# A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR

arXiv: https://arxiv.org/abs/2107.12689 · HTML: https://mlsanity.com/paper/2107.12689/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 51 references that are in the map.
- First posted on arXiv 27 Jul 2021.

## Builds on (most cited first, 23 of 23)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [nnU-Net: Self-adapting Framework for U-Net-Based Medical Image Segmentation](https://mlsanity.com/paper/1809.10486/) (2018)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Deep learning for cardiac image segmentation: A review](https://mlsanity.com/paper/1911.03723/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Anatomically Constrained Neural Networks (ACNN): Application to Cardiac Image Enhancement and Segmentation](https://mlsanity.com/paper/1705.08302/) (2017)
- [Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge](https://mlsanity.com/paper/1902.07880/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Cardiac Segmentation with Strong Anatomical Guarantees](https://mlsanity.com/paper/2006.08825/) (2020)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [Cubical Ripser: Software for computing persistent homology of image and volume data](https://mlsanity.com/paper/2005.12692/) (2020)
- [ImageCHD: A 3D Computed Tomography Image Dataset for Classification of Congenital Heart Disease](https://mlsanity.com/paper/2101.10799/) (2021)
- [Cardiac MRI Segmentation with Strong Anatomical Guarantees](https://mlsanity.com/paper/1907.02865/) (2019)
- [A persistent homology-based topological loss function for multi-class CNN segmentation of cardiac MRI](https://mlsanity.com/paper/2008.09585/) (2020)
- [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)
- [MvMM-RegNet: A new image registration framework based on multivariate mixture model and neural network estimation](https://mlsanity.com/paper/2006.15573/) (2020)
- [Uncertainty Quantification in CNN-Based Surface Prediction Using Shape Priors](https://mlsanity.com/paper/1807.11272/) (2018)
- [Duality in Persistent Homology of Images](https://mlsanity.com/paper/2005.04597/) (2020)
- [Deep Small Bowel Segmentation with Cylindrical Topological Constraints](https://mlsanity.com/paper/2007.08674/) (2020)

## Built on by (most cited first, 21 of 21)

- [Topologically Faithful Multi-class Segmentation in Medical Images](https://mlsanity.com/paper/2403.11001/) (2024)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [Topology-Preserving Image Segmentation with Spatial-Aware Persistent Feature Matching](https://mlsanity.com/paper/2412.02076/) (2024)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [Universal Topology Refinement for Medical Image Segmentation with Polynomial Feature Synthesis](https://mlsanity.com/paper/2409.09796/) (2024)
- [Universal Vessel Segmentation for Multi-Modality Retinal Images](https://mlsanity.com/paper/2502.06987/) (2025)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Topology-Aware Loss for Aorta and Great Vessel Segmentation in Computed Tomography Images](https://mlsanity.com/paper/2307.03137/) (2023)
- [A Diffusion Model for Simulation Ready Coronary Anatomy with Morpho-skeletal Control](https://mlsanity.com/paper/2407.15631/) (2024)
- [TopoMortar: A dataset to evaluate image segmentation methods focused on topology accuracy](https://mlsanity.com/paper/2503.03365/) (2025)
- [Disconnect to Connect: A Data Augmentation Method for Improving Topology Accuracy in Image Segmentation](https://mlsanity.com/paper/2503.05541/) (2025)
- [GloResNet: A lightweight 3D CNN with global topological features for preterm brain injury prediction](https://mlsanity.com/paper/2606.02498/) (2026)
- [Teaching AI the Anatomy Behind the Scan: Addressing Anatomical Flaws in Medical Image Segmentation with Learnable Prior](https://mlsanity.com/paper/2403.18878/) (2024)
- [PI-Att: Topology Attention for Segmentation Networks through Adaptive Persistence Image Representation](https://mlsanity.com/paper/2408.08038/) (2024)
- [A Multi-Resolution Benchmark Framework for Spatial Reasoning Assessment in Neural Networks](https://mlsanity.com/paper/2508.12741/) (2025)
- [Preserving instance continuity and length in segmentation through connectivity-aware loss computation](https://mlsanity.com/paper/2509.03154/) (2025)
- [Anatomica: Localized Control over Geometric and Topological Properties for Anatomical Diffusion Models](https://mlsanity.com/paper/2511.20587/) (2025)
- [Topological Conditioning for Mammography Models via a Stable Wavelet-Persistence Vectorization](https://mlsanity.com/paper/2512.10151/) (2025)
- [Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation](https://mlsanity.com/paper/2608.21332/) (2026)
- [Sparse cubical complexes for efficient topology-preservation in image data](https://mlsanity.com/paper/2609.37177/) (2026)
- [SILSA: Sliding-Window Slice Latents for Topology-Preserving High-Resolution 3D Generation](https://mlsanity.com/paper/2610.02201/) (2026)

## Abstract (arXiv)

Multi-class segmentation of cardiac magnetic resonance (CMR) images seeks a separation of data into anatomical components with known structure and configuration. The most popular CNN-based methods are optimised using pixel wise loss functions, ignorant of the spatially extended features that characterise anatomy. Therefore, whilst sharing a high spatial overlap with the ground truth, inferred CNN-based segmentations can lack coherence, including spurious connected components, holes and voids. Such results are implausible, violating anticipated anatomical topology. In response, (single-class) persistent homology-based loss functions have been proposed to capture global anatomical features. Our work extends these approaches to the task of multi-class segmentation. Building an enriched topological description of all class labels and class label pairs, our loss functions make predictable and statistically significant improvements in segmentation topology using a CNN-based post-processing framework. We also present (and make available) a highly efficient implementation based on cubical complexes and parallel execution, enabling practical application within high resolution 3D data for the first time. We demonstrate our approach on 2D short axis and 3D whole heart CMR segmentation, advancing a detailed and faithful analysis of performance on two publicly available datasets.