# Sparse cubical complexes for efficient topology-preservation in image data

arXiv: https://arxiv.org/abs/2609.37177 · HTML: https://mlsanity.com/paper/2609.37177/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 49 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 30 of 33)

- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [nnU-Net Revisited: A Call for Rigorous Validation in 3D Medical Image Segmentation](https://mlsanity.com/paper/2404.09556/) (2024)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Dynamic Snake Convolution based on Topological Geometric Constraints for Tubular Structure Segmentation](https://mlsanity.com/paper/2307.08388/) (2023)
- [Deep learning to achieve clinically applicable segmentation of head and neck anatomy for radiotherapy](https://mlsanity.com/paper/1809.04430/) (2018)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [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)
- [Fast and Accurate Tumor Segmentation of Histology Images using Persistent Homology and Deep Convolutional Features](https://mlsanity.com/paper/1805.03699/) (2018)
- [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)
- [Analysis of the MICCAI Brain Tumor Segmentation -- Metastases (BraTS-METS) 2025 Lighthouse Challenge: Brain Metastasis Segmentation on Pre- and Post-treatment MRI](https://mlsanity.com/paper/2504.12527/) (2025)
- [A skeletonization algorithm for gradient-based optimization](https://mlsanity.com/paper/2309.02527/) (2023)
- [Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation](https://mlsanity.com/paper/2411.03228/) (2024)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [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)
- [Persistent Homology with Improved Locality Information for more Effective Delineation](https://mlsanity.com/paper/2110.06295/) (2021)
- [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)
- [Pitfalls of topology-aware image segmentation](https://mlsanity.com/paper/2412.14619/) (2024)
- [Topology Optimization in Medical Image Segmentation with Fast Euler Characteristic](https://mlsanity.com/paper/2507.23763/) (2025)
- [Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images](https://mlsanity.com/paper/1705.02045/) (2017)

## Abstract (arXiv)

Persistent homology (PH) is a frequently used tool for extracting and preserving topological information from image data, particularly in image segmentation, where preservation of topological structures is important. However, despite its general applicability across dimensionality, domains, and target structures, the runtime cost of PH-based methods often makes their practical use infeasible. In this work, we argue that this runtime cost is largely driven by processing information that is unimportant for downstream application (e.g. as optimization objective). We propose sparse cubical filtrations as an alternative foundation for PH computation, reducing subsequent computational costs by factors of up to 100 on real datasets. We show close agreement with the optimization signal of the dense counterpart and empirically evaluate our solution's effectiveness as an optimization objective in realistic training regimes where other PH-based objectives can practically not operate (i.e., 3D data with large patch sizes). We show how our solution improves topological accuracy by up to 80% across six diverse datasets while maintaining pixel- and region-based accuracy.