## 3 Model Structure
### Construct Graph Structure
#### 3.2.2 Construct Inter-Frame Adjacency Matrix  
In target tracking tasks, methods that utilize the temporal sequence of image sequences to assist in tracking the target have been widely applied. For example, the inter-frame differencing method and the inter-frame differencing multiplication method were used to improve the tracking target detection accuracy by extracting and processing inter-frame information from the sequence of images. On the other hand, context-based algorithms utilized the changing patterns of the target object's shape and motion attributes over time to complete the target tracking task. This approach is mainly suitable for application scenarios that required high tracking accuracy and robustness. Graph convolutional networks could aggregate the neighbor information of nodes and extract the features of those nodes. In the previous section, we constructed the intra-frame adjacency matrix for the tracked target using superpixel blocks as nodes, thus extracting the spatial features of the tracked target. However, in addition to identifying the current spatial position of the target object, the model also needed to predict the possible future spatial positions of the target object. Therefore, it was necessary to use a graph structure to represent the relationship of the target object's motion direction across consecutive frames. To distinguish it from the previously introduced concept of the \"intra-frame adjacency matrix\", we referred to the adjacency matrix that describes the target object's motion direction information as the \"inter-frame adjacency matrix\". The corresponding graph structure was called the \"inter-frame graph structure\". Therefore, based on the definition of the graph structure, the inter-frame graph structure could be expressed as follows:  
\[G_{t-1,t}=\{A_{t-1,t},D_{t-1,t},F_{t-1,t}\},\quad m=1,\ldots,M \tag{12}\]  
where \(G_{t-1,t}\) represented the inter-frame graph structure composed of the superpixels from the \(t-1\)th frame and the \(t\)th frame image, and \(A_{t-1,t}\) was the inter-frame adjacency matrix that constituted \(G_{t-1,t}\). Its calculation process could be expressed as follows:  
\[A_{t-1,t}= \begin{bmatrix}\alpha_{11}&\cdots&\alpha_{1N}\\ \vdots&\ddots&\vdots\\ \alpha_{N1}&\cdots&\alpha_{NN}\end{bmatrix}  \tag{13}\]  
\[a_{ij}= \begin{cases}1,&\text{if }s_{t-1,i}\leftrightarrow s_{ij},\\ 0,&\text{if }s_{t-1,i}\leftrightarrow s_{ij},\end{cases} \quad i=1,\ldots,N, \quad j=1,\ldots,N \tag{14}\]  
where \(s_{t-1,i}\) was the \(i\)th superpixel block in the \(t-1\)th frame, \(s_{ij}\) was the \(j\)th superpixel block in the \(t\)th frame, and the connection criterion between \(s_{t-1,i}\) and \(s_{ij}\) could be expressed as follows:  
\[\left\{ \begin{array}{l}s_{t-1,i}\leftrightarrow s_{t,j},\,s_{t-1,i}>L_{t} \text{ and }s_{t,j}>L_{t},\\ s_{t-1,i}\leftrightarrow s_{t,j},\,s_{t-1,i}\leq L_{t}\text{ and }s_{t,j}\leq L_{t}. \end{array} \right. \tag{15}\]  
Here, \(L_{t-1}\) and \(L_{t}\), respectively, represented the average brightness of all superpixels in the \(t-1\)th frame and the \(t\)th frame. As shown in Equations (14) and (15), if the brightness of the \(n\)th superpixel block in the \(t-1\)th frame was greater than the average brightness of that frame, and the brightness of the \(i\)th and \(j\)th superpixel blocks in the \(t\)th frame was also greater than the average brightness of that frame, then the elements in the \(i\)th and \(j\)th columns of the \(n\)th row of the inter-frame adjacency matrix were assigned a value of 1. \(D_{t-1,t}\) represented the inter-frame degree matrix composed of the superpixels from the \(t-1\)th frame and the \(t\)th frame image. \(D_{t-1,t}\) could be obtained from \(A_{t-1,t}\), and its calculation process is referred to in Equation (8). \(F_{t-1,t}\) represented the inter-frame feature matrix between the \(t-1\)th and \(t\)th frames. Unlike the calculation method of the intra-frame feature matrix, the inter-frame feature matrix was not directly expressed by the intra-frame superpixel information.