# MD-TransUNet: An Image Segmentation Network for Car Front Face Design
## 2 Materials and Methods
### DCGCN Module  
The shape and edge information of the headlights and grille regions in front images of different car models can vary significantly. Graph convolutional networks (GCNs) map various features onto nodes and edges within a complex topological graph structure [28], allowing them to capture dynamic changes in sample intensity sensitively. GCNs are widely used in deep learning tasks for their ability to dynamically adjust learnable parameters to extract spatial information efficiently. By utilizing GCNs, dynamic topological relationships between channels can be captured, optimizing intra-channel functions. To better adjust segmentation areas and capture dynamic topological information between image feature channels, this paper employs the DCGCN module, as shown in Figure 3. By dynamically adjusting the convolution kernels, the DCGCN module adapts flexibly to these differences, thereby improving feature extraction effectiveness and enhancing the fine segmentation performance of headlights and grilles. Prior to executing graph convolution, the feature map with dimensions \(512\times 64\times 64\) is converted into a graph. After global average pooling, the initial feature map is transformed into a comprehensive \(512\)-dimensional column vector representation \(w\) for further processing. To achieve a detailed understanding of the topological relationships among different image channels, the column vector \(w\) is first processed using a specialized matrix activation function. Following this step, the learned parameter matrix is multiplied to obtain and improve the dynamic properties of the neighborhood matrix. Consequently, the derivation process of adjacency matrix \(A\in R^{512\times 512}\) is derived as follows:  
\[T(w)_{ij}=||\frac{1-e^{-(w_{i}-w_{j})}}{1+e^{-(w_{i}-w_{j})}}-1||(i,j=1,2,\dots,5 12), \tag{9}\]  
\[A_{ij}=\frac{T(w)_{ij}+T(w)_{ji}}{2}\times\Theta_{ij}(i,j=1,2,\dots,512) \tag{10}\]  
In the equation, where \(w_{i}\) and \(w_{j}\) are the \(i\)-th and \(j\)-th elements of \(w\), the function of \(T\) represents the matrix activation mechanism used in this context. Additionally, \(\Theta\in R^{512\times 512}\) denotes the learnable dynamic parameter structure, which plays a crucial role in the process. The matrix activation mechanism in Equation (9) is defined as a continuous function ranging from 0 to 1, ensuring smooth transitions. It maps the vector of calculated channel weights to a configuration in which the edge weights between the corresponding channel nodes are expressed by non-diagonal elements and the diagonal elements are set to 1 for stability. Equation (10) dynamically adjusts the connection weights based on the input parameters, with \(\Theta\) fine-tuning the topological relationships to obtain the final matrix configuration. In a graph convolution network, each node is associated with a corresponding feature vector. In this study, we represent the attributes of each node in a graph-structured data format by reconstructing the features of a single channel into 4096 \(\times\) 1 vectors. The result of this transformation is the construction of a graph with 512 nodes, where each node has a 4096 \(\times\) 1 feature vector. To perform an efficient convolution of the processed graph data, a method known as spectral filtering is used [29]. Spectral filtering can be briefly described as a process in which an \(x\) signal is modified by a \(g\) filter in the Fourier domain:  
\[g*x=U\text{diag}(g)U^{\top}x, \tag{11}\]  
In this formula, the matrix \(U\), encompassing the eigenvectors derived from the normalized graph Laplacian, is described by the equation \(L=I-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}=U\Lambda\U^{\top}\). The matrix \(\mathbf{\Lambda}\) is the diagonal matrix containing the eigenvalues of \(L\), where \(D\) represents the degree matrix, \(A\) denotes the adjacency matrix, and \(l\) is the identity matrix. The function \(g\) is treated as a function of \(\mathbf{\Lambda}\) denoted \(g(\mathbf{\Lambda})\).