Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints

2026-01-16

AI Summary

This work addresses the longstanding challenge in image segmentation of simultaneously preserving topological properties—such as connectivity and genus—and geometric width attributes like line thickness and length, which conventional methods often fail to reconcile due to topology-aware representations typically neglecting scale information. To bridge this gap, we propose a novel topological modeling framework that explicitly integrates geometric width into topological characterization for the first time. By synergistically combining persistent homology, PDE-based smoothing, and a variational segmentation model—augmented with topological energy constraints and a custom loss function—we jointly optimize local extrema of upper level sets. Experimental results demonstrate that our approach rigorously maintains target topological invariants while effectively preserving essential geometric width features, thereby significantly enhancing both topological correctness and geometric fidelity in segmentation outcomes.

Technology Category

• Computer Vision: Segmentation
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from partial differential equations (PDEs), to modify local extrema of upper-level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.

Problem

• image segmentation
• topological priors
• width constraints
• connectivity
• genus

Innovation

• topology-guaranteed segmentation
• persistent homology
• width-aware topology
• variational image segmentation
• PDE-based smoothing