# A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology

arXiv: https://arxiv.org/abs/1910.01877 · HTML: https://mlsanity.com/paper/1910.01877/

## Key facts

- Cited by 134 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2019.

## Built on by (most cited first, 30 of 134)

- [Recent advances and clinical applications of deep learning in medical image analysis](https://mlsanity.com/paper/2105.13381/) (2021)
- [Learning with Limited Annotations: A Survey on Deep Semi-Supervised Learning for Medical Image Segmentation](https://mlsanity.com/paper/2207.14191/) (2022)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging](https://mlsanity.com/paper/2203.12215/) (2022)
- [Artificial Text Detection via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2109.04825/) (2021)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Skeleton Recall Loss for Connectivity Conserving and Resource Efficient Segmentation of Thin Tubular Structures](https://mlsanity.com/paper/2404.03010/) (2024)
- [Trigger Hunting with a Topological Prior for Trojan Detection](https://mlsanity.com/paper/2110.08335/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Directional Connectivity-based Segmentation of Medical Images](https://mlsanity.com/paper/2304.00145/) (2023)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [Centerline Boundary Dice Loss for Vascular Segmentation](https://mlsanity.com/paper/2407.01517/) (2024)
- [Structure-Aware Image Segmentation with Homotopy Warping](https://mlsanity.com/paper/2112.07812/) (2021)
- [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)
- [Learning Probabilistic Topological Representations Using Discrete Morse Theory](https://mlsanity.com/paper/2206.01742/) (2022)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [TA-Net: Topology-Aware Network for Gland Segmentation](https://mlsanity.com/paper/2110.14593/) (2021)
- [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024)
- [A survey on shape-constraint deep learning for medical image segmentation](https://mlsanity.com/paper/2101.07721/) (2021)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [Capturing Shape Information with Multi-Scale Topological Loss Terms for 3D Reconstruction](https://mlsanity.com/paper/2203.01703/) (2022)
- [TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancer](https://mlsanity.com/paper/2105.06049/) (2021)
- [Topology-Aware Latent Diffusion for 3D Shape Generation](https://mlsanity.com/paper/2401.17603/) (2024)
- [TopoFR: A Closer Look at Topology Alignment on Face Recognition](https://mlsanity.com/paper/2410.10587/) (2024)
- [TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection in Chest X-Ray Images](https://mlsanity.com/paper/2101.08398/) (2021)
- [DTU-Net: Learning Topological Similarity for Curvilinear Structure Segmentation](https://mlsanity.com/paper/2205.11115/) (2022)

## Abstract (arXiv)

We introduce a method for training neural networks to perform image or volume segmentation in which prior knowledge about the topology of the segmented object can be explicitly provided and then incorporated into the training process. By using the differentiable properties of persistent homology, a concept used in topological data analysis, we can specify the desired topology of segmented objects in terms of their Betti numbers and then drive the proposed segmentations to contain the specified topological features. Importantly this process does not require any ground-truth labels, just prior knowledge of the topology of the structure being segmented. We demonstrate our approach in three experiments. Firstly we create a synthetic task in which handwritten MNIST digits are de-noised, and show that using this kind of topological prior knowledge in the training of the network significantly improves the quality of the de-noised digits. Secondly we perform an experiment in which the task is segmenting the myocardium of the left ventricle from cardiac magnetic resonance images. We show that the incorporation of the prior knowledge of the topology of this anatomy improves the resulting segmentations in terms of both the topological accuracy and the Dice coefficient. Thirdly, we extend the method to 3D volumes and demonstrate its performance on the task of segmenting the placenta from ultrasound data, again showing that incorporating topological priors improves performance on this challenging task. We find that embedding explicit prior knowledge in neural network segmentation tasks is most beneficial when the segmentation task is especially challenging and that it can be used in either a semi-supervised or post-processing context to extract a useful training gradient from images without pixelwise labels.