## 3 Method  
We propose a CNN-GRU-SA deduction network, as shown in Figure 2. The architecture combines the GRU and SA modules and is a multi-input single-output network architecture. Mainly, two methods exist for a multi-input network architecture to perform data fusion: feature-level fusion and decision-level fusion. Feature-level fusion involves extracting features from different input data and completing subsequent learning tasks by performing data fusion at the feature level. Decision-level fusion results from a deep learning model that is first trained with different input features and then fused with the outputs of multiple models [13]. The proposed CNN-GRU-SA deduction network adopts a decision-level fusion method, and the training process can be divided into three stages: input feature extraction, attention mechanism fusion, and output features. (a) Input feature extraction stage  
The pre-change control point features (i.e., 3D coordinates \(C_{0}\)) and post-change control point features (i.e., 3D coordinates \(C_{t}\)) of the network architecture input are expanded using the GRU module for feature data. The reset gate of the GRU can control the intensity of the information of the input feature being ignored, whereas the update gate helps preserve the input information. Using the GRU, the long-term dependence of the control point features can be captured better and faster, and the convergence speed of the training can be accelerated. The feature operation is as follows:  
\[{C_{0}}^{\prime}=G(C_{0}), \tag{1}\]  
\[{C_{t}}^{\prime}=G(C_{t}), \tag{2}\]  
Figure 2: CNN-GRU-SA network architecture. where \(G\) is the GRU feature extraction module, \(C_{0}{{}^{\prime}}\) is the feature after \(C_{0}\) expands its dimension, and \(C_{t}{{}^{\prime}}\) is the feature after \(C_{t}\) expands its dimensions. Simultaneously, the pre-change point cloud features (i.e., the overall shape \(S_{0}\)) input by the network architecture are expanded with feature data through a feature extraction module comprising four sets of CNN, batch normalization (BN), and ReLU activation functions. First, the feature dimensions are expanded using a CNN to enhance the robustness of the data. Second, to prevent excessive deviations and offsets, BN is used to standardize the features and unify the data distribution. Finally, the rectified linear unit activation function is used to enhance the ability of the features to change nonlinearly. The feature operation is as follows:  
\[S_{1}=R(B(F(S_{0}))), \tag{3}\]  
\[S_{2}=R(B(F(S_{1}))), \tag{4}\]  
\[S_{3}=R(B(F(S_{2}))), \tag{5}\]  
\[{S_{0}}{{}^{\prime}}=R(B(F(S_{3}))), \tag{6}\]  
where \(F\) is a 1D convolution operation with a convolution kernel size and step size of one. \(B\) is the batch operation and \(R\) is the activation function ReLU. \(S_{1}\), \(S_{2}\), \(S_{3}\), and \({S_{0}}{{}^{\prime}}\) are the features of dimension expansion. At this stage, the number of input features is increased from 3 to 256, and the number of features in the first dimension remains equal. Attention mechanism fusion stage  
First, the three new features \(C_{0}{{}^{\prime}}\), \(C_{t}{{}^{\prime}}\), and \({S_{0}}{{}^{\prime}}\) obtained during the input feature extraction stage are concatenated in the first dimension to form new features. The feature operation is as follows:  
\[S_{con}=Con({C_{0}}{{}^{\prime}},C_{t}{{}^{\prime}},S_{0}{{}^{\prime}}), \tag{7}\]  
where \(Con\) is the concatenation function and \(S_{con}\) is the new feature obtained after concatenation. Subsequently, the new features are sent to the SA module to obtain the attention weight. The feature operation is as follows:  
\[S_{SA}=SA(S_{con}), \tag{8}\]  
where \(SA\) is the self-attention mechanism module and \(S_{SA}\) is the obtained attention weight. Finally, the new feature \(S_{con}\), formed by splicing, and the attention weight \(S_{SA}\) are added. The feature operation is as follows:  
\[{C_{0}}^{\prime}=G(C_{0}), \tag{1}\]0  
where \({S_{t}}{{}^{\prime}}\) represents a new feature under the attention mechanism.