Deep Spectral Graph-Convolutional Frameworks for Unsupervised Medical Image Segmentation: Topological Priors and Eigenvalue-Based Structural Reasoning.

Medical image segmentation is frequently limited by the cost, time, and inter-observervariability associated with dense expert annotation. This paper develops a mathematicallyexplicit framework for unsupervised medical image segmentation in which an image isrepresented by a weighted graph and its global structural information is encoded through thespectrum of an appropriate graph Laplacian. The framework combines graph constructionfrom spatial and intensity features, spectral embedding, graph convolution, and unsupervisedobjectives designed to encourage reconstruction fidelity, spectral smoothness, and nondegenerate cluster assignments. In particular, Laplacian eigenvectors provide a globalcoordinate system for image regions, while graph convolutions propagate information alongstructurally meaningful edges. The formulation also incorporates normalized-cut reasoningand a spectral regularizer, and we prove an explicit two-sided spectral bound (Proposition 1)showing how this regularizer is controlled by the algebraic connectivity of the affinitygraph, together with a corollary on why entropy and balance terms remain necessarywhen that connectivity is weak. The paper is deliberately formulated as a methodologicalframework and experimental protocol: it does not claim numerical results that have notbeen independently generated and documented. We specify an evaluation design basedon Dice similarity coefficient, intersection over union, boundary F1 score, and Hausdorffdistance, together with robustness and ablation studies. The resulting formulation providesa reproducible mathematical foundation for investigating whether spectral information canimprove unsupervised segmentation under noise, low contrast, and limited annotation.