[Paper Review] Topology-Preserving Deep Image Segmentation

Xiao Hu, Fuxin Li+2 more|arXiv (Cornell University)|Jun 12, 2019

Topological and Geometric Data AnalysisComputer Science38 references86 citations

TL;DR

TopoNet learns to segment with correct topology by incorporating a differentiable topological loss based on persistent homology, improving Betti-number related metrics while maintaining pixel accuracy.

ABSTRACT

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.

Motivation & Objective

• Motivate segmentation with guaranteed topological correctness in fine-scale structures such as membranes, vessels, and roads.
• Introduce a differentiable topological loss computed from persistent diagrams to guide end-to-end training.
• Demonstrate improved topology-related metrics (Betti error, ARI, VI) while preserving per-pixel accuracy across diverse datasets.

Proposed method

• Define a likelihood map f from a neural network and a ground truth g and optimize L = L_bce + λ L_topo.
• Use persistent homology to compute a topology-aware, continuous loss via persistence diagrams and a best-match (Wasserstein-like) distance.
• Derive a differentiable gradient for L_topo with respect to network parameters using critical points of f.
• Train on small patches (e.g., 65x65) with relative persistent homology to localize topology and reduce computation.
• Provide an architecture-agnostic topological loss that can be integrated into standard segmentation networks.

Experimental results

Research questions

• RQ1 Can a differentiable topological loss based on persistence diagrams enforce topology-consistent segmentations?
• RQ2 Does incorporating L_topo improve topology-sensitive metrics (Betti number error, ARI, VOI) without sacrificing pixel accuracy?
• RQ3 Is the approach effective across diverse natural and biomedical datasets with fine-scale structures?
• RQ4 What are practical considerations (patch size, training dynamics) when integrating topological loss into end-to-end learning?

Key findings

• TopoNet achieves superior Betti number error across multiple datasets compared to baselines.
• TopoNet improves topology-relevant metrics ARI and VOI while maintaining competitive pixel accuracy.
• Topological loss is Lipschitz and yields a correct segmentation when L_topo equals zero.
• Training with patch-based, relative persistent homology is efficient and effective for fine-structure segmentation.
• Experiments span CREMI, ISBI12/13, CrackTree, Road, and DRIVE, showing consistent topology improvements.