# Topologically faithful image segmentation via induced matching of persistence barcodes

arXiv: https://arxiv.org/abs/2211.15272 · HTML: https://mlsanity.com/paper/2211.15272/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 28 Nov 2022.

## Builds on (most cited first, 19 of 19)

- [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)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Unified Focal loss: Generalising Dice and cross entropy-based losses to handle class imbalanced medical image segmentation](https://mlsanity.com/paper/2102.04525/) (2021)
- [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)
- [Localization in the Crowd with Topological Constraints](https://mlsanity.com/paper/2012.12482/) (2020)
- [Promoting Connectivity of Network-Like Structures by Enforcing Region Separation](https://mlsanity.com/paper/2009.07011/) (2020)
- [TA-Net: Topology-Aware Network for Gland Segmentation](https://mlsanity.com/paper/2110.14593/) (2021)
- [Capturing Shape Information with Multi-Scale Topological Loss Terms for 3D Reconstruction](https://mlsanity.com/paper/2203.01703/) (2022)
- [A Learning-based Framework for Topology-Preserving Segmentation using Quasiconformal Mappings](https://mlsanity.com/paper/2210.03299/) (2022)
- [Cycle Registration in Persistent Homology with Applications in Topological Bootstrap](https://mlsanity.com/paper/2101.00698/) (2021)
- [Induced Matchings and the Algebraic Stability of Persistence Barcodes](https://mlsanity.com/paper/1311.3681/) (2013)
- [Duality in Persistent Homology of Images](https://mlsanity.com/paper/2005.04597/) (2020)
- [Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images](https://mlsanity.com/paper/1705.02045/) (2017)
- [Efficient Computation of Image Persistence](https://mlsanity.com/paper/2201.04170/) (2022)
- [Fast Topological Signal Identification and Persistent Cohomological Cycle Matching](https://mlsanity.com/paper/2209.15446/) (2022)

## Built on by (most cited first, 30 of 49)

- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Centerline Boundary Dice Loss for Vascular Segmentation](https://mlsanity.com/paper/2407.01517/) (2024)
- [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [Topologically Faithful Multi-class Segmentation in Medical Images](https://mlsanity.com/paper/2403.11001/) (2024)
- [Semi-supervised Segmentation of Histopathology Images with Noise-Aware Topological Consistency](https://mlsanity.com/paper/2311.16447/) (2023)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [Enhancing Boundary Segmentation for Topological Accuracy with Skeleton-based Methods](https://mlsanity.com/paper/2404.18539/) (2024)
- [VSR-Net: Vessel-like Structure Rehabilitation Network with Graph Clustering](https://mlsanity.com/paper/2312.13116/) (2023)
- [Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology](https://mlsanity.com/paper/2407.04683/) (2024)
- [Topology-Preserving Image Segmentation with Spatial-Aware Persistent Feature Matching](https://mlsanity.com/paper/2412.02076/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [TopoLoRA-SAM: Topology-Aware Parameter-Efficient Adaptation of Foundation Segmenters for Thin-Structure and Cross-Domain Binary Semantic Segmentation](https://mlsanity.com/paper/2601.02273/) (2026)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH](https://mlsanity.com/paper/2310.16437/) (2023)
- [Universal Topology Refinement for Medical Image Segmentation with Polynomial Feature Synthesis](https://mlsanity.com/paper/2409.09796/) (2024)
- [TopoTxR: A topology-guided deep convolutional network for breast parenchyma learning on DCE-MRIs](https://mlsanity.com/paper/2411.03464/) (2024)
- [Universal Vessel Segmentation for Multi-Modality Retinal Images](https://mlsanity.com/paper/2502.06987/) (2025)
- [MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation](https://mlsanity.com/paper/2510.01532/) (2025)
- [Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds](https://mlsanity.com/paper/2402.15058/) (2024)
- [BackSplit: The Importance of Sub-dividing the Background in Biomedical Lesion Segmentation](https://mlsanity.com/paper/2511.19394/) (2025)
- [Integrating Multi-scale and Multi-filtration Topological Features for Medical Image Classification](https://mlsanity.com/paper/2512.07190/) (2025)
- [FMS2: Unified Flow Matching for Segmentation and Synthesis of Thin Structures](https://mlsanity.com/paper/2603.13659/) (2026)
- [Complex Wavelet Mutual Information Loss: A Multi-Scale Loss Function for Semantic Segmentation](https://mlsanity.com/paper/2502.00563/) (2025)
- [TopoMortar: A dataset to evaluate image segmentation methods focused on topology accuracy](https://mlsanity.com/paper/2503.03365/) (2025)
- [Skelite: Compact Neural Networks for Efficient Iterative Skeletonization](https://mlsanity.com/paper/2503.07369/) (2025)
- [TopoSculpt: Betti-Steered Topological Sculpting of 3D Fine-grained Tubular Shapes](https://mlsanity.com/paper/2509.03938/) (2025)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [Topo-R1: Detecting Topological Anomalies via Vision-Language Models](https://mlsanity.com/paper/2603.13054/) (2026)

## Abstract (arXiv)

Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap, an objective that is insufficient for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the correct topology of the segmented structures. However, so far, none of the existing approaches achieve a spatially correct matching between the topological features of ground truth and prediction. In this work, we propose the first topologically and feature-wise accurate metric and loss function for supervised image segmentation, which we term Betti matching. We show how induced matchings guarantee the spatially correct matching between barcodes in a segmentation setting. Furthermore, we propose an efficient algorithm to compute the Betti matching of images. We show that the Betti matching error is an interpretable metric to evaluate the topological correctness of segmentations, which is more sensitive than the well-established Betti number error. Moreover, the differentiability of the Betti matching loss enables its use as a loss function. It improves the topological performance of segmentation networks across six diverse datasets while preserving the volumetric performance. Our code is available in https://github.com/nstucki/Betti-matching.