# Explicit topological priors for deep-learning based image segmentation using persistent homology

arXiv: https://arxiv.org/abs/1901.10244 · HTML: https://mlsanity.com/paper/1901.10244/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2019.

## Built on by (most cited first, 30 of 30)

- [Medical Image Segmentation with Limited Supervision: A Review of Deep Network Models](https://mlsanity.com/paper/2103.00429/) (2021)
- [A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR](https://mlsanity.com/paper/2107.12689/) (2021)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [A survey on shape-constraint deep learning for medical image segmentation](https://mlsanity.com/paper/2101.07721/) (2021)
- [Topologically Faithful Multi-class Segmentation in Medical Images](https://mlsanity.com/paper/2403.11001/) (2024)
- [A Geometry-Informed Deep Learning Framework for Ultra-Sparse 3D Tomographic Image Reconstruction](https://mlsanity.com/paper/2105.11692/) (2021)
- [Persistent Homology with Improved Locality Information for more Effective Delineation](https://mlsanity.com/paper/2110.06295/) (2021)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [CP-loss: Connectivity-preserving Loss for Road Curb Detection in Autonomous Driving with Aerial Images](https://mlsanity.com/paper/2107.11920/) (2021)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [Combining Deep Learning and Mathematical Morphology for Historical Map Segmentation](https://mlsanity.com/paper/2101.02144/) (2021)
- [Revisiting Point Cloud Completion: Are We Ready For The Real-World?](https://mlsanity.com/paper/2411.17580/) (2024)
- [Deep Statistic Shape Model for Myocardium Segmentation](https://mlsanity.com/paper/2207.10607/) (2022)
- [Universal Topology Refinement for Medical Image Segmentation with Polynomial Feature Synthesis](https://mlsanity.com/paper/2409.09796/) (2024)
- [Efficient Connectivity-Preserving Instance Segmentation with Supervoxel-Based Loss Function](https://mlsanity.com/paper/2501.01022/) (2025)
- [Optimisation of Spectral Wavelets for Persistence-based Graph Classification](https://mlsanity.com/paper/2101.05201/) (2021)
- [A Topology-Attention ConvLSTM Network and Its Application to EM Images](https://mlsanity.com/paper/2202.03430/) (2022)
- [GLiDR: Topologically Regularized Graph Generative Network for Sparse LiDAR Point Clouds](https://mlsanity.com/paper/2312.00068/) (2023)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [Component Tree Loss Function: Definition and Optimization](https://mlsanity.com/paper/2101.08063/) (2021)
- [Detecting Hypo-plastic Left Heart Syndrome in Fetal Ultrasound via Disease-specific Atlas Maps](https://mlsanity.com/paper/2107.02643/) (2021)
- [Image Segmentation with Topological Priors](https://mlsanity.com/paper/2205.06197/) (2022)
- [Train-Free Segmentation in MRI with Cubical Persistent Homology](https://mlsanity.com/paper/2401.01160/) (2024)
- [TopoMortar: A dataset to evaluate image segmentation methods focused on topology accuracy](https://mlsanity.com/paper/2503.03365/) (2025)
- [Effect of Prior-based Losses on Segmentation Performance: A Benchmark](https://mlsanity.com/paper/2201.02428/) (2022)
- [Multi-task learning for joint weakly-supervised segmentation and aortic arch anomaly classification in fetal cardiac MRI](https://mlsanity.com/paper/2311.07234/) (2023)
- [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (2026)
- [Steering Optimisation Trajectories in Diffusion Representation Learning](https://mlsanity.com/paper/2607.05319/) (2026)
- [Sparse cubical complexes for efficient topology-preservation in image data](https://mlsanity.com/paper/2609.37177/) (2026)

## Abstract (arXiv)

We present a novel method to explicitly incorporate topological prior knowledge into deep learning based segmentation, which is, to our knowledge, the first work to do so. Our method uses the concept of persistent homology, a tool from topological data analysis, to capture high-level topological characteristics of segmentation results in a way which is differentiable with respect to the pixelwise probability of being assigned to a given class. The topological prior knowledge consists of the sequence of desired Betti numbers of the segmentation. As a proof-of-concept we demonstrate our approach by applying it to the problem of left-ventricle segmentation of cardiac MR images of 500 subjects from the UK Biobank dataset, where we show that it improves segmentation performance in terms of topological correctness without sacrificing pixelwise accuracy.