# Advancing Brain Tumor Segmentation with Spectral-Spatial Graph Neural Networks
## 2 Methods and Material
### GNN Layers  
Our Spatial Graph Neural Network (GNN) utilizes a multi-layer GraphSAGE architecture with pooling layers and ReLU activations for enhanced feature learning. This design progressively aggregates information from each node's neighborhood, allowing the network to capture complex patterns within the graph structure. The transformation applied to node features is illustrated by the following equation:  
\[h^{(l+1)}=\sigma\bigg{(}W^{(l)}\cdot\bigg{(}h^{(l)}||\max_{\forall\text{vs.}\in V (u)}\sigma(W_{\text{pool}}\cdot h^{(l)})\bigg{)}\bigg{)} \tag{3}\]  
where each parameter represents the following:  
* \(h^{(l)}\) is the feature vector of a node at layer \(l\). * \(h^{(l+1)}\) is the updated feature vector of the node at layer \(l+1\). * \(\sigma\) denotes a differentiable, non-linear activation function, such as ReLU. * \(W^{(l)}\) is the weight matrix specific to layer \(l\), applied during feature transformation. * \(W_{\text{pool}}\) is a global trainable weight matrix used in the pooling operation. * The concatenation operator \(||\) combines the node's original features with the aggregated neighborhood features. * \(V(u)\) represents the set of nodes that are neighbors of node \(u\). * \(\max_{\forall\text{vs.}\in V(u)}\) applies a max pooling operation over all features of the neighbors \(v\) of node \(u\) after applying the activation function and weight matrix \(W_{\text{pool}}\). For Spectral GNN, we utilize Chebyshev Convolution (ChebConv). ChebConv represents a form of Spectral Graph Neural Networks (GNNs) [27] and leverages the eigen-decomposition of the graph Laplacian to interpret graph convolution in the Fourier domain. Among these methods, the Chebyshev Convolution (ChebConv) layer stands out due to its efficient approach in capturing complex patterns in graph data by employing Chebyshev polynomials of the first kind for spectral filtering. Chebyshev polynomials [28] of the first kind, \(T_{k}(x)\), are defined by the recurrence relation:  
\[T_{0}(x) =1,\] \[T_{1}(x) =x,\] \[T_{k+1}(x) =2 xT_{k}(x)-T_{k-1}(x)\text{ for }k\geq 1.\]  
These polynomials form an orthogonal basis in the space of continuous functions and are particularly suitable for approximating functions in a truncated series. The ChebConv layer utilizes these polynomials to approximate the spectral graph convolutional operator, thereby avoiding the computationally intensive eigen-decomposition of the graph Laplacian. The graph convolution operation in ChebConv is executed in the spectral domain using Chebyshev polynomials. Specifically, the convolution of a graph signal \(x\) with a filter \(g\), parameterized by weights \(\theta\), is approximated using the first \(K\) Chebyshev polynomials of the scaled graph Laplacian \(\tilde{L}\):  
\[g_{\theta}(L)\cdot x\approx\sum_{k=0}^{K-1}\theta_{k}T_{k}(\tilde{L})x,\]  
where \(\tilde{L}=2L/\lambda_{max}-I\) is the scaled Laplacian, \(L\) is the graph Laplacian, \(\lambda_{max}\) is the largest eigenvalue of \(L\), and \(I\) is the identity matrix. This approximation facilitates a \(K\)-localized filter, meaning that each node aggregates information from its \(K\)-hop neighborhood, enabling the ChebConv layer to capture local structures in the graph efficiently. ChebConv layers offer significant advantages, including scalability to large graphs and flexibility in handling varying graph structures without the need for re-training, making them particularly suitable for inductive learning tasks. They are parameterized by the number of input and output features (in_feats and out_feats), the order of the Chebyshevpolynomial \(K\) (determining the size of the filter and the extent of the neighborhood in the convolution), and optionally the activation function and bias term. The integration of ChebConv layers into deep GNNs allows for the extraction and utilization of rich topological and feature-based information from graphs, making their spectral approach a robust mechanism for learning on graph-structured data.