# A Learning-based Framework for Topology-Preserving Segmentation using Quasiconformal Mappings

arXiv: https://arxiv.org/abs/2210.03299 · HTML: https://mlsanity.com/paper/2210.03299/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 22 of 22)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [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)
- [The state of the art in kidney and kidney tumor segmentation in contrast-enhanced CT imaging: Results of the KiTS19 Challenge](https://mlsanity.com/paper/1912.01054/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [DoubleU-Net: A Deep Convolutional Neural Network for Medical Image Segmentation](https://mlsanity.com/paper/2006.04868/) (2020)
- [Prior-aware Neural Network for Partially-Supervised Multi-Organ Segmentation](https://mlsanity.com/paper/1904.06346/) (2019)
- [Beyond the Pixel-Wise Loss for Topology-Aware Delineation](https://mlsanity.com/paper/1712.02190/) (2017)
- [Autoregressive Unsupervised Image Segmentation](https://mlsanity.com/paper/2007.08247/) (2020)
- [nnDetection: A Self-configuring Method for Medical Object Detection](https://mlsanity.com/paper/2106.00817/) (2021)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization](https://mlsanity.com/paper/2103.16768/) (2021)
- [Convexity Shape Prior for Level Set based Image Segmentation Method](https://mlsanity.com/paper/1805.08676/) (2018)
- [TEDS-Net: Enforcing Diffeomorphisms in Spatial Transformers to Guarantee Topology Preservation in Segmentations](https://mlsanity.com/paper/2107.13542/) (2021)
- [A unifying framework for n-dimensional quasi-conformal mappings](https://mlsanity.com/paper/2110.10437/) (2021)
- [A Deep Learning Framework for Diffeomorphic Mapping Problems via Quasi-conformal Geometry applied to Imaging](https://mlsanity.com/paper/2110.10580/) (2021)
- [A Generalized Asymmetric Dual-front Model for Active Contours and Image Segmentation](https://mlsanity.com/paper/2006.07839/) (2020)
- [Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification](https://mlsanity.com/paper/2103.16411/) (2021)

## Built on by (most cited first, 10 of 10)

- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [QIS : Interactive Segmentation via Quasi-Conformal Mappings](https://mlsanity.com/paper/2402.14695/) (2024)
- [SCOPE: Structural Continuity Preservation for Medical Image Segmentation](https://mlsanity.com/paper/2304.14572/) (2023)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [A Symmetric Dynamic Learning Framework for Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2411.02888/) (2024)
- [Quasi-Conformal Convolution : A Learnable Convolution for Deep Learning on Simply Connected Open Surfaces](https://mlsanity.com/paper/2502.01356/) (2025)
- [Circular Quasiconformal Deturbulence: Geometry-Based Restoration from Multiple Turbulent Frames](https://mlsanity.com/paper/2504.13432/) (2025)
- [Coupled Reconstruction of 2D Blood Flow and Vessel Geometry from Noisy Images via Physics-Informed Neural Networks and Quasi-Conformal Mapping](https://mlsanity.com/paper/2508.11216/) (2025)
- [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (2026)
- [Learning Task-Aware Sampling with Shared Saliency through Density-Equalizing Mappings](https://mlsanity.com/paper/2606.12869/) (2026)

## Abstract (arXiv)

We propose the Topology-Preserving Segmentation Network, a deformation-based model that can extract objects in an image while maintaining their topological properties. This network generates segmentation masks that have the same topology as the template mask, even when trained with limited data. The network consists of two components: the Deformation Estimation Network, which produces a deformation map that warps the template mask to enclose the region of interest, and the Beltrami Adjustment Module, which ensures the bijectivity of the deformation map by truncating the associated Beltrami coefficient based on Quasiconformal theories. The proposed network can also be trained in an unsupervised manner, eliminating the need for labeled training data. This is achieved by incorporating an unsupervised segmentation loss. Our experimental results on various image datasets show that TPSN achieves better segmentation accuracy than state-of-the-art models with correct topology. Furthermore, we demonstrate TPSN's ability to handle multiple object segmentation.