## 3 Methodology
### Network Architecture
#### 3.3.3 Deformation Decoder  
The deformation function \(f\) can be learnt via the deformation decoder, as illustrated in Figure 3. The deformation decoder firstly unifies the multi-scale local features through a bilinear interpolation and then retrieves the local features for the query point \(P\) at all scales in a pixel-aligned manner. Let \(p=\pi(P)\) be the projection of \(P\) on the image. Following the similar idea of Ladybird [26], two pixel-aligned local features \(h_{1}\) and \(h_{2}\) are provided for \(P\). Specifically, \(h_{1}\) and \(h_{2}\) are extracted from the projection of \(P\) and its self-reflecting symmetric point \(P_{s}\) on \(l_{i}\), respectively, which are then concatenated as the final pixel-aligned local feature \(l_{i}(p)\) of \(P\). Finally, the continuous pixel-aligned local feature pairs are usedto refine the coarse implicit field \(g_{co}\) through the core lightweight components of the deformation decoder, i.e., implicit field deformation block (IFDB). There exist three IFDBs in our implementation of the deformation decoder and the last one generates the deformed implicit field \(g_{st}\) (see Algorithm 1):  
\[g_{st}(P)=f(g_{co}(P),l_{0},l_{1},l_{2},l_{3},P,p), \tag{8}\]  
\[s_{1}(P),c_{1}=IFDB(g_{co}(P),l_{2}(p),l_{3}(p),P), \tag{9}\]  
\[s_{2}(P),c_{2}=IFDB(s_{1}(P),l_{1}(p),l_{2}(p),P,c_{1}), \tag{10}\]  
\[g_{st}(P),c_{3}=IFDB(s_{2}(P),l_{0}(p),l_{1}(p),P,c_{2}), \tag{11}\]  
where \(s_{j}(P)\) and \(c_{j}\) (\(j\in\{1,2,3\}\)) stand for the intermediate implicit field at \(P\) and the state code generated by the \(j^{th}\) IFDB, respectively, in particular, \(g_{st}(P)=s_{3}(P)\).  
```
0: coarse implicit field \(g_{co}\), multi-scale local features \(l_{i}(i\in\{0,1,2,3\})\), query point \(P\) and its projection \(p\) on the image
0: deformed implicit field \(g_{st}\)
1:functionDEFORMATION(\(g_{co}\), \(l_{i}\), \(P\), \(p\))
2:\(P_{s}\leftarrow\mathit{Find\_Symmetry\_Point}(P)\) // Find the symmetry point of the query point \(P\)
3:\(p_{s}\leftarrow\mathit{Find\_Symmetry\_Point\_Projection}(P_{s})\) // Find the projection of the symmetry point of the query point \(P\) on the image
4:for\(i=0\to 3\)do
5:\(res\gets 224\)
6:\(l_{i}\leftarrow\mathit{Bilinear\_Interpolation}(l_{i},res)\) // Unification of multi-scale local features to res through bilinear interpolation
7:\(h_{1}\leftarrow\mathit{Grid\_Sampling}(l_{i},p)\)
8:\(h_{2}\leftarrow\mathit{Grid\_Sampling}(l_{i},p_{s})\)
9:\(l_{i}(p)\leftarrow\mathit{Concatenate}(h_{1},h_{2})\)
10:endfor
11:for\(j=1\to 3\)do
12:if\(j==1\)then
13:\(s_{0}(P)=g_{co}(P)\)
14:\(c_{0}=0\)
15:endif
16:\(s_{j}(P),c_{j}\gets IFDB(s_{j-1}(P),l_{3-j}(p),l_{4-j}(p),P,c_{j-1})\)
17:if\(j==3\)then
18:\(g_{st}(P)\gets j_{i}(P)\)
19:endif
20:endfor
21:return\(g_{st}(P)\)
22:endfunction
```  
**Algorithm 1** Deformation  
Pixel-aligned local feature pairs not only enable the deformed implicit field to reconstruct the finer-grained topological structure details aligned with the image, but also guarantee that the surface details can be correctly recovered using the surface detail decoder. This is achieved by incorporating additional information about the query point and its symmetry point in the object shape into the features.  
Implicit Field Deformation Block  
It is known that the local features with larger scale tend to produce an overall shape, while the ones with smaller scale can keep fine-grained structure details. The IFDB takes advantage of this characteristic, which can be seen in Figure 4. To ensure a smooth implicit field deformation, IFDB deforms the input implicit field according to the variations of the pixel-aligned local features between adjacent scales. Moreover, a state code is used to record all the information of the current implicit field deformation, which will be updated at the end of each IFDB for the next IFDB. In contrast, the coordinates of the query points are input into the deformation module instead of inputting point features in MDISN [25], and our policy performs better than the latter.