Project Description
This project advances topology‑preserving medical image segmentation from 2D to full volumetric data by developing high‑performance tools grounded in persistent homology. Building on a Python proof‑of‑concept, we create an optimized C++ framework that enables efficient computation of induced matchings, meeting the demands of large‑scale convolutional neural network training. To this end, we introduce Betti‑Matching‑3D, a highly accelerated implementation that makes topology‑aware 3D segmentation feasible and significantly improves topological correctness. On top of this, we have developed a loss function for topologically accurate multi‑class medical segmentation, multiclass-BettiMatching, extending Betti Matching to complex anatomical settings by decomposing multi‑class tasks into single‑class persistent‑homology problems. Validated on diverse medical datasets, this approach enhances topological fidelity in cardiac, vascular, and cellular structures while remaining computationally practical. Together, these advances lay the foundation for robust, topology‑aware segmentation in 3D medical imaging.
Results
- Efficient C++/Python implementation drastically accelerates Betti Matching computation.
- Enables practical 3D segmentation network training with topology‑aware loss.
- Demonstrates improved topological correctness across several datasets.
- Outperforms the Cubical Ripser implementation in speed and performance.
- Supports arbitrary‑dimensional inputs with optimized 1D, 2D, and 3D performance.
- Betti-Matching-3D on GitHub
- The proposed multi‑class loss outperforms all baselines in Betti Matching error across all datasets.
- Achieves superior Betti number errors (0 and 1) and clDice on Platelet, TopCoW, and ACDC datasets.
- Matches or exceeds baselines in Dice score, despite being topology‑aware.
- Corrects topological inconsistencies in irregular ACDC slices where post‑processing fails.
- Excels in Platelet segmentation, avoiding structural merging seen in baselines.
- Reduces discontinuous vessel segments in OCTA‑500, crucial for downstream analyses.
- Shows robust performance in TopCoW’s complex 13‑class setting, with lowest variance among methods.
- Ablation reveals that weighting matched vs. unmatched features strongly affects topological accuracy with minimal impact on Dice.
- Demonstrates strong generalization, providing consistently improved topology without harming pixel‑wise performance.
- Topograph on GitHub
Follow-up
As a next step, Betti‑Matching‑3D could be accelerated through GPU‑based implementations, further optimizing C++ performance, and refining the loss function by studying the influence of matched vs. unmatched features and dimension‑specific components. Additional research is needed to understand the behavior of Betti Matching loss during fundamental topological events—such as the formation or removal of connected components, loops, or cavities—to guide improved weighting strategies. For multi‑class Betti Matching, dataset‑adaptive weighting schemes should be developed, as the importance of topological features varies across medical datasets. Further investigation is required to address incorrect spatial matchings seen in existing baselines and to understand dataset‑dependent behavior in complex anatomical structures. This research aims to improve the robustness, scalability, and interpretability of topology‑aware medical image segmentation.
Berger, A.H. et al. (2024). Topologically Faithful Multi-class Segmentation in Medical Images. In: Linguraru, M.G., et al. Medical Image Computing and Computer Assisted Intervention – MICCAI 2024. MICCAI 2024. Lecture Notes in Computer Science, vol 15008. Springer, Cham. https://doi.org/10.1007/978-3-031-72111-3_68
Stucki, N., Bürgin, V., Paetzold, J. C., & Bauer, U. (2024). Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology (arXiv:2407.04683). arXiv. https://doi.org/10.48550/arXiv.2407.04683
Laurin Lux, Alexander H Berger, Alexander Weers, Nico Stucki, Daniel Rueckert, Ulrich Bauer, Johannes Paetzold (2025): Topograph: An Efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation, Poster at the International Conference on Learning Representations (ICLR). https://iclr.cc/virtual/2025/poster/29716
Björn Menze, Biomedical Image Analysis and Machine Learning, Department of Quantitative Biomedicine, Universität Zürich
Johannes Paetzold, Artificial Intelligence in Radiology, Radiology , Weill Cornell Medical College
Suprosanna Shit, Biomedical Image Analysis and Machine Learning, Department of Quantitative Biomedicine, Universität Zürich
Alexander Berger, Institut für KI und Informatik in der Medizin, MRI, TUM
Laurin Lux, Institut für KI und Informatik in der Medizin, MRI, TUM
Vincent Bürgin, Chair of Foundations of Deep Neural Networks, TUM