## 3 Methodology
### Disentanglement Method  
The variations of the detailed information around the surface of the object (i.e., surface details) affect the Laplacian of the SDF [55]. Inspired by this, the surface details of an object can be detected through Laplacian operators and the remaining topological structures can be reconstructed according to an appropriate loss function, thus disentangling the implicit field reconstruction into topological structure reconstruction and surface detail recovery. As shown in Figure 2, the ground-truth SDF is disentangled into the deformed implicit field with structure details and the enhanced displacement field containing surface details of the object. The most similar work to ours is D\({}^{2}\)IM-Net [29], which only disentangles the ground-truth SDF into the sum of a coarse implicit field and a displacement field. Unlike D\({}^{2}\)IM-Net, our disentangled deformed implicit field is based on the coarse implicit field, where the deformed implicit field contains most of the topological structures. Given a query point \(P\), our disentanglement solution can be denoted as:  
\[SDF(P)=g_{su}(P)+g_{st}(P), \tag{1}\]  
\[f:g_{co}\in\mathbb{R}\mapsto g_{st}\in\mathbb{R}, \tag{2}\]  
where \(SDF\) denotes the ground-truth SDF, \(g_{su}\), \(g_{st}\), and \(g_{co}\) represent the enhanced displacement field with surface details, the deformed implicit field containing most of topological structures, and the coarse implicit field, respectively, and \(f\) defines the deformation function from \(g_{co}\) to \(g_{st}\).  
Actually, we can suppose that the shape embedded in the deformed implicit field is smooth and the reconstructed shape from the deformed implicit field can only approximate the object surface. Therefore, the surface details can be further represented with the enhanced displacement field. As the enhanced displacement field is attached onto the smooth deformed implicit field near the iso-surface of the object, the Laplacian of the enhanced displacement field is approximately equal to the Laplacian of \(SDF\):  
\[\triangle\,SDF(P)=\triangle\,g_{su}(P). \tag{3}\]  
In order to accelerate the network training, Laplacian for only the sampling points, whose minimum distance to the object shape \(T\) is less than a predefined threshold \(\alpha\), will be taken into consideration.  
Motivated by the works [29; 31; 56] related to inference on the visible and invisible surfaces of objects, the forward and backward displacement maps are introduced for the visible and occluded parts of the object, respectively. Our forward displacement map recovers the visible surface details of the object based on a Laplacian loss and the backward displacement map is used to fine-tune the deformed implicit field, further compensating for unreconstructed structure details and fixing incorrect topological structures. In short, we have  
\[SDF(P)=\begin{cases}g_{st}(P)+g_{suF}(p),ifP\in P_{V},\\ g_{st}(P)+g_{suB}(p),otherwise,\end{cases} \tag{4}\]  
\[\triangle\,g_{suF}(p)=\triangle\,SDF(P),P\in P_{V}, \tag{5}\]  
where \(g_{suF}\) and \(g_{suB}\) represent the forward and backward displacement maps, respectively. \(p=\pi(P)\) is the projection of \(P\) on the single-view image. \(P_{V}\) is the point set which consists of points close to the visible surface of the object.  
Indeed, 3D displacement fields are more direct and are also defined in 3D space. However, displacement maps are applied instead of 3D displacement fields because they enable alignment of the input image with the details, making it possible to calculate the Laplacian loss term. Additionally, it is more intuitive for us to observe the detailed information of the object in the displacement maps.