Continuous-tone Simple Points: Differentiable Topology-Preserving Learning with an l0-Norm Cyclic Gradient

Forum topic · 小凯 · 2026-05-02

Summary

This paper introduces a novel method for computing simple points directly on continuous-valued (gray-scale) images, enabling differentiable topological inference for deep learning. Existing simple point detection methods are limited to binary images and are non-differentiable, making them incompatible with gradient-based optimization in modern deep learning pipelines. The authors derive an l0-norm-based cyclic gradient formulation that extends simple point detection to continuous tones, and build an efficient skeletonization algorithm that preserves topology in both binary and continuous-valued images. They further design a variational model that enforces topological constraints by preserving topologically non-removable (non-simple) points, which can be seamlessly integrated into any deep neural network segmentation model with softmax or sigmoid outputs. Experiments across multiple benchmarks show improved topological integrity and structural accuracy compared to prior approaches. The work bridges topological data analysis and deep learning, addressing a key limitation of morphological and purely data-driven methods that fail to guarantee topological consistency. Code is released as open source. Paper: arXiv 2604.28159.

Paper Overview

Field: Computer Vision (CV)
Authors: Le Dung, Toshiaki Kondo, Munehiro Nakamura et al.
Published: 2026-04-30
arXiv: 2604.28159

Summary (translated)

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 guarantee topological consistency.

To address these limitations, the authors propose a new method that computes simple points directly on continuous-valued images, enabling differentiable topological inference. The key idea is an l0-norm formulation of the cyclic gradient that characterizes simple points without requiring binarization.

Contributions

A theory for detecting simple points on continuous-tone images, making topological reasoning differentiable.
An efficient skeleton extraction algorithm that preserves topology for both binary and continuous-valued images.
A variational model that enforces topological constraints by preserving topologically non-removable (non-simple) points, which can be seamlessly integrated into any deep neural network segmentation model with softmax or sigmoid outputs.

Results

Experiments on multiple benchmarks demonstrate that the proposed method improves both topological integrity and structural accuracy. Code is open-sourced.

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