# Fast Implicit Surface Reconstruction for the Radial Basis Functions Interpolant
## 7 Conclusions and Discussion  
In this paper, we improve an efficient implicit surface reconstruction method for the radial basis functions interpolant. The growing strategy of the surface following method combines both the evaluation and reconstruction processes. It ensures a smaller number of evaluation points and higher evaluation efficiency in surface following. The single point FMM is improved to evaluate points one by one. Based on the improved voxel growing method without manually specifying the seed points, the leaf growing method is developed to avoid a mass of redundant computation. The black-box FMM is utilized to evaluate the RBF interpolant with different kernels in a wide range of applications. The numerical results showed that one point evaluation in surface following obtains an optimum efficiency to the original approach. Compared with the existing methods, the improved method performs a better time and space efficiency. The main limitation of the method is that the evaluations of the target points are no longer mutually independent because of the surface following process. It increases the difficulty of the parallel execution of the algorithm to a certain extent. To overcome this limitation, the space should be divided into several blocks. The reconstruction process between blocks does not affect each other. If the blocks are divided according to the octree structure in FMM, the evaluation process of each block can be also performed independently. Therefore, the leaf growing method is used to reconstruct the local implicit surface in parallel for each block. Finally, the results of each block are spliced into a complete surface. To further improve the reconstruction efficiency and mesh quality, there are still some extensions that need to be studied. For example, the cells used in voxel growing are the same size, and the operations of single point FMM can be optimized for parallel implementation. To reduce the evaluation points, the function values and the gradient values can be combined in leaf growing to reconstruct the implicit surface adaptively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{\(N_{c}\)} & \multirow{2}{*}{\(d\)} & \multirow{2}{*}{\(p\)} & \multicolumn{4}{c}{**Evaluation Points**} & \multicolumn{4}{c}{**Time (Seconds)**} \\ \cline{5-8}  & & & & \multicolumn{2}{c}{**MC**} & \multicolumn{1}{c}{**VG**} & \multicolumn{1}{c}{**LG**} & \multicolumn{1}{c}{**MC**} & \multicolumn{1}{c}{**VG**} & \multicolumn{1}{c}{**LG**} \\ \hline Figure 10a & 21,975 & 2.0 & 10 & 5,972,274 & 167,858 & 167,858 & 11.4 & 4.5 & 4.1 \\ Figure 10b & 21,792 & 2.0 & 10 & 16,310,430 & 422,615 & 422,615 & 19.1 & 8.2 & 7.5 \\ Figure 10c & 45,975 & 1.0 & 10 & 7,180,920 & 173,552 & 173,552 & 41.3 & 10.8 & 10.3 \\ Figure 11a & 29,770 & 1.0 & 10 & 44,498,160 & 759,525 & 759,525 & 61.7 & 14.5 & 12.7 \\ Figure 11b & 29,397 & 1.0 & 10 & 261,766,151 & 1,320,267 & 1,320,267 & 63.3 & 23.8 & 19.9 \\ Figure 11c & 19,065 & 3.0 & 10 & 56,107,920 & 738,284 & 738,284 & 73.2 & 11.8 & 10.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the evaluation time using the marching cubes method (MC), the voxel growing method (VG) and the leaf growing method (LG). Also showing the number of domain constraints \(N_{c}\), the size of reconstruction cell \(d\) and the expansion order \(p\). **Author Contributions:** [PERSON] and [PERSON] conceived, designed, and performed the experiments; [PERSON] analyzed the data and revised the methodology; [PERSON] wrote the paper; all authors discussed the results and revised the paper. **Funding:** This work was financially supported by the National Key R&D Program of China (2017 YFC0602905) and the National Natural Science Foundation of China (41572317). **Acknowledgments:** Thanks to the public datasets used in this research. We also thank the reviewers for their comments and suggestions to improve the quality of the paper. **Conflicts of Interest:** The authors declare no conflict of interest.