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Topology-Preserving Image Segmentation with Spatial-Aware Persistent Feature Matching

Bo Wen, Haochen Zhang, Dirk-Uwe G. Bartsch, William R. Freeman, Truong Q. Nguyen, Cheolhong An

TL;DR

This work tackles the problem of topology-preserving segmentation for tubular structures by addressing the ill-posed matching problem in pure persistent-homology-based losses. It introduces SATLoss, a Spatial-Aware Topological Loss that couples persistent feature matching with spatial information from the original image, using creators’ locations to weight transport in the persistent diagram. SATLoss is shown to improve topological accuracy across diverse tubular-structure datasets (CREMI, DRIVE, Roads, CrackTree) while maintaining competitive Dice scores and reducing computational cost relative to Betti-matching approaches like BMLoss. The approach integrates smoothly with standard segmentation networks (e.g., U-Net) and relies on efficient PH computation via GUDHI, offering practical gains for large-scale tubular structure segmentation and potential for broader topology-aware learning tasks.

Abstract

Topological correctness is critical for segmentation of tubular structures, which pervade in biomedical images. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the persistent features from the segmentation with the persistent features from the ground truth and minimize the difference between them. However, these methods suffer from an ambiguous matching problem since the matching only relies on the information in the topological space. In this work, we propose an effective and efficient Spatial-Aware Topological Loss Function that further leverages the information in the original spatial domain of the image to assist the matching of persistent features. Extensive experiments on images of various types of tubular structures show that the proposed method has superior performance in improving the topological accuracy of the segmentation compared with state-of-the-art methods. Code is available at https://github.com/JRC-VPLab/SATLoss.

Topology-Preserving Image Segmentation with Spatial-Aware Persistent Feature Matching

TL;DR

This work tackles the problem of topology-preserving segmentation for tubular structures by addressing the ill-posed matching problem in pure persistent-homology-based losses. It introduces SATLoss, a Spatial-Aware Topological Loss that couples persistent feature matching with spatial information from the original image, using creators’ locations to weight transport in the persistent diagram. SATLoss is shown to improve topological accuracy across diverse tubular-structure datasets (CREMI, DRIVE, Roads, CrackTree) while maintaining competitive Dice scores and reducing computational cost relative to Betti-matching approaches like BMLoss. The approach integrates smoothly with standard segmentation networks (e.g., U-Net) and relies on efficient PH computation via GUDHI, offering practical gains for large-scale tubular structure segmentation and potential for broader topology-aware learning tasks.

Abstract

Topological correctness is critical for segmentation of tubular structures, which pervade in biomedical images. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the persistent features from the segmentation with the persistent features from the ground truth and minimize the difference between them. However, these methods suffer from an ambiguous matching problem since the matching only relies on the information in the topological space. In this work, we propose an effective and efficient Spatial-Aware Topological Loss Function that further leverages the information in the original spatial domain of the image to assist the matching of persistent features. Extensive experiments on images of various types of tubular structures show that the proposed method has superior performance in improving the topological accuracy of the segmentation compared with state-of-the-art methods. Code is available at https://github.com/JRC-VPLab/SATLoss.

Paper Structure

This paper contains 29 sections, 7 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Method
  4. Background
  5. Topology in Digital Images
  6. Persistent Homology
  7. Persistent Barcodes and Diagrams
  8. Spatial-Aware Topological Loss Function
  9. Spatial-Aware Matching and Loss Formulation
  10. Gradient of SATLoss
  11. Computational Cost
  12. Experiments
  13. Datasets
  14. Implementation Details
  15. Evaluation Metrics
  16. ...and 14 more sections

Figures (15)

  • Figure 1: Overview of our method. 'Dgm' is the persistent diagram of the images and the dotted lines indicate the spatial correspondences. Colored triangles indicate creators of topological features and are used as spatial location references for topological feature matching.
  • Figure 2: A toy example for persistent homology in digital images: (a) A grayscale image; (b) A visualization of the super-level filtration by representing each sub-complex by its 0-cubes (pixels), the value under each image is the filtration (threshold) value. The triangles and crosses (best zoom in to view) represents the creators and destroyers of selected persistent features (green for 0-a, sky blue for 0-f and red for 1-a), respectively; (c) The persistent barcodes, blue for 0-features and red for 1-feature; (d) The persistent diagram.
  • Figure 3: Visualization of persistent features matching. A matched 0/1-feature is marked with the same color with the feature in GT.
  • Figure 4: Qualitative comparison with SOTA methods. From left to right: image, ground truth, BCELoss, WTLoss, He, clDice, SATLoss. Green, orange and red arrows indicate good, moderate and bad segmentation (topologically), respectively.
  • Figure 5: Qualitative comparison with BMLoss. From left to right: image; ground truth; BMLoss, SATLoss.
  • ...and 10 more figures