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

Rohit Kumar Salla, Neelesh Gupta, Xingjian Li, Min Xu

arXiv:2609.29717v1cs.CV

TL;DR

Cryo-ET segmentation can be voxel-accurate yet topologically broken, while topology-aware losses encourage correct structure without ensuring it. TopoFuse uses a differentiable projection with sparse edits and a learned prior to predict correction targets. Across three cryo-ET benchmarks, it reduces Betti error and improves Dice while editing a small share of voxels.

  • Problem

    Cryo-ET segmentation can produce voxel-accurate but topologically broken masks, and topology-sensitive losses reduce violations without eliminating them.

    Source: §1 · pp. 1–2

  • Method

    TopoFuse applies PH-guided sparse critical-voxel edits to enforce topology targets, using a learned prior head to predict inference-time targets.

    Source: §1 · p. 2, §4.5 · p. 6

  • Results

    54% lower BE0 than the strongest soft-loss baseline (p < 0.001), 4.6 pp higher Dice, and 3.1% voxel edits across three cryo-ET benchmarks.

    Source: §7 · p. 14

  • Takeaways & Limitations

    The repair certificate records corrections and supports quality-controlled downstream analysis.

    Source: §7 · p. 14

    The projection enforces constraints only for d∈{0, 2} on the downsampled grid (s=2); d=1 tunnels are excluded because of 3D persistent-homology cost.

    Source: §4.5 · p. 6

Abstract

from arXiv · show

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 $\mathrm{Proj}_T$ (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 $d \in \{0,2\}$. 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.

1 Introduction

TopoFuse addresses cryo-ET masks that can be voxel-accurate yet structurally incorrect, and replaces topology-loss pressure with sparse edits that enforce a topology target. It combines tri-planar features, topology-aware fusion, projection, and a learned prior for predicting the correction target.

1 Introduction

  • Cryo-ET segmentation can yield voxel-accurate predictions that fragment membranes, split mitochondria, or collapse vesicle lumens.These errors can invalidate surface-area estimates and break contact-site analysis architectures.Source: §1 · pp. 1–2
  • Topology-sensitive losses reduce violations but cannot eliminate them because they encourage correct structure through gradients rather than enforcing it.Weak gradients, poorly tuned weights, or pathological persistence pairings can leave violations that persist at test time.Source: §1 · p. 2
  • The PH-guided sparse edit operator changes critical voxels associated with unmatched features to make predictions satisfy a topology target.Edits are sparse, almost-everywhere differentiable, and logged; the paper uses “projection” to mean feasibility enforcement, not a Euclidean projection.Source: §1 · p. 2
  • TopoFuse combines tri-planar SAM encoders, topology-aware FiLM fusion, a projection layer, and a learned topology prior head.At inference, the prior head predicts the correction target from features.Source: §1 · pp. 2–3
  • Across three public cryo-ET benchmarks, TopoFuse consistently and significantly outperforms the best topology-loss baselines on voxel and topological metrics.The introduction reports validation on EMD-0506, EMPIAR-10499, and EMPIAR-10045.Source: §1 · p. 3

2 Related Work

Prior cryo-ET segmentation methods focus on voxel supervision or image priors, while topology-aware losses penalize rather than prevent topological violations. Earlier projection approaches have different constraints, and cryo-ET presents a demanding setting for topology enforcement.

2 Related Work

  • Existing cryo-ET segmentation methods use voxel-level objectives or slice image priors, and none of the listed methods reasons about topological correctness.3D U-Net and nnU-Net are described as dominant volumetric baselines; DeepFinder targets particle picking rather than dense segmentation.Source: §2 · p. 3
  • Persistent-homology and related topology-aware learning methods treat topology as a soft regularizer, penalizing violations without structurally preventing them.The cited approaches include discrete Morse theory, homotopy warping, centreline Dice, and Betti matching.Source: §2 · p. 3
  • Compared with prior projection methods, TopoFuse is described as sparse, certificate-equipped, and self-contained at test time through a learned prior.The cited cardiac-segmentation formulation lacks sparsity guarantees and a certificate, and requires ground-truth topology at inference.Source: §2 · p. 3
  • Thin membranes, severe missing-wedge anisotropy, and extreme noise make cryo-ET a stress test for topology enforcement.The paper also states that its projection framework applies wherever persistent-homology targets can be specified.Source: §2 · p. 3

3 Background: Persistent Homology

Persistent homology tracks topological features across a threshold filtration and records their births and deaths. In TopoFuse, critical cells identify locations where logit edits can change topology.

3 Background: Persistent Homology

  • A sublevel-set filtration tracks connected components and enclosed voids as a scalar-field threshold increases.The paper records features in dimensions d=0 and d=2 as they appear and disappear.Source: §3 · p. 3
  • Birth–death pairs record when features appear and disappear, and their lifetime measures feature significance.The lifetime is |d_i − b_i|.Source: §3 · p. 4
  • Birth and death cells anchor each feature to critical voxels, where logit edits can change the topology.Persistent-homology gradients with respect to the scalar field are non-zero only at these critical voxels, enabling their use for edit localization rather than loss construction.Source: §3 · p. 4

4 Method

TopoFuse combines three axis-specific feature streams through topology-conditioned fusion, then uses persistent-homology-guided sparse edits to enforce topology targets. A learned prior supplies inference-time targets, while the training objective supervises both projected and pre-projection predictions.

4.1 Notation

  • The architecture encodes a tomogram as a volume V, processes it in the canonical xy, xz, and yz planes, and represents predictions before and after projection as probability maps.The repair certificate records iteration count, edited voxel positions, and spatial edit sparsity.Source: §4.1 · p. 4

4.2 Architecture Overview

  • The four-stage architecture extracts tri-planar features, fuses them with topology-aware weights, applies projection edits, and predicts correction targets from features at inference.The projection produces a repair certificate alongside each prediction.Source: §4.2 · p. 4

4.3 Tri-Planar Foundation Encoder

  • Three SAM encoders extract axis-specific features from orthogonal slice stacks, with the views helping address cryo-ET anisotropy and missing-wedge blur.The encoder uses pretrained SAM ViT-B weights as a 2D image encoder, not its promptable-segmentation mechanism.Source: §4.3 · pp. 4–5

4.4 Topology-Conditioned FiLM Fusion

  • A topology descriptor summarizes predicted feature counts and persistence mass, and FiLM uses it to condition the fusion weights for the three planar logits.The descriptor represents the current topological state; correctness is learned indirectly through segmentation supervision.Source: §4.4 · pp. 5–6

4.5 Differentiable Topology Projection

  • On the s=2 grid, the PH-guided projection approximates its ideal objective through iterative sparse logit edits that change only identified critical voxels, enforcing constraints for dimensions d∈{0,2}.Non-convergence (dB > ϵ after Tmax steps) is flagged and occurs in 4% of volumes; median spatial sparsity across test sets is 3.1%.Source: §4.5 · pp. 6–7

4.6 Topology Prior Head

  • The topology prior head predicts Betti counts and persistence budgets from fused features, constructing a target diagram without ground-truth topology at inference.A 0.35 BE0 residual gap between learned and oracle priors is attributed to prior-prediction error.Source: §4.5 · p. 6, §4.6 · p. 8

4.7 Training Objective

  • The total objective combines post-projection Dice and cross-entropy, a pre-projection Wasserstein topology loss, prior regression, and auxiliary pre-projection Dice and cross-entropy.The topology loss is warmed up over 5,000 steps, and varying its weight by ±0.5× changes Dice by less than 0.3 and BE0 by less than 0.04.Source: §4.7 · pp. 8–9

5 Experiments

Experiments compare TopoFuse with topology-loss baselines across three cryo-ET benchmarks and test its components, controlled topology recovery, computational overhead, and repair certificates. The results show lower topology error and higher Dice, alongside sparse edits and explicit detection of some over-corrections.

5.1 Setup

  • The evaluation uses EMD-0506, EMPIAR-10499, and EMPIAR-10045, with topology baselines sharing the tri-planar backbone and training budget.Topology metrics use binarized predictions at s = 2; training runs for 200k steps and reports means and standard deviations across five seeds.Source: §5.1 · p. 9

5.3 Main Results

  • Across three benchmarks, TopoFuse reduces BE0 54% versus BettiMatching (p < 0.001) and improves Dice by 4.6 pp.The topology baselines share the same tri-planar backbone and training budget.Source: Table 2 · p. 10, §5.3 · p. 11

5.2 Computational Cost

  • TopoFuse adds 66M parameters and 157 GFLOPs; PH computation is 5.9% of FLOPs, while CPU–GPU synchronization adds about 15 ms per crop.In a representative slice, 5.13% edits reduce connected components from 7,090 to 36.Source: §5.2 · p. 10, Table 1 · p. 10, Figure 2 · p. 11

5.4 Ablation Study

  • BE0 falls from 2.41 to 1.11 after projection; learned fusion helps before enforcement, FiLM adds a marginal improvement, and T = 5 reaches 96% convergence.T = 10 adds negligible improvement, while T = 3 achieves 89% convergence with marginal quality loss.Source: §5.4 · p. 12, Table 3 · p. 12

5.5 Synthetic Controlled Benchmark

  • On SYN, exact recovery is 84% versus 61% for BettiMatching at SNR = 0.10, and 71% versus 44% for the next-best method at SNR = 0.05.SYN controls topology; median edit sparsity is 3.1±1.4% across conditions.Source: §5.1 · p. 9, §5.5 · p. 13

5.6 Certificate and Failure Analysis

  • When a prior predicts one component for two adjacent organelles, projection can merge them; certificates flag residual error, and identify over-correction by elevated sparsity in 83% of observed failures.Successful repairs typically have 3.1% sparsity, compared with more than 8% in the flagged cases.Source: §5.6 · p. 13

6 Discussion

The 54% BE0 gap under matched conditions supports the shift from loss-based encouragement to structural correction, while the discussion qualifies the projection's grid, homology, prior, and crop-level scope.

6 Discussion

  • The 54% BE0 gap over the strongest soft-loss baseline under matched backbone, training budget, and inference protocol supports structural correction through a forward-pass operator.Source: §6 · p. 13
  • The projection enforces a specified PH target on the downsampled representation (s=2) for d∈{0, 2}, conditional on convergence and target correctness.Source: §6 · p. 13
  • It does not claim full-resolution correctness, cover tunnels (d=1), or compensate for a substantially wrong prior.Source: §6 · p. 13
  • d=1-like perforations affect ≤8.2% of volumes, concentrate in thin cristae and perforated membrane sheets, and remain future work; d=1 enforcement has O(n^1.5) cost.Source: §6 · p. 14
  • Wrong-prior over-correction occurs in ≈4% of volumes and concentrates in the most anisotropic, thin-membrane EMPIAR-10045 volumes.Source: §6 · p. 14
  • Independent per-class edits introduce conflicts in <0.3% of edited voxels (mean/median/max 0.30/0.24/1.61%), resolved by argmax with negligible topological impact (BE0 shift ≤0.01).Source: §6 · p. 14

7 Conclusion

TopoFuse treats topology as a differentiable projection that makes sparse PH-guided edits and records them in a repair certificate. Across three cryo-ET benchmarks, it reduces BE0 by 54% over the strongest soft-loss baseline, improves Dice by 4.6 pp, and edits 3.1% of voxels.

7 Conclusion

  • TopoFuse treats topology as a differentiable projection operator and uses sparse PH-guided edits to enforce a target on downsampled predictions, recording corrections in a repair certificate.Source: §7 · p. 14
  • Across three cryo-ET benchmarks, TopoFuse reduces BE0 by 54% over the strongest soft-loss baseline (p < 0.001), improves Dice by 4.6 pp, and edits 3.1% of voxels.Source: §7 · p. 14
  • The repair certificate enables quality-controlled downstream analysis that gradient-regularised methods cannot offer.Source: §7 · p. 14
  • Validation is limited to cryo-ET; future work includes d=1 homology, uncertainty-aware prior estimation, and cross-tile consistency.Source: §7 · p. 14
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