# Topology-Aware Uncertainty for Image Segmentation

arXiv: https://arxiv.org/abs/2306.05671 · HTML: https://mlsanity.com/paper/2306.05671/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 75 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 30 of 38)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Semantic Image Segmentation with Deep Convolutional Nets and Fully Connected CRFs](https://mlsanity.com/paper/1412.7062/) (2014)
- [A Survey of Deep Active Learning](https://mlsanity.com/paper/2009.00236/) (2020)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [A Probabilistic U-Net for Segmentation of Ambiguous Images](https://mlsanity.com/paper/1806.05034/) (2018)
- [Aleatoric uncertainty estimation with test-time augmentation for medical image segmentation with convolutional neural networks](https://mlsanity.com/paper/1807.07356/) (2018)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Bayesian Convolutional Neural Networks with Bernoulli Approximate Variational Inference](https://mlsanity.com/paper/1506.02158/) (2015)
- [Exploring Uncertainty Measures in Deep Networks for Multiple Sclerosis Lesion Detection and Segmentation](https://mlsanity.com/paper/1808.01200/) (2018)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [PHiSeg: Capturing Uncertainty in Medical Image Segmentation](https://mlsanity.com/paper/1906.04045/) (2019)
- [Learning in an Uncertain World: Representing Ambiguity Through Multiple Hypotheses](https://mlsanity.com/paper/1612.00197/) (2016)
- [Beyond the Pixel-Wise Loss for Topology-Aware Delineation](https://mlsanity.com/paper/1712.02190/) (2017)
- [ROSE: A Retinal OCT-Angiography Vessel Segmentation Dataset and New Model](https://mlsanity.com/paper/2007.05201/) (2020)
- [DeepVesselNet: Vessel Segmentation, Centerline Prediction, and Bifurcation Detection in 3-D Angiographic Volumes](https://mlsanity.com/paper/1803.09340/) (2018)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Exploiting Epistemic Uncertainty of Anatomy Segmentation for Anomaly Detection in Retinal OCT](https://mlsanity.com/paper/1905.12806/) (2019)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [Efficient automatic segmentation for multi-level pulmonary arteries: The PARSE challenge](https://mlsanity.com/paper/2304.03708/) (2023)
- [Towards safe deep learning: accurately quantifying biomarker uncertainty in neural network predictions](https://mlsanity.com/paper/1806.08640/) (2018)

## Built on by (most cited first, 30 of 32)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [RUN: Reversible Unfolding Network for Concealed Object Segmentation](https://mlsanity.com/paper/2501.18783/) (2025)
- [Centerline Boundary Dice Loss for Vascular Segmentation](https://mlsanity.com/paper/2407.01517/) (2024)
- [Uncertainty in Graph Neural Networks: A Survey](https://mlsanity.com/paper/2403.07185/) (2024)
- [Calibrating Uncertainty for Semi-Supervised Crowd Counting](https://mlsanity.com/paper/2308.09887/) (2023)
- [TopoCellGen: Generating Histopathology Cell Topology with a Diffusion Model](https://mlsanity.com/paper/2412.06011/) (2024)
- [Representing Topological Self-Similarity Using Fractal Feature Maps for Accurate Segmentation of Tubular Structures](https://mlsanity.com/paper/2407.14754/) (2024)
- [Semi-supervised Segmentation of Histopathology Images with Noise-Aware Topological Consistency](https://mlsanity.com/paper/2311.16447/) (2023)
- [Reversible Unfolding Network for Concealed Visual Perception with Generative Refinement](https://mlsanity.com/paper/2508.15027/) (2025)
- [TopoDiffusionNet: A Topology-aware Diffusion Model](https://mlsanity.com/paper/2410.16646/) (2024)
- [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)
- [Spatial Diffusion for Cell Layout Generation](https://mlsanity.com/paper/2409.03106/) (2024)
- [Neurovascular Segmentation in sOCT with Deep Learning and Synthetic Training Data](https://mlsanity.com/paper/2407.01419/) (2024)
- [MATCH: Multi-faceted Adaptive Topo-Consistency for Semi-Supervised Histopathology Segmentation](https://mlsanity.com/paper/2510.01532/) (2025)
- [Scale-Free Image Keypoints Using Differentiable Persistent Homology](https://mlsanity.com/paper/2406.01315/) (2024)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [FMS2: Unified Flow Matching for Segmentation and Synthesis of Thin Structures](https://mlsanity.com/paper/2603.13659/) (2026)
- [MorphoSeg: An Uncertainty-Aware Deep Learning Method for Biomedical Segmentation of Complex Cellular Morphologies](https://mlsanity.com/paper/2409.17110/) (2024)
- [TopoMortar: A dataset to evaluate image segmentation methods focused on topology accuracy](https://mlsanity.com/paper/2503.03365/) (2025)
- [Evaluation of Blood Vessel Segmentation Methods on Hard-to-Detect Vascular Structures](https://mlsanity.com/paper/2406.13128/) (2024)
- [Scalar Function Topology Divergence: Comparing Topology of 3D Objects](https://mlsanity.com/paper/2407.08364/) (2024)
- [Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures](https://mlsanity.com/paper/2412.20608/) (2024)
- [Leveraging Textual Anatomical Knowledge for Class-Imbalanced Semi-Supervised Multi-Organ Segmentation](https://mlsanity.com/paper/2501.13470/) (2025)
- [Dynamic Exploration on Segment-Proposal Graphs for Tubular Centerline Tracking](https://mlsanity.com/paper/2506.18930/) (2025)
- [QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems](https://mlsanity.com/paper/2507.14760/) (2025)
- [Rethinking the Detail-Preserved Completion of Complex Tubular Structures based on Point Cloud: a Dataset and a Benchmark](https://mlsanity.com/paper/2508.17658/) (2025)
- [Bridging 3D Deep Learning and Curation for Analysis and High-Quality Segmentation in Practice](https://mlsanity.com/paper/2511.22236/) (2025)
- [Topology-Driven Fusion of nnU-Net and MedNeXt for Accurate Brain Tumor Segmentation on Sub-Saharan Africa Dataset](https://mlsanity.com/paper/2604.15964/) (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)

## Abstract (arXiv)

Segmentation of curvilinear structures such as vasculature and road networks is challenging due to relatively weak signals and complex geometry/topology. To facilitate and accelerate large scale annotation, one has to adopt semi-automatic approaches such as proofreading by experts. In this work, we focus on uncertainty estimation for such tasks, so that highly uncertain, and thus error-prone structures can be identified for human annotators to verify. Unlike most existing works, which provide pixel-wise uncertainty maps, we stipulate it is crucial to estimate uncertainty in the units of topological structures, e.g., small pieces of connections and branches. To achieve this, we leverage tools from topological data analysis, specifically discrete Morse theory (DMT), to first capture the structures, and then reason about their uncertainties. To model the uncertainty, we (1) propose a joint prediction model that estimates the uncertainty of a structure while taking the neighboring structures into consideration (inter-structural uncertainty); (2) propose a novel Probabilistic DMT to model the inherent uncertainty within each structure (intra-structural uncertainty) by sampling its representations via a perturb-and-walk scheme. On various 2D and 3D datasets, our method produces better structure-wise uncertainty maps compared to existing works. Code available at https://github.com/Saumya-Gupta-26/struct-uncertainty