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Topograph: An Efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation
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Proposes Topograph, a linear-time loss built on a superpixel-style combined component graph of prediction and ground truth that targets topologically critical regions, plus a DIU metric comparing union and intersection homotopy.
Problem
Pixel-wise losses such as Dice ignore topological correctness in segmentation, while existing topology-aware losses either lack strict guarantees, are task-specific, or are computationally expensive (persistent homology).
Approach
Build a combined component graph from the binarized prediction and ground truth (4-class TP/TN/FP/FN components); mark incorrectly predicted nodes that are not "regular" (exactly one correct foreground and one correct background neighbor) as critical, and apply a loss averaging class scores over critical regions. Zero loss implies homotopy equivalence of prediction, ground truth, union and interse
Key results
- On Roads, Topograph achieves the lowest DIU (6.521) and BM (5.635) errors among compared losses, and on Platelet it reaches the best clDice (0.843) and DIU (10.906).
- Loss computation averages 95.94 ms versus 327.64 ms for Betti matching and 602.99 ms for HuTopo, a roughly 3-6x speedup on average loss calculation.
- On the multi-class TopCoW dataset, Topograph obtains the best DIU (10.477) and BM (0.658) while matching Dice (0.735) of the best baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Topograph instead of Betti Matching | U-Net segmentation · Roads | DIU: 6.733 -> 6.521 | Helped |
| Use Topograph instead of the strongest baseline | U-Net segmentation · TopCoW | BM: 0.678 -> 0.658 | Helped |
| Use Topograph instead of ClDice | U-Net segmentation · Cremi | BM: 10.368 -> 10.392 | Hurt |
| Add Gaussian threshold variation instead of fixed binarization (ablation) | Topograph · Buildings validation | DIU: 51.5625 -> 46.125 | Helped |
| Set threshold variation to 0.1 instead of 0. (ablation) | Topograph · Buildings validation | BM: 40.6250 -> 35.5625 | Helped |
| Set topological loss weight alpha to 0.2 instead of 0.0 (ablation) | Topograph · TopCoW validation | DIU: 10.944 -> 6.5556 | Helped |
| Set alpha to 0.5 instead of 0.2 (ablation) | Topograph · TopCoW validation | DIU: 6.5556 -> 11.3889 | Hurt |
| Use RMS instead of mean aggregation (ablation) | Topograph · Roads validation | DIU: 4.70 -> 4.25 | Helped |
| Use max instead of mean aggregation (ablation) | Topograph · Buildings validation | DIU: 47.7500 -> 45.7500 | Helped |
| Use max instead of mean aggregation (ablation) | Topograph · Platelet validation | DIU: 11.25722 -> 10.86111 | Helped |
| Increase training data from 12.5% to 100% (ablation) | Topograph · Buildings validation | BM: 72.8929 -> 59.2381 | Helped |
| Use Topograph instead of Betti Matching for loss calculation | Platelet training · Platelet | Average loss calculation time: 327.64 ms -> 95.94 ms | Helped |
Surprising findings
- Dice loss yields the highest Dice and clDice on several datasets yet is worse on topological metrics, showing that pixel accuracy does not reflect topological correctness.
- HuTopo attains very low B0 error on Buildings but high BM and DIU errors, indicating correct component counts without spatial alignment.
Limitations
- Currently restricted to 2D images; 3D extension left to future work.
- Binarization discards topological information across thresholds compared with filtration-based persistent homology methods.
- Loss is only tested with a single U-Net architecture trained from scratch.
Experimental setup
Datasets: CREMI Roads Buildings Platelet TopCoW
Benchmarks: Buildings Roads CREMI Platelet TopCoW
Metrics: DIU Betti matching error B0 error B1 error Dice clDice
Baselines: Dice loss clDice HuTopo Betti matching loss Mosin loss
Builds on: Betti matching clDice HuTopo U-Net
Closest papers by meaning
- Structure-Aware Image Segmentation with Homotopy Warping (NeurIPS 2022 · similarity 0.92)
- Topology-Preserving Deep Image Segmentation (NeurIPS 2019 · similarity 0.91)
- Topologically Faithful Image Segmentation via Induced Matching of Persistence Barcodes (ICML 2023 · similarity 0.91)
- TopoDiffusionNet: A Topology-aware Diffusion Model (ICLR 2025 · similarity 0.88)
- SegGraph: Leveraging Graphs of SAM Segments for Few-Shot 3D Part Segmentation (NeurIPS 2025 · similarity 0.86)
- CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves (NeurIPS 2026 · similarity 0.86)
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