# Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation

arXiv: https://arxiv.org/abs/2604.28159 · HTML: https://mlsanity.com/paper/2604.28159/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 66 references that are in the map.
- First posted on arXiv 30 Apr 2026.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [SegFormer: Simple and Efficient Design for Semantic Segmentation with Transformers](https://mlsanity.com/paper/2105.15203/) (2021)
- [SAM 2: Segment Anything in Images and Videos](https://mlsanity.com/paper/2408.00714/) (2024)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [UNet++: Redesigning Skip Connections to Exploit Multiscale Features in Image Segmentation](https://mlsanity.com/paper/1912.05074/) (2019)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Dynamic Snake Convolution based on Topological Geometric Constraints for Tubular Structure Segmentation](https://mlsanity.com/paper/2307.08388/) (2023)
- [Beyond the Pixel-Wise Loss for Topology-Aware Delineation](https://mlsanity.com/paper/1712.02190/) (2017)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Dynamic Feature Integration for Simultaneous Detection of Salient Object, Edge and Skeleton](https://mlsanity.com/paper/2004.08595/) (2020)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [Directional Connectivity-based Segmentation of Medical Images](https://mlsanity.com/paper/2304.00145/) (2023)
- [Promoting Connectivity of Network-Like Structures by Enforcing Region Separation](https://mlsanity.com/paper/2009.07011/) (2020)
- [A skeletonization algorithm for gradient-based optimization](https://mlsanity.com/paper/2309.02527/) (2023)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [BiconNet: An Edge-preserved Connectivity-based Approach for Salient Object Detection](https://mlsanity.com/paper/2103.00334/) (2021)
- [DeepSkeleton: Learning Multi-task Scale-associated Deep Side Outputs for Object Skeleton Extraction in Natural Images](https://mlsanity.com/paper/1609.03659/) (2016)
- [Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization](https://mlsanity.com/paper/2103.16768/) (2021)
- [Representing Topological Self-Similarity Using Fractal Feature Maps for Accurate Segmentation of Tubular Structures](https://mlsanity.com/paper/2407.14754/) (2024)
- [A Learning-based Framework for Topology-Preserving Segmentation using Quasiconformal Mappings](https://mlsanity.com/paper/2210.03299/) (2022)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)

## Abstract (arXiv)

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.