# TopoFuse: Topology-Aware Tri-Planar Fusion for 3D Cryo-Electron Tomography Segmentation

arXiv: https://arxiv.org/abs/2609.29717 · HTML: https://mlsanity.com/paper/2609.29717/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [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)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Beyond the Pixel-Wise Loss for Topology-Aware Delineation](https://mlsanity.com/paper/1712.02190/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [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)
- [Cubical Ripser: Software for computing persistent homology of image and volume data](https://mlsanity.com/paper/2005.12692/) (2020)
- [Adapting Segment Anything Model (SAM) through Prompt-based Learning for Enhanced Protein Identification in Cryo-EM Micrographs](https://mlsanity.com/paper/2311.16140/) (2023)

## Abstract (arXiv)

Automated segmentation of cryo-electron tomograms routinely produces masks that are voxel-accurate but topologically broken: membranes fragment, organelles merge into one another, and enclosed cavities collapse. Existing topology-aware losses reduce these violations but cannot eliminate them, because topology is encouraged through gradient pressure rather than structurally enforced. We introduce TopoFuse, which reframes topology as a differentiable projection operator rather than a loss penalty. At each forward pass, the projection operator ProjT (a PH-guided sparse edit) identifies the critical voxels responsible for topological violations via bottleneck matching and applies sparse edits to satisfy a specified topology target (diagram feature counts and lifetime budgets) for dimensions d ∈ 0,2. If the projection converges, the output satisfies those constraints on the downsampled grid (s=2); when it does not, a repair certificate exposes this explicitly, enabling downstream filtering. A topology prior head predicts the correction target directly from input features, removing any dependence on ground-truth topology at inference. Across three cryo-ET benchmarks, TopoFuse reduces Betti number error by 54% over the strongest soft-loss baseline (p < 0.001), improves Dice by 4.6 points, and edits only 3.1% of voxels to achieve this.