# A Two-Step Surface Reconstruction Method Using Signed Marching Cubes
## 3 Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{p}\)} & \multicolumn{3}{c}{\(\mathbf{N_{e}}\)} \\ \cline{2-4}  & **10,000** & **100,000** & **1,000,000** \\ \hline
12 & \(3.92\times 10^{-9}\) & \(5.94\times 10^{-9}\) & \(5.65\times 10^{-9}\) \\
11 & \(3.17\times 10^{-8}\) & \(4.64\times 10^{-8}\) & \(3.54\times 10^{-8}\) \\
10 & \(2.88\times 10^{-7}\) & \(3.32\times 10^{-7}\) & \(4.54\times 10^{-7}\) \\
9 & \(4.83\times 10^{-6}\) & \(5.50\times 10^{-6}\) & \(4.47\times 10^{-6}\) \\
8 & \(2.94\times 10^{-5}\) & \(2.50\times 10^{-5}\) & \(2.55\times 10^{-5}\) \\
7 & \(1.66\times 10^{-4}\) & \(1.65\times 10^{-4}\) & \(1.58\times 10^{-4}\) \\
6 & \(3.24\times 10^{-3}\) & \(2.62\times 10^{-3}\) & \(3.02\times 10^{-3}\) \\
5 & \(9.88\times 10^{-3}\) & \(7.78\times 10^{-3}\) & \(1.00\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The max residual of the RBF interpolant (triharmonic spline) using the fast multipole method with different expansion orders (the number of source points \(N_{s}\) is 10,000). \(N_{e}\) is the number of evaluation points. Similar to the original marching cubes method, if the size of the target objects is even smaller than the specified reconstruction size, it may lead to a failure reconstruction where the small objects cannot be detected. Figure 7a demonstrates a failure example. The small object marked in the black box is separated from the main part (seen as Figure 7b) and cannot be detected. In this case, a smaller reconstruction size should be specified to reconstruct the complete model. In case of the example with the same number of source points, the number of evaluated points represents the reconstruction performance. We selected three kinds of voxel sizes for each model. Voxel size \(S_{c}\) is determined according to the size of different models, and the specific method is as follows: before reconstructing the model, the minimum bounding box of the reconstructed model needs to be determined according to the sampling data. It is  
Figure 6: The reconstruction results of six geological body models based on the interpolation points using the signed marching cubes method. Among the interpolation constraints, cyan points represent the interpolation constraints exterior of the isosurface, blue points represent the interpolation constraints interior of the isosurface, and red points represent the interpolation constraints on the isosurface. (**a–f**) are six reconstructed models and the corresponding interpolation constraints. Figure 7: A reconstruction result when part of the model is smaller than the specified reconstruction size. (**a**) a failure reconstruction result; (**b**) a small object that do not be reconstructed. assumed that the x, y, and z axis lengths of the model minimum bounding box are \(l_{1}\), \(l_{2}\), and \(l_{3}\), respectively, and the reference axis lengths are determined with the same reference ratio (e.g., 0.1 times). Assuming that the relationship between the lengths of the three reference axes is \(l_{1}^{\prime}<l_{2}^{\prime}<l_{3}^{\prime}\), the voxel reference length \(l_{0}\) is determined by the following formula  
\[\left\{ \begin{array}{ll}l_{0}=0.1l_{2}&\quad if\;l_{3}<0.1l_{2}\\ l_{0}=0.05l_{3}&\quad if\;0.1l_{2}\leq l_{3}\leq 0.5l_{2}\\ l_{0}=0.025l_{3}&\quad if\;l_{3}>0.5l_{2}\end{array} \right. \tag{14}\]  
where 0.1, 0.05, and 0.025 are reference values determined according to experience, and can be finally determined according to the manual adjustment. Then, according to the modeler's requirements for different reconstruction precision, the voxel size \(S_{c}\) can be determined approximately as  
\[\left\{ \begin{array}{ll}S_{c}=5l_{0}&\quad coarse\;accuracy\\ S_{c}=l_{0}&\quad general\;accuracy\\ S_{c}=0.3l_{0}&\quad fine\;accuracy\end{array} \right. \tag{15}\]  
It is worth noting that the actual size should be adjusted according to the specific applications.