Leveraging Persistence Image to Enhance Robustness and Performance in Curvilinear Structure Segmentation

Abstract: Segmenting curvilinear structures in medical images is essential for analyzing morphological patterns in clinical applications. Integrating topological properties, such as connectivity, improves segmentation accuracy and consistency. However, extracting and embedding such properties - especially from Persistence Diagrams (PD) - is challenging due to their non-differentiability and computational cost. Existing approaches mostly encode topology through handcrafted loss functions, which generalize poorly across tasks. In this paper, we propose PIs-Regressor, a simple yet effective module that learns persistence image (PI) - finite, differentiable representations of topological features - directly from data. Together with Topology SegNet, which fuses these features in both downsampling and upsampling stages, our framework integrates topology into the network architecture itself rather than auxiliary losses. Unlike existing methods that depend heavily on handcrafted loss functions, our approach directly incorporates topological information into the network structure, leading to more robust segmentation. Our design is flexible and can be seamlessly combined with other topology-based methods to further enhance segmentation performance. Experimental results show that integrating topological features enhances model robustness, effectively handling challenges like overexposure and blurring in medical imaging. Our approach on three curvilinear benchmarks demonstrate state-of-the-art performance in both pixel-level accuracy and topological fidelity.

• The paper proposes a novel framework using a persistence image to directly incorporate topological information into neural networks, enhancing curvilinear structure segmentation accuracy (e.g., improving Dice score by 1.18 on DRIVE dataset and reducing β₀ error substantially).
• The model demonstrates improved robustness to imaging degradations such as overexposure, reducing β₀ error significantly compared to existing methods (e.g., a 0.72 Dice increase under overexposure conditions on DRIVE set).
• The approach can be integrated with other topology-aware losses (such as PH loss or clDice), offering flexibility and expandability for various topological structure segmentations at different resolution stages.

Motivation and positioning

Segmentation of curvilinear structures—retinal vessels, endoplasmic reticulum (ER), coronary arteries—must satisfy two criteria that pixel-wise metrics such as Dice do not jointly capture: pixel-level accuracy and topological fidelity (connectivity, loop structure). Broken vessels or missing branches can survive a high Dice score while invalidating downstream morphological analysis. Existing topology-aware methods fall into three categories, each with drawbacks: PH-driven loss design (e.g., Hu et al.'s topology-preserving loss) requires differentiating through persistence diagrams (PDs), which are discrete and non-differentiable, making integration computationally expensive; indirect PH annotations (fractal feature maps, adjacent anatomy labels) demand heavy manual annotation and provide only surrogate encodings; and skeleton-based losses (clDice, cbDice) capture connectivity but not the broader space of topological features, and skeletonization itself can introduce errors that degrade segmentation quality.

The paper proposes an alternative: rather than encoding topology through auxiliary loss functions, it embeds topological information directly into the network architecture. The core observation is that the persistence image (PI)—a finite-dimensional, differentiable vectorization of persistent homology obtained by discretizing the persistence surface over a grid—is well suited to neural computation, provided it can be produced efficiently from raw images.

Method

The framework consists of two cooperating subnetworks.

PIs-Regressor: A compact encoder–decoder that regresses an approximation $p' \in \mathbb{R}^{b \times 1 \times h \times w}$ of the ground-truth persistence image directly from the input image. The encoder is a pre-trained ResNet-50; the decoder uses a deconvolution block followed by a 1×1 convolution and Sigmoid activation. Training uses MSE against PIs computed from ground-truth masks via the standard pipeline: filtration of the foreground point cloud over increasing scale $\epsilon$, extraction of $H_1$ persistent homology (chosen because loops are the most discriminative feature for vascular structures across scales), transformation of PD points to (birth, persistence) coordinates, construction of the persistence surface $\rho_B(z) = \sum_{u \in T(B)} f(u)\phi_u(z)$, and grid integration. Gaussian smoothing in PI generation confers error tolerance: local discrepancies in $p'$ do not destabilize the global topological signal.

Topology SegNet: The predicted PI is replicated to three channels, concatenated with the input image, and passed through a U-Net-style segmentation network in which topological features are additionally fused at every downsampling stage (with max-pooled features) and every upsampling stage (with deconvolved features). This repeated injection ensures topological cues influence features at multiple resolutions.

Training uses a combined Dice + cross-entropy loss ($\alpha = 0.5$); the two subnetworks are optimized separately (Adam at $10^{-3}$ for the regressor, SGD at $10^{-2}$ with 0.9 decay for the segmenter), 500 epochs, batch size 4. Betti numbers are computed with GUDHI.

A notable design claim is modularity: since the module operates on the architecture side, it can be composed with existing topology-aware losses (PH loss, clDice, cbDice) rather than replacing them.

Results on three benchmarks

Experiments cover DRIVE, ER (fluorescence microscopy), and a private ultra-widefield OPTOS dataset, evaluated with Dice, clDice, mIoU, and Betti errors ($\beta_0$, $\beta_1$). Representative comparisons:

Dataset	Baseline	Ours + same loss	Δ Dice	Δ β₀
DRIVE	U-Net + CE (80.73 / 217.2)	81.91 / 126.8	+1.18	−90.4
ER	U-Net + clDice (84.23 / 184.2 β₀)	84.52 / 30.70	+0.29	−153.5
Private	U-Net + cbDice (75.15 Dice)	Ours + CE+Dice: 76.99	+1.84	−4.23

On DRIVE, adding the PI module to a CE-trained U-Net reduces $\epsilon$0 error by 90.4—a roughly 42% reduction—while improving Dice by 1.18 points, indicating that connectivity preservation improves without sacrificing pixel accuracy. On ER, where fragmented segments and isolated misclassifications dominate, the largest gains appear: combining the module with clDice yields +2.88 clDice and reductions of 53.5 ($\epsilon$1) and 46.02 ($\epsilon$2). On the private OPTOS data, the best configuration (Ours + CE+Dice) exceeds the strongest baseline (cbDice) by 1.84 Dice, 1.69 clDice, 0.92 mIoU, with $\epsilon$3 reduced from 8.503 to 4.278.

Two observations qualify these results. First, gains are not uniform across all loss combinations—for example, on DRIVE, "Ours + cbDice" slightly worsens clDice relative to cbDice alone (81.82 vs. 82.57), and on ER, "Ours + PH Loss" underperforms PH Loss alone on several metrics. The paper does not analyze when composition helps or hurts. Second, the private dataset has very small absolute Betti errors (single digits), so topological improvements there are less dramatic than on DRIVE/ER.

Robustness under perturbation

A central empirical claim is that architectural topology injection improves robustness to imaging degradation. Models pretrained on clean DRIVE were tested on blurred, low-contrast, underexposed, and overexposed variants. The most striking result concerns overexposure: the CE-trained U-Net's $\epsilon$4 error rises by 1039.9 (to 1257.1) and $\epsilon$5 by 43.6, whereas the proposed model's $\epsilon$6 increases by only 76.9 and its $\epsilon$7 actually decreases by 0.8. Against the best competing method under overexposure (U-Net + cbDice), the proposed model leads by 0.72 Dice, 0.06 clDice, and 0.68 mIoU. Under blur—the harshest condition for all methods (Dice drops to ~65 across the board)—the proposed model attains 66.48 Dice and 65.83 clDice versus 56.52 clDice for the CE baseline, a gain of over 9 points in clDice. These results support the hypothesis that global topological features, being invariant to smooth intensity deformations, act as a stabilizing prior when local appearance cues are corrupted.

Limitations and open questions

Several limitations are evident. The ground-truth PI supervision requires computing persistent homology on masks offline; while this avoids differentiating through PDs during training, the pipeline still depends on GUDHI computations whose cost at scale is not analyzed. The restriction to $\epsilon$8 is justified for curvilinear structures but leaves higher-dimensional homology unexplored. The ablation over loss combinations shows non-monotonic behavior (some pairings degrade performance), and no principled explanation or selection rule is offered. Evaluation is confined to 2D retinal and microscopy imagery; generalization to 3D tubular structures (e.g., coronary CT angiography) remains untested. Finally, the robustness study uses synthetic perturbations of DRIVE; whether the observed stability transfers to naturally degraded clinical acquisitions is an open question the paper does not address.

Conclusion

This work reframes topology-aware segmentation as an architectural rather than loss-design problem: a lightweight regressor approximates persistence images from raw images, and a fusion-augmented U-Net consumes them at multiple scales. Across three benchmarks the approach delivers consistent pixel-level gains alongside substantial reductions in Betti error—most notably a ~42% $\epsilon$9 reduction on DRIVE and a drop from 184.2 to 30.7 in $H_1$0 on ER—and exhibits markedly improved robustness under blur and overexposure. Its composability with existing topology-aware losses is a practical advantage, though the conditions under which such composition helps remain insufficiently characterized.