# Learning Probabilistic Topological Representations Using Discrete Morse Theory

arXiv: https://arxiv.org/abs/2206.01742 · HTML: https://mlsanity.com/paper/2206.01742/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 68 references that are in the map.
- First posted on arXiv 3 Jun 2022.

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

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [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)
- [Bayesian SegNet: Model Uncertainty in Deep Convolutional Encoder-Decoder Architectures for Scene Understanding](https://mlsanity.com/paper/1511.02680/) (2015)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Uncertainty Estimates and Multi-Hypotheses Networks for Optical Flow](https://mlsanity.com/paper/1802.07095/) (2018)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Confidence-Aware Learning for Deep Neural Networks](https://mlsanity.com/paper/2007.01458/) (2020)
- [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)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Trigger Hunting with a Topological Prior for Trojan Detection](https://mlsanity.com/paper/2110.08335/) (2021)
- [Structure-Aware Image Segmentation with Homotopy Warping](https://mlsanity.com/paper/2112.07812/) (2021)
- [Road Network Reconstruction from Satellite Images with Machine Learning Supported by Topological Methods](https://mlsanity.com/paper/1909.06728/) (2019)
- [Graph Reconstruction by Discrete Morse Theory](https://mlsanity.com/paper/1803.05093/) (2018)
- [A Manifold View of Adversarial Risk](https://mlsanity.com/paper/2203.13277/) (2022)

## Built on by (most cited first, 20 of 20)

- [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)
- [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)
- [Enhancing Modality-Agnostic Representations via Meta-Learning for Brain Tumor Segmentation](https://mlsanity.com/paper/2302.04308/) (2023)
- [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)
- [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)
- [Universal Topology Refinement for Medical Image Segmentation with Polynomial Feature Synthesis](https://mlsanity.com/paper/2409.09796/) (2024)
- [Efficient Connectivity-Preserving Instance Segmentation with Supervoxel-Based Loss Function](https://mlsanity.com/paper/2501.01022/) (2025)
- [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)
- [Probabilistic U-Net with Kendall Shape Spaces for Geometry-Aware Segmentations of Images](https://mlsanity.com/paper/2410.14017/) (2024)
- [TopoMortar: A dataset to evaluate image segmentation methods focused on topology accuracy](https://mlsanity.com/paper/2503.03365/) (2025)
- [Disconnect to Connect: A Data Augmentation Method for Improving Topology Accuracy in Image Segmentation](https://mlsanity.com/paper/2503.05541/) (2025)
- [Topology-Aware CLIP Few-Shot Learning](https://mlsanity.com/paper/2505.01694/) (2025)
- [TopoSculpt: Betti-Steered Topological Sculpting of 3D Fine-grained Tubular Shapes](https://mlsanity.com/paper/2509.03938/) (2025)
- [Learning Topological Representations for Deep Image Understanding](https://mlsanity.com/paper/2403.15361/) (2024)
- [Graph Representation via Elements of Discrete Morse and Cobordism Theories](https://mlsanity.com/paper/2610.01937/) (2026)

## Abstract (arXiv)

Accurate delineation of fine-scale structures is a very important yet challenging problem. Existing methods use topological information as an additional training loss, but are ultimately making pixel-wise predictions. In this paper, we propose the first deep learning based method to learn topological/structural representations. We use discrete Morse theory and persistent homology to construct an one-parameter family of structures as the topological/structural representation space. Furthermore, we learn a probabilistic model that can perform inference tasks in such a topological/structural representation space. Our method generates true structures rather than pixel-maps, leading to better topological integrity in automatic segmentation tasks. It also facilitates semi-automatic interactive annotation/proofreading via the sampling of structures and structure-aware uncertainty.