Source-linked AI summary
Topology-Preserving Deep Image Segmentation
Xiaoling Hu, Li Fuxin, Dimitris Samaras, Chao Chen
TL;DR
Fine-scale segmentation methods can make topological errors such as broken connections, even when pixel-wise accuracy is high. The paper introduces TopoNet, which trains an end-to-end deep network with a differentiable persistent-homology-based loss matching predicted and ground-truth Betti numbers. Across natural and biomedical datasets, it reports better topology-relevant performance, while its scope focuses on binary segmentation and 1-dimensional topology.
Problem
Fine-scale structures and thin connections remain vulnerable to topological segmentation errors despite satisfactory per-pixel accuracy.
Method
TopoNet uses a continuous differentiable topological loss based on persistent homology to match a prediction’s topology with the ground truth during end-to-end neural-network training.
Results
TopoNet significantly outperforms existing methods on topology-aware metrics, including Betti number error, ARI, and VOI, without sacrificing pixel accuracy across natural and biomedical datasets.
Takeaways & Limitations
The persistent-homology loss is generic and can be incorporated into different deep segmentation architectures for fine-structure segmentation.
Takeaways & Limitations
The formulation assumes binary segmentation with one likelihood function, and evaluation and training focus on 1-dimensional topology.
Abstract
from arXiv · showhide
Segmentation algorithms are prone to make topological errors on fine-scale structures, e.g., broken connections. We propose a novel method that learns to segment with correct topology. In particular, we design a continuous-valued loss function that enforces a segmentation to have the same topology as the ground truth, i.e., having the same Betti number. The proposed topology-preserving loss function is differentiable and we incorporate it into end-to-end training of a deep neural network. Our method achieves much better performance on the Betti number error, which directly accounts for the topological correctness. It also performs superiorly on other topology-relevant metrics, e.g., the Adjusted Rand Index and the Variation of Information. We illustrate the effectiveness of the proposed method on a broad spectrum of natural and biomedical datasets.
1 Introduction
TopoNet addresses topology errors in fine-scale segmentation by using a differentiable topological loss derived from persistent homology. It targets topological fidelity while retaining pixel-level accuracy and improves topology-relevant evaluation across natural and biomedical settings.
- Motivation: Fine-scale structures such as thin connections can suffer segmentation errors despite strong per-pixel accuracy, compromising analyses that depend on their structure.Examples include ropes and handles in robotics and thin objects in biomedical images.
- Proposed method: TopoNet enforces agreement between predicted and ground-truth topology by matching their Betti numbers, which count connected components and handles.The method seeks high topological fidelity without sacrificing per-pixel accuracy.
- Proposed method: Persistent homology summarizes the neural network’s likelihood function across all thresholds instead of evaluating topology only after thresholding at 0.5.This yields a unified differentiable measurement of topological similarity.
- Results: TopoNet is presented as an end-to-end network with guaranteed topological correctness when its topological loss reaches zero.The paper validates it against state-of-the-art methods on natural and biomedical datasets with fine-scale structures.
- Relation to prior work: Compared with prior topology-aware approaches, TopoNet directly computes topology and is designed to generalize beyond elongated shapes or fixed topological constraints.Earlier approaches may lack topological guarantees even at zero loss or cannot generate segmentations preserving learned topological priors.
2 TopoNet
TopoNet combines per-pixel segmentation with a differentiable topological loss based on persistent homology, targeting agreement between predicted and ground-truth topology. Its zero-loss condition guarantees matching Betti numbers after thresholding, while training uses likelihood maps and topology-aware gradients.
- 2 TopoNet: TopoNet trains a deep network with a topological loss alongside per-pixel cross-entropy to achieve pixel accuracy and topological correctness.The likelihood map f is predicted by the network, while g denotes the ground truth; λ controls the topological-loss weight.
- 2.1 Topology and Persistent Homology: Fixed-threshold Betti-number comparison is discrete and cannot distinguish likelihood maps with identical segmentations but different structural repair difficulty.Deeper gaps near broken bridges or handles require more modification, motivating a continuous-valued measure.
- 2.1 Topology and Persistent Homology: Persistent homology summarizes topological structures across all thresholds in persistence diagrams, enabling comparison of predicted likelihood f with ground truth g.As thresholds decrease, structures are born and killed; each diagram dot records a structure’s birth and death thresholds.
- 2.2 Topological loss and the Gradient: The topological loss measures the minimal effort to move persistence-diagram dots to matched points, treating unmatched predicted dots as removable noise on the diagonal.The correspondence is dimension-specific for connected components and handles, and the loss is piecewise differentiable except on measure-zero cases.
- 2.2 Topological loss and the Gradient: When Ltopo(f, g) reaches zero, thresholding f at 0.5 produces a segmentation with the same Betti number as g.This theorem provides the method’s stated topological-correctness guarantee.
- 2.2 Topological loss and the Gradient: TopoNet computes topological information around critical points and uses the resulting gradient to optimize the network’s predicted likelihood function.The likelihood is piecewise linear in practice, with pixel values controlling its linear interpolation over the image domain.
3 Experiments
Experiments evaluate TopoNet on six natural and biomedical datasets using pixel accuracy and topology-aware metrics. Results show stronger topological accuracy and improved structural consistency, while training analysis exposes a trade-off between topological and pixel-wise losses.
- Datasets: TopoNet is evaluated on six natural and biomedical datasets spanning neuron membranes, road cracks, roads, and retinal vessels.The datasets are CREMI6, ISBI12, ISBI13, CrackTree, Road, and DRIVE.
- Metrics: Evaluation uses pixel-wise accuracy, Betti number error, Adjusted Rand Index, and Variation of Information.Betti number error directly compares the number of handles between segmentation and ground truth, while ARI and VOI compare induced partitions.
- Quantitative and qualitative results: TopoNet significantly outperforms existing methods on all three topology-aware metrics without sacrificing pixel accuracy.The comparison includes DIVE, U-Net, and Mosin.; Tables 1 and 2 report results across the evaluated datasets.
- Quantitative and qualitative results: Qualitative comparisons show TopoNet more consistently segments fine structures such as membranes, roads, and vessels than the other methods.Figure 5 compares sample images, ground truth, and outputs from DIVE, U-Net, Mosin., and TopoNet.
- Training analysis: After about 30 epochs, total loss stabilizes while cross-entropy rises slightly as topological loss decreases.The authors attribute this behavior to topological loss concentrating training on critical pixels, which may slightly reduce overall pixel accuracy.
- Training analysis: As training progresses, thresholded membrane predictions become more topologically consistent, illustrating the effect of the topological loss.Figure 7 shows results at 10, 20, 30, 40, and 50 epochs.
4 Conclusion
The paper introduces a persistent-homology-driven topological loss for end-to-end deep segmentation. The generic loss is particularly suited to fine-structure segmentation and improves topology-relevant performance across architectures.
- 4 Conclusion: The proposed topological loss is driven by persistent homology and incorporated into end-to-end training of deep neural networks.The loss term is generic and can be incorporated into different architectures.