# SCGFormer: Semantic Chebyshev Graph Convolution Transformer for 3D Human Pose Estimation
## 3 Main Work
### Preliminaries  
A human skeleton can be represented as a graph and predicted using graph convolution. Let \(J\) denote the number of joints, and \(D_{l}\) represent the dimension of the input data. \(X^{l}\in R^{l\times D_{l}}\) is the input of the L-th layer. The two-dimensional joint coordinates are initialized as \(X^{0}\in R^{l\times 2}\), which is the input of SCGFormer. Then, the output \(X^{l+1}\) of the L-th layer is:  
\[X^{l+1}=\sigma(\hat{D}^{-\frac{1}{2}}\tilde{A}\hat{D}^{-\frac{1}{2}}X^{l}W), \tag{1}\]  
where \(\sigma\) is the activation function RELU, \(\tilde{A}=A+I\), \(A\in R^{l\times J}\) denotes the adjacency matrix of the joints, \(I\) is the unit matrix, \(\hat{D}\) means the diagonal matrix whose elements are the degrees of the nodes, and \(W\in R^{D_{l}\times D_{l+1}}\) indicates a learnable parameter matrix  
Figure 1: The general framework and core module structure of SCGFormer. By leveraging the provided 2D joint information, SCGFormer establishes a vital linkage between 2D and 3D skeletal information, facilitated by the incorporation of SGraAttention and the AcChebGConv block. This orchestrated process culminates in the accurate prediction of intricate 3D skeletal details. of the layer. The adjacency matrix \(A\) cannot aggregate the feature information of its joint, so the matrix \(\bar{A}\) is obtained from \(A\) by adding the self-loop, which can learn the feature information of the joint itself. In the following two graph convolutions, we use \(\bar{A}\), which helps to understand the correlation between joints. Inspired by the work [27], we add a semantic graph convolution (SemGconv) to the SGraAttention module, which better uses the adjacent node relations and graph structure. The general output \(X^{l+1}\) of the L-th layer of the semantic graph convolution is as follows:  
\[X^{l+1}=\sigma(WX^{l}\rho_{i}(M\odot\bar{A})), \tag{2}\]  
\[M\odot\bar{A}= \begin{cases}-\infty&\text{ if }a_{ij}\
eq 1\\ m_{ij}&\text{ if }a_{ij}=1^{\prime},\end{cases}  \tag{3}\]  
In Equation (2), \(\rho_{i}\) is the activation function to normalize the numerical weights; \(M\in R^{I\times I}\) is a learnable weight transformation matrix that performs elemental operations \(\odot\) on the adjacency matrix \(\bar{A}=[a_{ij}]\), the specific formula of which is as shown in Equation (3). If \(a_{ij}=1\), then the value of the corresponding position \(m_{ij}\) is returned; otherwise, negative infinity is returned. In the actual network layer, a smaller value (\(-9\times 10^{15}\)) will be taken instead of negative infinity. \(W\) is the parameter matrix of the layer and is divided into two parts, \(W_{0}\) and \(W_{1}\). \(W_{0}\) relates to the feature-transformed representation of the node itself, while \(W_{1}\) learns the feature representations of all nodes except its own. Then, the detailed output \(X^{l+1}\) of the L-th layer in the semantic graph convolution is as follows:  
\[X^{l+1}=\sigma(I\otimes W_{0}X^{l}\rho_{i}(M\odot\bar{A})+(1-I)\otimes W_{1}X^ {l}\rho_{i}(M\odot\bar{A})), \tag{4}\]  
where \(\otimes\) represents the element-by-element multiplication in the matrix, all nodes of the previous graph convolution share a parameter matrix [29], and semantic graph convolution [27] expresses the correlation between individual nodes better. The Transformer model, which is now the mainstream for natural language, has achieved good results in computer vision [16; 44; 45; 46] due to its self-attention mechanism [41]. The self-attention mechanism processes the feature tensor \(X^{l}\) through three multilayer perceptions (MLPs) [42] to output three matrices: \(Q^{l}\), \(K^{l}\), and \(V^{l}\). Then, the output feature is obtained by matrix multiplication. The output of the L-th layer of the Transformer is formulated as follows:  
\[X^{l+1}=\sigma(\frac{Q^{l}K^{l^{\prime}}}{\sqrt{d_{k}}})V^{l}, \tag{5}\]  
where \(\sigma\) represents softmax [47; 48]. Its elements are scaled individually with \(\sqrt{d_{k}}\) to avoid a sharp distribution of matrix elements.