# Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic

arXiv: https://arxiv.org/abs/2507.23763 · HTML: https://mlsanity.com/paper/2507.23763/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 55 references that are in the map.
- First posted on arXiv 31 Jul 2025.

## Builds on (most cited first, 24 of 24)

- [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)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [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)
- [Boundary loss for highly unbalanced segmentation](https://mlsanity.com/paper/1812.07032/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Medical SAM 2: Segment medical images as video via Segment Anything Model 2](https://mlsanity.com/paper/2408.00874/) (2024)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Anatomically Constrained Neural Networks (ACNN): Application to Cardiac Image Enhancement and Segmentation](https://mlsanity.com/paper/1705.08302/) (2017)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [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)
- [Skeleton Recall Loss for Connectivity Conserving and Resource Efficient Segmentation of Thin Tubular Structures](https://mlsanity.com/paper/2404.03010/) (2024)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [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)
- [A persistent homology-based topological loss for CNN-based multi-class segmentation of CMR](https://mlsanity.com/paper/2107.12689/) (2021)
- [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)
- [De-LightSAM: Modality-Decoupled Lightweight SAM for Generalizable Medical Segmentation](https://mlsanity.com/paper/2407.14153/) (2024)
- [Segmentation of the cortical plate in fetal brain MRI with a topological loss](https://mlsanity.com/paper/2010.12391/) (2020)
- [Topology-Guided Multi-Class Cell Context Generation for Digital Pathology](https://mlsanity.com/paper/2304.02255/) (2023)

## Built on by (most cited first, 5 of 5)

- [TopoSculpt: Betti-Steered Topological Sculpting of 3D Fine-grained Tubular Shapes](https://mlsanity.com/paper/2509.03938/) (2025)
- [Topo-R1: Detecting Topological Anomalies via Vision-Language Models](https://mlsanity.com/paper/2603.13054/) (2026)
- [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (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)

Deep learning-based medical image segmentation techniques have shown promising results when evaluated based on conventional metrics such as the Dice score or Intersection-over-Union. However, these fully automatic methods often fail to meet clinically acceptable accuracy, especially when topological constraints should be observed, e.g., continuous boundaries or closed surfaces. In medical image segmentation, the correctness of a segmentation in terms of the required topological genus sometimes is even more important than the pixel-wise accuracy. Existing topology-aware approaches commonly estimate and constrain the topological structure via the concept of persistent homology (PH). However, these methods are difficult to implement for high dimensional data due to their polynomial computational complexity. To overcome this problem, we propose a novel and fast approach for topology-aware segmentation based on the Euler Characteristic (chi). First, we propose a fast formulation for chi computation in both 2D and 3D. The scalar chi error between the prediction and ground-truth serves as the topological evaluation metric. Then we estimate the spatial topology correctness of any segmentation network via a so-called topological violation map, i.e., a detailed map that highlights regions with chi errors. Finally, the segmentation results from the arbitrary network are refined based on the topological violation maps by a topology-aware correction network. Our experiments are conducted on both 2D and 3D datasets and show that our method can significantly improve topological correctness while preserving pixel-wise segmentation accuracy.