# Advancing Brain Tumor Segmentation with Spectral-Spatial Graph Neural Networks
## 3 Results and Discussion  
Our comprehensive evaluation of supervoxel segmentation algorithms for 3D brain MRI scans demonstrates the superior performance of the Voxel Cloud Connectivity Segmentation (VCCS) method across multiple metrics. The results, summarized in Table 1, illustrate VCCS's leading edge in homogeneity (0.03), moment of inertia (1.2), shape uniformity (0.10), size uniformity (0.10), and time complexity (O(N log N)), affirming its efficiency and effectiveness in generating precise supervoxels. VCCS exhibited the highest homogeneity, indicating a consistent intensity within supervoxels, crucial for distinguishing between various brain tissues accurately. Its moment of inertia was the lowest, reflecting compact and well-defined supervoxels essential for detailed anatomical analysis. Additionally, VCCS achieved the best scores in shape and size uniformity, ensuring the segmentation's reliability and facilitating subsequent image processing tasks. Comparatively, other algorithms such as SLIC, Watershed, Mean Shift, Felzenszwalb-Huttenlocher, and Region Growing showed varying performance levels but did not match VCCS's balanced excellence across all evaluated aspects. For instance, while some algorithms like SLIC were fast (O(N)), they could not provide the same level of detail and accuracy in segmentation as VCCS. Figure 5: Accuracy vs. epochs for three learning rates. The time complexity of VCCS, O(N log N), balances between computational efficiency and segmentation quality, making it a practical choice for large-scale medical image analysis applications. These findings underscore VCCS as the most suitable algorithm for supervoxel segmentation in 3D brain MRI scans, offering a compelling balance of accuracy, computational efficiency, and segmentation quality. Following the development of the graph structure derived from brain MRI data, spectral analysis was conducted to evaluate its structural integrity, as illustrated in Figure 6. The \"Histogram of Laplacian Eigenvalues\" reveals a distribution with a significant concentration of values in the lower range, indicative of the presence of many low-frequency components in the graph. These components suggest gradual variations in voxel intensity across the graph, as in the main images, highlighting coherent structural patterns. The detection of extremely small eigenvalues points to the existence of connected components or a notable community structure within the graph, reflecting distinct anatomical regions of the brain. The Spy Plot of the Laplacian Matrix visually depicts the non-zero elements of the graph's Laplacian matrix. The dense clustering of dots along the diagonal highlights the connectivity of nodes to themselves or their immediate neighbors, underscoring spatial adjacency in terms of voxel intensity and structural coherence. In contrast, the sparse presence of dots elsewhere in the matrix indicates a lack of direct connections between most voxel pairs. This pattern implies that the graph may be extensive and sparse, containing several localized clusters or communities that echo the brain's complex structural organization. Our investigation delves into the comparative performance of Spatial GNN, Spectral GNN, and Spectral-Spatial GNN approach, aiming to highlight the significant advancements and insights these methods offer in brain tumor segmentation. The Spatial GNN demonstrates commendable performance with an accuracy of 92.5%. This highlights the effectiveness of spatial features in identifying tumor regions within MRI scans. Conversely, the Spectral GNN, which processes data in the spectral domain, shows a higher accuracy of 95.3%, indicating the additional benefits of spectral analysis in enhancing segmentation outcomes (See Table 4). Figure 6: Figure (**a**) on the left displays the Spy Plot of the Laplacian matrix, highlighting sparse connectivity. The middle Figure (**b**) showcases the Enhanced Incidence Matrix Visualization, revealing intricate graph connections.