# A persistent homology-based topological loss function for multi-class CNN segmentation of cardiac MRI

arXiv: https://arxiv.org/abs/2008.09585 · HTML: https://mlsanity.com/paper/2008.09585/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2020.

## Built on by (most cited first, 13 of 13)

- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [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)
- [A survey on shape-constraint deep learning for medical image segmentation](https://mlsanity.com/paper/2101.07721/) (2021)
- [Multi-class point cloud completion networks for 3D cardiac anatomy reconstruction from cine magnetic resonance images](https://mlsanity.com/paper/2307.08535/) (2023)
- [Rethinking Persistent Homology for Visual Recognition](https://mlsanity.com/paper/2207.04220/) (2022)
- [Enforcing connectivity of 3D linear structures using their 2D projections](https://mlsanity.com/paper/2207.06832/) (2022)
- [CAPE: Connectivity-Aware Path Enforcement Loss for Curvilinear Structure Delineation](https://mlsanity.com/paper/2504.00753/) (2025)
- [Train-Free Segmentation in MRI with Cubical Persistent Homology](https://mlsanity.com/paper/2401.01160/) (2024)
- [Decorrelating neurons using persistence](https://mlsanity.com/paper/2308.04870/) (2023)
- [Multi-task learning for joint weakly-supervised segmentation and aortic arch anomaly classification in fetal cardiac MRI](https://mlsanity.com/paper/2311.07234/) (2023)
- [Modality-AGnostic Image Cascade (MAGIC) for Multi-Modality Cardiac Substructure Segmentation](https://mlsanity.com/paper/2506.10797/) (2025)
- [Quality-Guided Semi-Supervised Learning for Medical Image Segmentation](https://mlsanity.com/paper/2606.01753/) (2026)

## Abstract (arXiv)

With respect to spatial overlap, CNN-based segmentation of short axis cardiovascular magnetic resonance (CMR) images has achieved a level of performance consistent with inter observer variation. However, conventional training procedures frequently depend on pixel-wise loss functions, limiting optimisation with respect to extended or global features. As a result, inferred segmentations can lack spatial coherence, including spurious connected components or holes. Such results are implausible, violating the anticipated topology of image segments, which is frequently known a priori. Addressing this challenge, published work has employed persistent homology, constructing topological loss functions for the evaluation of image segments against an explicit prior. Building a richer description of segmentation topology by considering all possible labels and label pairs, we extend these losses to the task of multi-class segmentation. These topological priors allow us to resolve all topological errors in a subset of 150 examples from the ACDC short axis CMR training data set, without sacrificing overlap performance.