GeoTop: Advancing Image Classification with Geometric-Topological Analysis

TL;DR

GeoTop is introduced, a mathematically principled framework that unifies Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) to resolve topological equivalence ambiguity in diagnostic imaging.

Abstract

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.

Figures (6)

• Figure 1: Conceptual overview of the GeoTop framework. The pipeline integrates (left) image filtration, (middle) topological feature extraction via persistent homology, and (right) geometric characterisation through LKCs, with fused features enabling enhanced classification.
• Figure 2: (a) Distribution of prediction scores for TDA, LKC, and GeoTop methods, showing certain performance overlap between TDA and LKC (mean AUC = $0.84$) and improved accuracy for GeoTop (AUC = $0.87$). (b) Average confusion matrices from bootstrapped random forest classification, demonstrating reduced false positives/negatives in the combined GeoTop approach. Data derived from $500$ iterations of $80/20$ train-test splits on skin lesion images.
• Figure 3: Comparative topological analysis of benign and malignant lesions. (a-b) Persistence barcode and diagram for benign cases showing discriminative $H_1$ features. (c-d) Corresponding topological features for malignant lesions highlighting characteristic $H_0$ patterns.
• Figure 4: Geometric characterisation of skin lesions via LKCs. (a) Comparative profiles of normalised LKCs--Area ($\hat{\mathcal{A}}$), Perimeter ($\hat{\mathcal{P}}$), and Euler-Poincaré characteristic ($\hat{\mathcal{E}}$)--for benign (teal) and malignant (crimson) lesions across normalised intensity thresholds [0,1]. Shaded regions represent $\pm1$ standard deviation around the mean (solid line), computed across $1,800$ benign and $1,497$ malignant cases. Malignant lesions exhibit significantly greater variance in mid-range thresholds ($0.4-0.6$, highlighted in gold, $p<0.01$, two-tailed $t$-test), particularly in perimeter measurements, reflecting their irregular, invasive boundaries. The Euler characteristic profiles further differentiate lesion types by quantifying dynamic transitions between connected components and holes. Rectangles indicate diagnostically critical thresholds where geometric features achieve maximal class separation (see Methods). (b) Distribution of each LKC value at the diagnostically critical threshold ($t=0.5$). Violin plots show the kernel probability density, with white dots indicating the mean. Malignant lesions show statistically significant divergence from benign cases in all three geometric descriptors ($p$ values shown, two-tailed $t$-test), confirming that geometric heterogeneity is a robust marker of malignancy. The non-Gaussian distributions underscore the necessity of threshold-dependent morphological analysis over static feature extraction. The integrated geometric analysis provided by LKCs captures critical morphological patterns--such as boundary irregularity and topological complexity--that are frequently missed by topological or deep learning approaches alone.
• Figure 5: Comparative topological analysis of benign and malignant lesions. (a) Upper -- Examples of images that are well labeled by GeoTop but wrongly classified by TDA and LKC. Lower -- Examples of images that are well labeled by LKC and TDA but wrongly classified by GeoTop. (b) Upper -- Examples of images that are well labeled by TDA but wrongly classified by LKC. Lower -- Examples of images that are well labeled by LKC but wrongly classified by TDA. (c) Examples of bad predictions for all three methods.

Theorems & Definitions (2)

• lemma 1: Geometric Separability beyond Topological Equivalence
• proof