## 4 Methods
### Feature Construction Module (FCM)  
In graph neural networks, the number of nodes and edges is fixed, allowing for the construction of relevant features. This process serves as a means of data preprocessing and dimensionality reduction, thereby improving the accuracy and interpretability of model training and guiding models to learn more effective data representations. Subsequently, features are designed and extracted from the graph data of the geometric point clouds. For the graphs \(G_{src}\) and \(G_{tgt}\) to be registered, sorting is performed in the order of \(G_{src1}\), \(G_{tgt1}\), \(G_{src2}\), and \(G_{tgt2}\) to facilitate subsequent feature construction.  
Node features are categorized into two main types: geometric features and complex network features. Geometric features, due to their intuitiveness, can effectively capture the geometric attributes of building structures and are computationally simple and efficient. Specifically, geometric features include degree, mean distance, minimum distance, maximum distance, mean angle, minimum angle, maximum angle, and mean orientation direction.  
The calculation formulas for degree, mean distance, mean orientation, and mean angle are provided in Table 1.  
The characteristics of complex networks can reveal the essence and laws of the entire system and highlight several important key nodes in the geometric point cloud. The relevant formulas and meanings are presented as Table 2:  
Based on these definitions, a 15-dimensional feature vector is constructed. Finally, all features are normalized to ensure consistency and comparability.  
For edge features, we similarly designed several important edge attributes, including the following:  
* Edge length: The Euclidean distance between the two endpoints of the edge.
* Edge direction: The normalized direction vector from one endpoint to the other (3 dimensions including xyz).
* Edge loop count: The number of loops occupied by an edge. The meaning of a ring is to characterize the number of connected edges that are connected head to tail to form a closed loop.  
These features collectively form a five-dimensional edge feature vector. This helps in understanding the structure and layout of objects. Finally, the edge features are also normalized.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Formula** & **Meaning** \\ \hline Degree & \(k_{i}=\sum\limits_{j=1}^{N}A_{ij}\) & The number of edges connected to vertex \(i\). \\ Average Distance & \(L=\dfrac{1}{k_{i}}\sum\limits_{i\
eq j}d_{ij}\) & The average distance of the sum of the lengths of all edges where the point is located. \\ Average Orientation & \(\vec{v}_{\text{avg}}=\dfrac{1}{k_{i}}\sum\limits_{i\
eq j}\vec{v}_{ij}\) & The value obtained by normalizing the sum of vectors extending in the direction of all edges where the point is located, starting from that point. \\ Average Angles & \(\theta_{avg_{i}}=\dfrac{2}{k_{i}(k_{i}-1)}\sum\limits_{j=1}^{N}\theta_{ij}\) & The average degree of angle between the vectors that extend from this point to all the edges where the point is located, measured as the angle between each pair of vectors. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Formula and meaning of geometric characteristics.