## 3 Methodology
### Loss Function and Sampling Strategy  
The total loss function of ED2 IF2-Net consists of four components \(L=L_{Coa}+L_{Def}+L_{Ove}+L_{Lap}\), where \(L_{Coa}\), \(L_{Def}\), \(L_{Ove}\), and \(L_{Lap}\) represent the coarse shape loss, deformation loss, overall shape loss, and Laplacian loss, respectively. More specifically, \(L_{2}\)-norm-based \(L_{Coa}\) is used to minimize the distance between the coarse implicit field \(g_{co}\) and the ground-truth SDF \(SDF\), and \(L_{1}\)-norm-based \(L_{Ove}\) is employed to minimize the distance between the fused implicit field \(g\) and \(SDF\), which can regularize the enhanced displacement field:  
\[L_{Coa}=\frac{1}{N}\sum_{i=1}^{N}||g_{co}(P_{i})-SDF(P_{i})||_{2}^{2}, \tag{12}\]  
\[L_{Ove}=\frac{1}{N}\sum_{i=1}^{N}|g(P_{i})-SDF(P_{i})|. \tag{13}\]  
The structure details are evaluated through a novel deformation loss \(L_{Def}\). Since the deformation decoder iteratively refines \(g_{co}\), the intermediate implicit field \(s\) generated by  
Figure 5: Architecture of the surface detail decoder.  
all IFDBs, and the deformed implicit field \(g_{st}\) are all taken into consideration, and their \(L_{1}\)-distances to \(SDF\) are accumulated through a weighted summation:  
\[\begin{split} L_{Def}&=\frac{1}{N}\sum_{i=1}^{N}\sum _{j=1}^{M}\omega_{j}|s_{j}(P_{i})-SDF(P_{i})|\\ &+\frac{1}{N}\sum_{i=1}^{N}\omega_{0}|g_{st}(P_{i})-SDF(P_{i})|, \end{split} \tag{14}\]  
\[\sum_{j=0}^{M}\omega_{j}=1, \tag{15}\]  
where \(P_{i}\) represents the \(i^{th}\) query point, \(N\) denotes the number of query points, \(M\) indicates the number of intermediate implicit fields, the \(j^{th}\) intermediate implicit field is defined as \(s_{j}\), \(\omega_{j}\) stands for the weight assigned to \(s_{j}\) in the deformation loss, specifically, and \(\omega_{0}\) is the weight of the deformed implicit field \(g_{st}\).  
\(L_{Lap}\) is the \(L_{2}\)-distance between the Laplacian of the forward displacement map \(g_{suF}\) and the Laplacian of \(SDF\). As these two Laplacians are not in the same space, the forward displacement map \(g_{suF}\) is a 2D image, whereas \(SDF\) is in 3D space. This problem is addressed using the Laplacian of \(SDF\) with respect to the projected points on the image. Suppose \(p_{i}(u_{i},v_{i})\) denotes the projection of \(P_{i}\) on the image and \(p_{i}^{\prime}\) represents the coordinates of \(P_{i}\) in the camera coordinate system. Similar to D2 IM-Net [29], Laplacian loss \(L_{Lap}\) can be formulated as:  
\[L_{Lap}=\frac{1}{|P_{V}|}\sum_{P_{i}\in P_{V}}||\bigtriangleup g_{suF}(p_{i})- l(p_{i})||_{2}^{2}. \tag{16}\]  
The Laplacian of the forward displacement map \(g_{suF}\) is:  
\[\bigtriangleup g_{suF}(p_{i})=\frac{\partial^{2}g_{suF}(p_{i})}{\partial u_{i }^{2}}+\frac{\partial^{2}g_{suF}(p_{i})}{\partial\gamma_{i}^{2}}. \tag{17}\]  
In case \(P_{i}\) is close to the visible surface of an object, \(N(p_{i})\) is the unit normal from the ground-truth normal map, equivalent to the gradient of the \(SDF\) with respect to \(p_{i}^{\prime}\) as:  
\[N(p_{i})=\frac{\partial SDF(P_{i})}{\partial p_{i}^{\prime}},P_{i}\in P_{V}. \tag{18}\]  
Then, the Laplacian of \(SDF\) can be defined as:  
\[l(p_{i})=N(p_{i})\frac{\partial p_{i}^{\prime}}{\partial^{2}u_{i}}+N(p_{i}) \frac{\partial p_{i}^{\prime}}{\partial^{2}v_{i}}. \tag{19}\]  
To enhance the fidelity of the reconstructed object and capture richer small-scale details, ED2 IF2-Net employs a weighted sampling strategy similar to D2 IM-Net [29]. This strategy assumes dense sampling of the object, where the density of each sampling point is determined by the number of surrounding sampling points within a specified radius. Inside and outside the object, a clipping policy defines compact sample densities. These densities, along with the same samples, serve as sampling weights during the training of ED2 IF2-Net. The effectiveness of the weighted sampling strategy in reconstructing small-scale details is demonstrated through ablation studies.