GeoTop: Advancing Image Classification with Geometric-Topological Analysis

A fundamental challenge in diagnostic imaging is the phenomenon of topological equivalence, where benign and malignant structures share global topology but differ in critical geometric detail, leading to diagnostic errors in both conventional and deep learning models. We introduce GeoTop, a mathematically principled framework that unifies Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) to resolve this ambiguity. Unlike hybrid deep learning approaches, GeoTop provides intrinsic interpretability by fusing the capacity of persistent homology to identify robust topological signatures with the precision of LKCs in quantifying local geometric features such as boundary complexity and surface regularity. The framework's clinical utility is demonstrated through its application to skin lesion classification, where it achieves a consistent accuracy improvement of 3.6% and reduces false positives and negatives by 15-18% compared to conventional single-modality methods. Crucially, GeoTop directly addresses the problem of topological equivalence by incorporating geometric differentiators, providing both theoretical guarantees (via a formal lemma) and empirical validation via controlled benchmarks. Beyond its predictive performance, GeoTop offers inherent mathematical interpretability through persistence diagrams and curvature-based descriptors, computational efficiency for large datasets (processing 224x224 pixel images in less or equal 0.5 s), and demonstrated generalisability to molecular-level data. By unifying topological invariance with geometric sensitivity, GeoTop provides a principled, interpretable solution for advanced shape discrimination in diagnostic imaging.