Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation

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. Our loss demonstrates state-of-the-art performance with up to fivefold faster loss computation compared to persistent homology methods.

@article{lux2025_2411.03228,
  title={ Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation },
  author={ Laurin Lux and Alexander H. Berger and Alexander Weers and Nico Stucki and Daniel Rueckert and Ulrich Bauer and Johannes C. Paetzold },
  journal={arXiv preprint arXiv:2411.03228},
  year={ 2025 }
}