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REVIEW 4 major objections 6 minor 57 references

Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Steering convolution offsets with persistent homology preserves connectivity in medical image segmentation without sacrificing pixel accuracy.

desk verdict PH-guided offsets are a genuinely new combination, but the evidence doesn't isolate the mechanism and some claims are contradicted by the paper's own tables. read the letter →

arxiv 2412.20608 v1 pith:Z3D5BIRC submitted 2024-12-29 eess.IV cs.CV

Yousef Yeganeh , Rui Xiao , Goktug Guvercin , Nassir Navab , Azade Farshad This is my paper · ORCID
classification eess.IVcs.CV MSC 55N3168U1092C55
keywords conformableconvolutiontopologicalposteriorgeneratorpersistenthomologycubicalcomplexesdeformabletopology-preservingsegmentationmedicalimageBettinumbers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Medical segmentation models can score well on pixel-level accuracy while splitting or merging fine anatomical structures such as retinal vessels and neuron membranes. This paper proposes a convolutional layer, Conformable Convolution, whose kernel sampling positions are steered by a topological prior instead of learned from raw image content alone. The prior is produced by the Topological Posterior Generator, which runs persistent homology on pooled feature maps, keeps only features whose persistence exceeds a threshold, and spreads the surviving generators with Gaussian dilation. On three datasets, retinal vessels, colon cancer cells, and neuron electron microscopy, the layer reduces Betti-number and Euler-characteristic errors compared with deformable and snake-convolution baselines while keeping Dice and AUC broadly comparable. The authors claim the layer is architecture-agnostic and can replace any convolution in a segmentation network.

What carries the argument

The Topological Posterior Generator is the load-bearing component: it converts a feature map into a spatial prior by channel pooling, cubical persistent homology, persistence-threshold filtering, and Gaussian dilation. It supplies the coordinates, the birth and death locations of 0-dimensional connected components, that the offset-generating convolution should attend to. Conformable Convolution is the second component: a deformable-style convolution in which the offsets are computed as $g(\text{TPG}(\phi_{\text{post}}))$, so the adaptive kernel samples input features at positions $p + p_c + \Delta\hat{p}_c$ that are biased toward topologically significant regions.

What would settle it

Retrain the CHASE vessel model with the TPG prior replaced by Gaussian-dilated random generator locations of the same density; if Betti-zero and Euler errors are no worse than with the true topological prior, the specific topological information is not what drives the improvement. Alternatively, sweep $\tau_0$ from zero to the 90th percentile of persistence values and check whether the reported error reductions depend sharply on the threshold.

Watch

Extended reading notes

Core claim

The central claim is that persistent homology computed on intermediate feature maps yields a spatial prior that makes adaptive convolution kernels focus on topologically important locations, and that this focus transfers into topologically more faithful segmentation maps. Concretely, the TPG block channel-pools the input feature map, builds cubical complexes, records 0-dimensional persistence pairs, filters generators by a persistence threshold $\tau_0$, and forms $\phi_{\text{post}} = \phi_{\text{dil}} \odot \phi_{\text{in}} + \phi_{\text{in}}$, where $\phi_{\text{dil}}$ is the Gaussian-dilated indicator of surviving generators. Conformable Convolution predicts kernel offsets from $\phi_{\text{post}}$ rather than from the raw feature map, so the deformation of the receptive field is biased toward birth and death locations of connected components. The paper reports that this lowers error in $\beta_0$, $\beta_1$, and Euler characteristic on CHASE DB1, HT29, and ISBI12 relative to deformable convolution and dynamic snake convolution, while pixel-level metrics stay comparable.

Load-bearing premise

The mechanism rests on the assumption that 0-dimensional persistent homology on a single channel-pooled feature map, with a fixed persistence threshold $\tau_0$, marks the anatomical structures that matter for segmentation at every layer where the module is used.

Editorial extensions

If this is right

  • Inserting one Conformable block at the bottleneck of a U-Net reduces Betti-zero error on CHASE from 26.9 to 21.6 and Euler-characteristic error from 28.5 to 20.6 while keeping Dice near 79 to 80 percent.
  • Adding the layer to Y-Net raises clDice from 0.76 to 0.79 and cuts Betti-zero error from 27.9 to 21.1 on the same dataset.
  • Because the module is architecture-agnostic, the same recipe of pooling, persistence, filtering, dilation, and offset prediction should transfer to other encoder-decoder or transformer-based segmentation backbones.
  • Ablations show that filtration, Gaussian dilation, and feature aggregation each contribute; removing any of them raises topological errors, so all three are needed for the reported gain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the authors do not pursue is to feed 1-dimensional persistence generators, loop births and deaths, into the same offset map; ISBI12's holes suggest this could push the reported $\beta_1$ error below 7.9.
  • The fixed threshold $\tau_0$ and the unspecified channel-pooling operator $\psi$ are hidden design choices; making both adaptive, learned per layer or per dataset, would clarify how much of the effect is generic topological guidance versus dataset-specific tuning.
  • Because TPG consumes any feature map, the same prior could modulate attention weights in transformers rather than convolution offsets, effectively giving topology-aware attention for segmentation and registration tasks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes Conformable Convolution, a convolutional layer whose kernel offsets are guided by a Topological Posterior Generator (TPG) that applies persistent homology to channel-pooled feature maps. The TPG computes birth/death generators in cubical complexes, filters them by a persistence threshold, applies Gaussian dilation, and aggregates the result with the original feature map. The resulting posterior is fed to a small offset-generating network to modulate a second convolution, in the style of deformable convolution. The authors evaluate the layer on three segmentation datasets (CHASE, HT29, ISBI12) using pixel-level metrics and topological/connectivity metrics, and they report ablations of filtration, Gaussian dilation, and feature aggregation, as well as an ablation of the number of Conform blocks.

Significance. If the central claim were established, the paper would contribute a genuinely architecture-agnostic way to inject topological priors into convolutional networks, with potential value for medical segmentation of tubular and networked structures. The idea of using persistent homology to guide kernel offsets is a reasonable departure from both topology-aware loss functions and purely learned deformable convolutions. The paper also makes a positive effort to evaluate with topological metrics beyond Dice, including Betti errors, Euler characteristic, clDice, ARI, and VI, across three datasets. However, the evidence as presented does not isolate the causal role of the PH-derived prior, several key components of the TPG are underspecified, and some reported results are inconsistent with the claim that pixel-level accuracy is not sacrificed. The strengths are the breadth of evaluation and the clarity of the proposed module; the weaknesses are concentrated in missing control experiments and parametric detail.

major comments (4)
  1. [Section 4.1, Tables 4 and 5] The causal role of the persistent-homology-derived prior is not isolated. Every ablation in Table 4 retains the PH-generated generator coordinates as the prior and only removes filtration, Gaussian dilation, or aggregation; there is no baseline in which the offset generator consumes a non-topological prior (e.g., the raw feature map, an edge map, or random coordinates) while keeping the rest of the Conformable block fixed. Since the Conformable block adds an extra convolutional branch and a Gaussian-dilation operation beyond the standard deformable convolution, the observed topological gains could plausibly come from those components rather than from the PH guidance. A control experiment with a non-topological prior is needed to support the abstract's claim that PH-guided offsets preferentially focus on topologically significant regions.
  2. [Section 4.1, Eqs. (4)-(6)] The channel pooling operator ψ and the persistence filtering threshold τ0 are never specified, and their sensitivity is not examined. Equation (4) defines φ_pooled = ψ(φ_in) but does not state whether ψ is a mean, max, learned linear combination, or something else. Equation (5) uses an indicator with threshold τ0, but the paper gives no value, no selection criterion, and no sweep over τ0. Because the surviving generator set G_M depends directly on τ0 and on the pooled map, the entire offset-guidance mechanism is not reproducible from the manuscript and its robustness to these free choices is unknown.
  3. [Section 3, Eqs. (5)-(6), Tables 3-4] There is an unexplained mismatch between the homology dimension used and the metrics reported. The method explicitly tracks 0-dimensional cubes and connected-component generators, yet the paper reports improvements in error β1 and discusses loops and holes (e.g., ISBI12 with 'numerous loops') without explaining how 0-dimensional persistence can directly target 1-dimensional topological features. If the reported β1 improvements are indirect effects of better connected-component preservation, that claim should be stated and supported; otherwise the connection between the TPG mechanism and the β1 results is unsupported.
  4. [Table 3, Table 5, Section 5.2.1] The statement that the method 'does not sacrifice the pixel-wise results' is not supported by the reported statistics. In Table 3, on HT29 the Conform layer has lower Dice (94.6±1.3) than both Deform (95.8±2.1) and DSC (95.8±2.0), and lower AUC (99.1±0.6) than Deform (99.6±0.2); on CHASE, Conform has lower AUC (94.2±0.2) than DSC (95.9±0.2). These are load-bearing for the claim that topological gains come without pixel-level cost. Additionally, Table 5 explicitly states that only the best model is selected and all standard deviations are zero, so the comparison across numbers of Conform layers cannot be assessed statistically. The authors should report all runs and either temper the pixel-level claim or provide a proper model-selection procedure.
minor comments (6)
  1. [Abstract] The word 'Topological' in the abstract is typeset with an erroneous space ('T opological').
  2. [Section 5.2.1] The phrase 'the conformal property of our method' appears to be a typo for 'conformable property', since the method is named Conformable Convolution throughout the paper.
  3. [Tables 1-3 and Section 5.2.1] The table numbering and cross-references are inconsistent: Table 1 and Table 3 have nearly identical captions ('Segmentation Performance Compared to SOTA Layers with Adaptive Kernel...'), and the text refers to 'Tab. 3 on ISBI12' for a model comparison that does not appear in Table 3. Please harmonize the table numbers and correct the references.
  4. [Eqs. (9) and (13)] Equation (9) defines φ_post = φ_dil ⊙ φ_in + φ_in, while the ablation description in Section 5.2.2 and Eq. (13) use φ_post = φ_dil ⊙ φ_in as the 'blocked aggregation' variant. The text should clarify exactly which operation is being ablated and why removing the residual addition is the intended ablation of 'feature aggregation'.
  5. [Section 5.1 and Introduction] The introduction mentions a new blood-flow-simulation metric in the supplementary materials, and Section 5.1 states that implementation details are reported in the supplement; however, no supplement was provided with the manuscript. Please include the supplement or remove the references to it.
  6. [Table 2] The clDice column in Table 2 mixes percentages (e.g., 0.75, 0.73) and apparently unscaled values without a clear legend, making it difficult to compare rows; please use a consistent format.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the topology-guided offsets are computed from PH on feature maps, and all topological metrics are external to the training objective.

full rationale

The paper's derivation chain is an empirical architecture rather than a closed-form derivation. TPG computes phi_post from the input feature map via channel pooling, persistent homology on cubical complexes, persistence filtering (Eq. 5-6), Gaussian dilation (Eq. 8), and feature aggregation (Eq. 9). Conformable convolution consumes phi_post to generate offsets (Eq. 12) in the same spirit as deformable convolution (Eq. 11). No equation defines the target metrics (Betti errors, Euler characteristic, clDice) in terms of the input prior; the evaluation metrics are computed between the predicted and ground-truth segmentation maps independently of the TPG construction. The persistence threshold tau_0 and pooling operator psi are hyperparameters, but their unspecified choice is a soundness/reproducibility concern, not a circular reduction. Self-citations to Y-Net [11] and SCOPE [53] are used as baselines and architectural components, not as justifications of the central claim, and the citation to Edelsbrunner et al. [9] for low-persistence noise filtering is an external, standard TDA result. The comparison against deformable convolution [7] provides a no-PH offset baseline, so the claimed role of the topological prior is empirically testable rather than forced by construction. No fitted parameter is renamed as a prediction, and no uniqueness claim is imported to forbid alternatives. The main weaknesses, such as unspecified tau_0, undefined psi, and the lack of a fully isolated no-TPG ablation, are correctness and evidence issues rather than circularity, so a score of 0 is appropriate.

Assumptions & free parameters 3 free parameters · 2 assumptions · 0 invented entities

No new physical entities or forces are introduced. The method relies on one hand-chosen threshold and unspecified pooling and placement choices, which are the main free parameters.

free parameters (3)
  • Persistence filtering threshold tau_0 = not reported
    Used in Eq. (5) to retain only generators with pers(tau_i, tau_j) > tau_0. The paper does not state how tau_0 is chosen or whether it is tuned per dataset or layer.
  • Channel pooling operation psi = not specified
    Eq. (4) pools the channel dimension of phi_in to a single map; the exact operator (mean, max, etc.) is not stated, which affects the computed topology.
  • Number and position of Conform layers = one bottleneck block (chosen after ablation)
    Table 5 and the supplement ablate layer count and insertion position, with the best configuration selected for the main comparisons.
assumptions (2)
  • domain assumption Persistent homology on a cubical complex of the pooled feature map captures the topology of the target anatomical structures.
    The entire TPG mechanism relies on this; no evidence is provided that the pooled feature map's 0-dimensional persistence aligns with semantic structure.
  • domain assumption A single persistence threshold tau_0 separates meaningful structure from noise across layers and datasets.
    Eq. (5) assumes a global filtering rule; the paper does not analyze sensitivity to tau_0.

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Cite this review

Pith. "Pith review of Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures." pith.science (2026). https://pith.science/paper/Z3D5BIRC

@misc{pith2026241220608,
  author       = {Pith},
  title        = {Pith review of: Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z3D5BIRC}},
  note         = {Machine review of arXiv:2412.20608}
}
read the original abstract

While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their successes, deep learning models often struggle to accurately capture the connectivity and continuity of fine, sometimes pixel-thin, yet critical structures due to their reliance on implicit learning from data. Such shortcomings can significantly impact the reliability of analysis results and hinder clinical decision-making. To address this challenge, we introduce Conformable Convolution, a novel convolutional layer designed to explicitly enforce topological consistency. Conformable Convolution learns adaptive kernel offsets that preferentially focus on regions of high topological significance within an image. This prioritization is guided by our proposed Topological Posterior Generator (TPG) module, which leverages persistent homology. The TPG module identifies key topological features and guides the convolutional layers by applying persistent homology to feature maps transformed into cubical complexes. Our proposed modules are architecture-agnostic, enabling them to be integrated seamlessly into various architectures. We showcase the effectiveness of our framework in the segmentation task, where preserving the interconnectedness of structures is critical. Experimental results on three diverse datasets demonstrate that our framework effectively preserves the topology in the segmentation downstream task, both quantitatively and qualitatively.

Figures

Figures reproduced from arXiv: 2412.20608 by the authors.

Figure 1
Figure 1. Our proposed layer comprises two modules: (a) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An example visualization on how PH applies a filtering function fτ (.) with changing τ (τ1, τ2, τ3) to the original image with vessel and noise , obtaining a nested set of cubical com￾plexes Q (Q1, Q2, Q3). As τ increases from τ1 to τ2, vessel is first born at Q1 and noise is later born at Q2 . Both of them die at Q3, as τ further raises to τ3. (a) Mean Feature Map (b) All generators (c) Topo Priors after filtering … view at source ↗
Figure 3
Figure 3. Visualization of Topological Priors in each layer of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Demonstration of the Gaussian dilation process on real [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Qualitative Segmentation Results corresponding to Tab. 3. errorβ0 (highlighting disconnected components) are in red squares, while errorβ1 (highlighting holes) are in red circles. count the number of connected components and indepen￾dent holes, respectively. The Euler …

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.