# Topologically Faithful Multi-class Segmentation in Medical Images

arXiv: https://arxiv.org/abs/2403.11001 · HTML: https://mlsanity.com/paper/2403.11001/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 16 Mar 2024.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Generalised Dice overlap as a deep learning loss function for highly unbalanced segmentations](https://mlsanity.com/paper/1707.03237/) (2017)
- [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)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [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)
- [OCTA-500: A Retinal Dataset for Optical Coherence Tomography Angiography Study](https://mlsanity.com/paper/2012.07261/) (2020)
- [The TopCoW Challenge -- Topology-Aware Circle of Willis Segmentation for CT and MR Angiography](https://mlsanity.com/paper/2312.17670/) (2023)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR](https://mlsanity.com/paper/2107.12689/) (2021)
- [Synthetic optical coherence tomography angiographs for detailed retinal vessel segmentation without human annotations](https://mlsanity.com/paper/2306.10941/) (2023)
- [Topology-Preserving Automatic Labeling of Coronary Arteries via Anatomy-aware Connection Classifier](https://mlsanity.com/paper/2307.11959/) (2023)
- [Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem](https://mlsanity.com/paper/1610.10085/) (2016)
- [Structure-Preserving Instance Segmentation via Skeleton-Aware Distance Transform](https://mlsanity.com/paper/2310.05262/) (2023)

## Built on by (most cited first, 12 of 12)

- [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [BrainMRDiff: A Diffusion Model for Anatomically Consistent Brain MRI Synthesis](https://mlsanity.com/paper/2504.04532/) (2025)
- [Universal Topology Refinement for Medical Image Segmentation with Polynomial Feature Synthesis](https://mlsanity.com/paper/2409.09796/) (2024)
- [MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation](https://mlsanity.com/paper/2510.01532/) (2025)
- [TopoSculpt: Betti-Steered Topological Sculpting of 3D Fine-grained Tubular Shapes](https://mlsanity.com/paper/2509.03938/) (2025)
- [Robust automatic brain vessel segmentation in 3D CTA scans using dynamic 4D-CTA data](https://mlsanity.com/paper/2602.00391/) (2026)
- [SAMSEM -- A Generic and Scalable Approach for IC Metal Line Segmentation](https://mlsanity.com/paper/2603.16548/) (2026)
- [Anatomical Prior-Driven Framework for Autonomous Robotic Cardiac Ultrasound Standard View Acquisition](https://mlsanity.com/paper/2603.21134/) (2026)
- [SEMIR: Topology-Preserving Graph Minors for Thin-Structure Segmentation](https://mlsanity.com/paper/2606.24935/) (2026)
- [Widest-Path Reachability Fields for Connectivity-Preserving Slender Structure Segmentation](https://mlsanity.com/paper/2607.07123/) (2026)
- [Sparse cubical complexes for efficient topology-preservation in image data](https://mlsanity.com/paper/2609.37177/) (2026)

## Abstract (arXiv)

Topological accuracy in medical image segmentation is a highly important property for downstream applications such as network analysis and flow modeling in vessels or cell counting. Recently, significant methodological advancements have brought well-founded concepts from algebraic topology to binary segmentation. However, these approaches have been underexplored in multi-class segmentation scenarios, where topological errors are common. We propose a general loss function for topologically faithful multi-class segmentation extending the recent Betti matching concept, which is based on induced matchings of persistence barcodes. We project the N-class segmentation problem to N single-class segmentation tasks, which allows us to use 1-parameter persistent homology, making training of neural networks computationally feasible. We validate our method on a comprehensive set of four medical datasets with highly variant topological characteristics. Our loss formulation significantly enhances topological correctness in cardiac, cell, artery-vein, and Circle of Willis segmentation.