Topological correctness plays a critical role in many image segmentation tasks, yet most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. Existing topology-aware methods often lack robust topological guarantees, are limited to specific use cases, or impose high computational costs. In this work, we propose a novel, graph-based framework for topologically accurate image segmentation that is both computationally efficient and generally applicable. Our method constructs a component graph that fully encodes the topological information of both the prediction and ground truth, allowing us to efficiently identify topologically critical regions and aggregate a loss based on local neighborhood information. Furthermore, we introduce a strict topological metric capturing the homotopy equivalence between the union and intersection of prediction-label pairs. We formally prove the topological guarantees of our approach and empirically validate its effectiveness on binary and multi-class datasets, demonstrating state-of-the-art performance with up to fivefold faster loss computation compared to persistent homology methods.

Segmentation models predominantly optimize pixel-overlap-based loss, an objective that is actually inadequate for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the topology of the segmented structures. However, so far, existing methods only consider global topological properties, ignoring the need to preserve topological features spatially, which is crucial for accurate segmentation. We introduce the concept of induced matchings from persistent homology to achieve a spatially correct matching between persistence barcodes in a segmentation setting. Based on this concept, we define the Betti matching error as an interpretable, topologically and feature-wise accurate metric for image segmentations, which resolves the limitations of the Betti number error. Our Betti matching error is differentiable and efficient to use as a loss function. We demonstrate that it improves the topological performance of segmentation networks significantly across six diverse datasets while preserving the performance with respect to traditional scores. Our code is publicly available (https://github.com/nstucki/Betti-matching/).