# Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology

arXiv: https://arxiv.org/abs/2407.04683 · HTML: https://mlsanity.com/paper/2407.04683/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 5 Jul 2024.

## Builds on (most cited first, 15 of 15)

- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [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)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [Structure-Aware Image Segmentation with Homotopy Warping](https://mlsanity.com/paper/2112.07812/) (2021)
- [Cubical Ripser: Software for computing persistent homology of image and volume data](https://mlsanity.com/paper/2005.12692/) (2020)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [giotto-ph: A Python Library for High-Performance Computation of Persistent Homology of Vietoris-Rips Filtrations](https://mlsanity.com/paper/2107.05412/) (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)

## Built on by (most cited first, 8 of 8)

- [Topology-Constrained Learning for Efficient Laparoscopic Liver Landmark Detection](https://mlsanity.com/paper/2507.00519/) (2025)
- [Topo-R1: Detecting Topological Anomalies via Vision-Language Models](https://mlsanity.com/paper/2603.13054/) (2026)
- [Preserving instance continuity and length in segmentation through connectivity-aware loss computation](https://mlsanity.com/paper/2509.03154/) (2025)
- [SAMSEM -- A Generic and Scalable Approach for IC Metal Line Segmentation](https://mlsanity.com/paper/2603.16548/) (2026)
- [AG-TAL: Anatomically-Guided Topology-Aware Loss for Multiclass Segmentation of the Circle of Willis Using Large-Scale Multi-Center Datasets](https://mlsanity.com/paper/2604.27357/) (2026)
- [Topology-Aware Training and Spatial Diagnostics for Fiber Bundle Segmentation in Tracer Histology](https://mlsanity.com/paper/2609.04454/) (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)

In this work, we propose an efficient algorithm for the calculation of the Betti matching, which can be used as a loss function to train topology aware segmentation networks. Betti matching loss builds on techniques from topological data analysis, specifically persistent homology. A major challenge is the computational cost of computing persistence barcodes. In response to this challenge, we propose a new, highly optimized implementation of Betti matching, implemented in C++ together with a python interface, which achieves significant speedups compared to the state-of-the-art implementation Cubical Ripser. We use Betti matching 3D to train segmentation networks with the Betti matching loss and demonstrate improved topological correctness of predicted segmentations across several datasets. The source code is available at https://github.com/nstucki/Betti-Matching-3D.