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

TL;DR

This paper introduces a novel mathematical framework that integrates width information into topological features for image segmentation, ensuring preservation of connectivity, genus, and width constraints through variational models and neural networks.

Contribution

It proposes a new approach combining persistent homology and PDE smoothing to incorporate width into topological analysis for segmentation.

Findings

Successfully preserves topological invariants like connectivity and genus.

Effectively embeds width properties such as thickness and length into segmentation.

Demonstrates improved topological fidelity in numerical experiments.

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.