published version of the manuscript. This research was partially supported by the Shanghai Alliance Program (LM2019043).  
Not applicable.  
The authors declare no conflict of interest.