Continuous-tone Simple Points: An $\ell_0$-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation

2026-04-30

AI Summary

Existing topology-preserving methods based on simple points are limited to binary images and are non-differentiable, making them incompatible with deep learning frameworks and thus unable to provide topological guarantees in data-driven segmentation. This work presents the first differentiable approach for detecting simple points directly on continuous grayscale images. By integrating a differentiable topological reasoning mechanism with a variational model that preserves topologically non-removable points, our method seamlessly embeds discrete topological theory into deep segmentation networks. The approach combines cyclic gradient-based ℓ₀ modeling with differentiable simple point identification and is compatible with standard softmax or sigmoid output architectures. Experiments across multiple benchmarks demonstrate significant improvements in both topological integrity and structural accuracy of segmentation results. The code is publicly available.

Abstract

Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple points into deep learning tasks remains challenging, as existing simple point detection methods are confined to binary images and are non-differentiable, rendering them incompatible with gradient-based optimization in modern deep learning. Moreover, morphological and purely data-driven approaches often fail to guaranty topological consistency. To address these limitations, we propose a novel method that directly computes simple points on continuous-valued images, enabling differentiable topological inference. Building on this theory, we develop an efficient skeleton extraction algorithm that preserves topological structures in binary and continuous-valued images. Furthermore, we design a variational model that enforces topological constraints by preserving topologically non-removable (i.e., non-simple) points, which can be seamlessly integrated into any deep neural network segmentation with softmax or sigmoid outputs. Experimental results demonstrate that the proposed approach effectively improves topological integrity and structural accuracy across multiple benchmarks. The codes are available in https://github.com/levnsio/CSP.

Problem

• topology-preserving
• simple points
• image segmentation
• differentiable
• continuous-tone

Innovation

• simple points
• topology-preserving
• differentiable topology
• continuous-tone images
• variational segmentation