# A Two-Step Surface Reconstruction Method Using Signed Marching Cubes
## 2 Method
### Two-Step Surface Reconstruction Method  
The ideal of the marching cubes method is to evaluate all partitioned voxels in a specified 3D space to extract the target isosurface for an implicit function, as shown in  
Figure 1: The voxel growing method with multiple seed points. (**a**) The zero-level set represents the implicit surface. (**b**) The initial seed voxels are selected from the interpolation constraints with zero values. Figure 2a. The method can recover a complete model. However, it should traverse and calculate all voxels in the 3D space, which consumes a lot of time to filter the non-boundary voxels that do not intersect with the target isosurface. To improve the reconstruction efficiency, the voxel growing (VG) method is developed to deal with the problem that the marching cube method takes a lot of time to calculate non-boundary voxels. The method tracks the boundary voxels by searching the adjacent voxels based on some initial seed voxels, as shown in Figure 2b. However, it also leads to a new problem, in that it depends on reliable seed voxels to recover a complete model. In general, at least one seed voxel should be determined to reconstruct each separate isosurface in the geometry space. Based on the above analysis, we present the signed marching cubes method (SMC) to accelerate the speed of surface reconstruction combined with the signed fast multipole method. It is useful to avoid the invalid evaluation of non-boundary voxels that do not intersect with the target isosurface. To improve the evaluation efficiency, the expansion order should be optimized to minimize the calculation and complexity on the premise of satisfying the reconstruction accuracy. The basic idea is to determine the optimized expansion order according to the evaluated function value and its corresponding evaluation error. For the exterior voxels, the evaluated accurate function values of the cube points satisfy \(s_{p^{\prime}}(\mathbf{x})=\sigma>0\) when using a higher expansion order. If only the sign of the function values should be determined, the optimized expansion order \(p^{\prime}\) can be calculated as follows  
\[\left\{ \begin{array}{l}\varepsilon_{p^{\prime}-1}>\sigma>\varepsilon_{p^{ \prime}}>\varepsilon_{e}\\ s_{p^{\prime}}(\mathbf{x})-\varepsilon_{p^{\prime}}>0\end{array} \right.,\;if\;s_{ p^{\prime}}(\mathbf{x})=\sigma>0 \tag{12}\]  
where \(\varepsilon_{e}\) is the specified evaluation error used to calculate the accurate function value. Similarly, for the interior voxels, the evaluated accurate function values of the cube points satisfy \(s_{p^{\prime}}(\mathbf{x})=\sigma<0\) when using a higher expansion order. If only the sign of the function values should be determined, the optimized expansion order \(p^{\prime}\) can be calculated as follows  
\[\left\{ \begin{array}{l}\varepsilon_{p^{\prime}-1}>|\sigma|>\varepsilon_{p^{ \prime}}>\varepsilon_{e}\\ s_{p^{\prime}}(\mathbf{x})+\varepsilon_{p^{\prime}}<0\end{array} \right.,\;if\;s_{ p^{\prime}}(\mathbf{x})=\sigma<0 \tag{13}\]  
Based on the above idea, it is clear that the optimized expansion orders of the evaluation points far from the target isosurface are lower than the evaluation points near the target isosurface. Therefore, to further improve the reconstruction efficiency, we can utilize different optimized expansion orders to find all the non-boundary voxels. To simplify the problem, we utilized two expansion orders, a lower expansion order \(p^{\prime}\) and a  
Figure 2: Comparison of evaluation points using different reconstruction methods. (**a**) The evaluation process using the marching cube method. (**b**) The evaluation process using the voxel growing method with one seed. higher expansion order \(p\), to obtain most of the non-boundary voxels instead of variable expansion orders. The basic steps of the two-step reconstruction method can be described as follows.