# A Two-Step Surface Reconstruction Method Using Signed Marching Cubes
## 4 Conclusions and Discussion  
In this paper, we present an efficient implicit surface reconstruction method using the signed marching cubes method. The basic idea of this method is to filter the invalid voxels that do not intersect with the target isosurface roughly and to then determine the reliable voxels that intersect with the target isosurface exactly. The signed marching cubes method is a two-step reconstruction method, including a rough evaluation step and an exact evaluation step. The coarse evaluation step evaluates the points of all voxels using the fast multipole method with a lower order. After the rough evaluation step, the voxels that intersect with the target isosurface are screened out. Then, the exact evaluation step evaluates the points of filtered voxels using the fast multipole method with a higher order. Compared with the traditional marching cubes method, the improved reconstruction method reduces the amount of calculations for invalid voxels that do not intersect with the target isosurfaces. Therefore, based on the two-step evaluation process, the efficiency of the surface reconstruction can be improved greatly. We analyzed the performance of the two-step surface reconstruction algorithm using several experiment datasets.  
The two-step reconstruction method using two expansion orders of the fast multipole method are investigated preliminarily. It is worth noting that the two-step surface reconstruction method can be applied to the existing improved marching cubes methods proposed by some scholars by modifying the algorithm simply. However, there are still some limitations in our work that need to be improved. One of them is the lack of theoretical analysis of the optimal expansion order and the balance efficiency, which should study the optimization of the relationship between the reconstruction efficiency and the expansion orders. For the two-step reconstruction method, the number of evaluation points is closely related to the selected two expansion orders. Besides the number of evaluation points, the optimal efficiency needs to comprehensively consider some other relevant factors. Another limitation is that, similar to the original marching cubes method, the two-step reconstruction method may fail to detect the objects that do not intersect with the voxels. In this case, a smaller size should be specified to recover the undetected objects. In addition, similar to the surface reconstruction method based on the marching cubes method, the two-step surface reconstruction method finds it difficult to recover the sharp geometry features of the target isosurface. Some strategies about dealing with the ambiguity of the face topology should be integrated into the signed marching cubes method directly to improve the mesh quality. Another direction for future extension, which we have already started to explore, is to develop a multi-step reconstruction method using multiple expansion orders to further improve the reconstruction efficiency.  
[PERSON] and [PERSON] conceived and designed the experiments; [PERSON], [PERSON] and [PERSON] analyzed the data and revised the methodology; [PERSON] wrote the paper; all of the authors discussed the results and revised the paper. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the Fundamental Research Funds for the Central Universities of Central South University (2020 zzts201).  
Not applicable.  
**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. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; and in the decision to publish the results.