Optimal Transport Driven Asymmetric Image-to-Image Translation for Nuclei Segmentation of Histological Images

Pradipta Maji; Suman Mahapatra

REVIEW 4 major objections 8 minor 31 references

Optimal Transport Driven Asymmetric Image-to-Image Translation for Nuclei Segmentation of Histological Images

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that asymmetric image-to-image translation for nuclei segmentation can be made both simpler and more accurate by adding an embedding space and an invertible generator, which removes the explicit cycle-consistency loss.

desk verdict The performance advantage over supervised baselines is mostly the H-channel SSIM prior, not the optimal transport / invertible generator claim. read the letter →

arxiv 2506.07023 v1 pith:KJK2M6JN submitted 2025-06-08 eess.IV cs.AIcs.CV

Suman Mahapatra , Pradipta Maji This is my paper

classification eess.IVcs.AIcs.CV

keywords nucleisegmentationimage-to-imagetranslationoptimaltransportinvertiblegeneratorcycle-consistencylosshistologicalimageanalysisgenerativeadversarialnetworks

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

This paper is trying to establish that nuclei segmentation in histological images can be treated as an image-to-image translation problem between asymmetric domains, and that the information gap can be managed without paying the usual cost of extra networks or cycle-consistency losses. The proposed model, Ostrich, inserts an embedding space between the information-rich image domain and the information-poor segmentation-map domain, then uses optimal transport and measure theory to derive an invertible generator whose sections act as inverses of the forward generators. That derivation removes the explicit cycle-consistency loss and reduces the architecture from three generators and three discriminators to two generators and one paired discriminator. The paper reports that Ostrich outperforms state-of-the-art nuclei segmentation methods on the TCGA and CoNIC datasets, with significant gains in 87.5% of evaluation cases, while using fewer parameters than CycleGAN-based alternatives. A careful reader would care because this suggests a principled route to lower model complexity in medical-image translation without sacrificing accuracy.

What carries the argument

The load-bearing object is the invertible generator $G_\theta$, whose sections $G_\theta^y$ and $G_\theta^z$ are treated as inverse operators of the forward generators $E_\omega$ and $F_\varphi$; Theorem 1 uses this invertibility to cancel the cycle-consistency terms and collapse three GAN objectives into one paired discriminator. The invertible generator is implemented with invertible $1\times 1$ convolutions and general coupling layers, together with a spatially constrained squeeze-unsqueeze operation that keeps neighboring pixels in the same sub-patch, and an SSIM-based regularization term built from the hematoxylin channel via Otsu thresholding and Voronoi labeling.

What would settle it

Measure reconstruction errors on held-out patches of the trained model, comparing an input image $x$ with $G_\theta^z(F_\varphi(x))$ and a segmentation map $y$ with $F_\varphi(G_\theta^z(y))$; if these errors are large, the exact-inverse assumption behind Theorem 1 fails, and the reported accuracy gains cannot be attributed to the cycle-consistency-free optimal-transport reduction.

Watch

Extended reading notes

Core claim

The central claim is an optimal-transport reduction that makes asymmetric image-to-image translation cheap enough for nuclei segmentation. The paper formalizes three maps: $F_\varphi: X \to Y$ from images to segmentation maps, $E_\omega: X \to Z$ from images to an embedding space, and $G_\theta: (Y \times Z) \to X$ that reconstructs images from map-plus-embedding pairs. Theorem 1 states that if the sections of $G_\theta$ are exact inverses of $F_\varphi$ and $E_\omega$, the cycle-consistency loss and the image-domain discriminator vanish, and the minimax problem reduces to two generators and one paired discriminator with loss $2\,l_{GAN}(F_\varphi) + l_{GAN}(E_\omega) + \lambda_1 l_{SSIM}(F_\varphi)$. The implementation approximates invertibility by letting the reconstruction network share the parameters of $F_\varphi$, builds the invertible generator from invertible $1\times 1$ convolutions and general coupling layers, and adds a spatially constrained squeeze operation to preserve spatial continuity. On TCGA and CoNIC, Ostrich outperforms the compared state-of-the-art methods in most evaluation cases and is significantly better in 87.5% of cases across the two datasets, using 7.27 million parameters versus 18.04 million for CycleGAN and 27.05 million for the asymmetric CycleGAN variant.

Load-bearing premise

The proof that cycle-consistency can be dropped assumes the sections of the generator are exact inverses of the forward generators, but the trained network only shares parameters between them, so exact invertibility is not guaranteed.

Editorial extensions

If this is right

On TCGA data, Ostrich reaches a Dice coefficient of 0.788, ahead of every compared state-of-the-art method on Dice, Jaccard, and precision, with recall slightly below BoNuS.
On CoNIC data, Ostrich outperforms all compared methods on all four metrics, with Dice 0.741 versus the best competitor HoVer-Net at 0.740.
Because the cycle-consistency loss and one discriminator are removed, the trained model uses 7.27 million parameters, far fewer than CycleGAN's 18.04 million and the asymmetric CycleGAN variant's 27.05 million.
The spatially constrained squeeze operation separates overlapping nuclei better than checkerboard masking, and the difference is statistically significant on TCGA.
The model performs both instance and binary semantic segmentation without assuming any particular nucleus shape, so it can capture heterogeneous nuclei across organs.

Reading between the lines

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

Beyond the paper, the same invertible-generator reduction should apply to other asymmetric translation tasks, such as stain transfer or low-dose CT denoising, where one domain contains far more information than the other; the paper only demonstrates histology segmentation.
Because the implementation only approximates exact invertibility by parameter sharing, a version that retains a light reconstruction or cycle-consistency loss might be more stable on domains where the hematoxylin-channel SSIM regularizer is unavailable or weak.
The SSIM regularization term injects a hand-built weak supervision signal derived from Otsu thresholding and Voronoi labeling; ablating it against a learned or data-driven prior would reveal how much of the reported gain comes from optimal-transport structure versus this domain-specific regularizer.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Checked the manuscript and the stress-test note. The stress-test lands: the headline 87.5% claim is not supported as stated. On TCGA, removing the SSIM regularization R drops Ostrich from 0.788 to 0.762 Dice, below BoNuS's 0.785. The authors added R to the three I2I baselines but not to the supervised segmentation baselines, so the comparison against U-Net, HoVer-Net, and BoNuS is asymmetric. That means the better trade-off is better attributed to the hand-crafted H-channel prior than to the optimal-transport/invertible-generator architecture. The paper says lambda_1 decreases during training but does not give a schedule; either way, the ablation shows R does heavy lifting in the final weights.

Credit where it is due: the embedding-space idea for asymmetric I2I is sensible, the spatial squeeze is a small but reasonable fix to checkerboard artifacts, and the ablation lets a reader see the confound. Comparing against a wide set of baselines on two public datasets is the right thing to do, and the parameter counts are useful.

The soft spots beyond the confound: Theorem 1 assumes G_theta^z = F_phi^{-1} and G_theta^y = E_omega^{-1}, but the implementation only shares parameters; it does not enforce exact invertibility. The proof also has an unjustified containment step (Phi_3 subset of Phi_1), so the claimed elimination of the cycle-consistency loss and the extra generator is not established. That matters because the architecture is justified by that theorem. Also, no code, no error bars, no multiple runs; the p-values are computed over test patches from a single split, which inflates apparent significance. There is no sensitivity analysis for lambda_1, lambda_2, or the embedding dimension. These are fixable.

Who is this for? Someone working on I2I-based nuclei segmentation will find the engineering combination worth knowing, especially the cautionary ablation. I would not rely on the theoretical reduction. If I were editing, I would send it to peer review, because the problem is relevant and the flaws are addressable, but I would expect the authors to re-frame the claims, add R to the supervised baselines, and release code. For my own work, I would not cite the headline result; I would cite the ablation as a cautionary example if at all.

Referee Report

4 major / 8 minor

Summary. The paper presents Ostrich, an asymmetric image-to-image translation model for nuclei segmentation from H&E histology images. It models translation between the image domain X and the segmentation domain Y through an embedding space Z, with forward generators Fφ and Eω and an invertible generator Gθ. The authors derive an optimal-transport GAN objective (Eq. 6), prove (Theorem 1) that under invertibility of the sections of Gθ the problem reduces to two generators and one discriminator (Eq. 9), and introduce an H-channel SSIM regularization l_ssim(Fφ) to guide early training. Experiments on TCGA and CoNIC compare Ostrich with CycleGAN variants and supervised segmentation networks, reporting the best Dice/Jaccard/precision/recall in most cases and fewer parameters than several I2I baselines.

Significance. The proposed direction is timely and the paper contains useful ingredients: an OT-inspired asymmetric I2I formulation with an embedding space, a parameter-sharing invertible-generator idea, a spatially constrained squeeze operation, and experiments on two public datasets with parameter counts and paired statistical tests. If the attribution issues were resolved, the result would be a comparatively lightweight I2I segmentation model with competitive Dice/Jaccard performance on TCGA and CoNIC. The main weaknesses, namely the uncontrolled auxiliary prior, the unproven theoretical reduction, and the lack of repeated-run uncertainty, currently prevent the paper from supporting its central claims. The contribution is nonetheless potentially valuable to the medical image analysis community and could be made publishable with a controlled experimental design and a corrected theoretical statement.

major comments (4)

[Section III-B-2, Eq. (3), Table I] The main empirical claim is confounded by an asymmetric evaluation setup. The SSIM regularization l_ssim(Fφ) of Eq. (3) is added to Ostrich and to the three I2I baselines (CycleGAN, OT-CycleGAN, Asym-CycleGAN), but it is not added to the supervised segmentation baselines (U-Net, Mask-R-CNN, U-Net++, HoVer-Net, MoNS, StarDist, Swin-MIL, BoNuS). Table I shows that removing R from Ostrich lowers Dice from 0.788035 to 0.761814, a drop of 0.026221, which is approximately 8 times the remaining margin of 0.003295 over BoNuS (0.784740). Without the H-channel prior, Ostrich would fall below BoNuS on Dice. Therefore the reported superiority cannot be attributed to the OT/invertible-generator architecture unless a controlled comparison is run in which the same auxiliary prior is given to the supervised baselines, or in which Ostrich without R is compared against baselines without the prior.
[Section II-B, Theorem 1, Eqs. (16)-(24)] The proof of Theorem 1 is not valid as written. The step claiming that Phi_3 is contained in Phi_1 is asserted rather than proved; membership in Phi_1 requires representation as phi o G_theta_z with phi in Lip_1(X), and the fact that an element of Phi_3 is 1-Lipschitz on Y does not imply such a representation, especially since a bound on G_theta_z does not transfer to a bound on its inverse F_phi. In addition, the tightness argument after Eq. (25) constructs phi(x)=psi*(F_phi(x)) and needs F_phi to be 1-Lipschitz, which is not assumed. Finally, the theorem assumes exact invertibility G_theta_z = F_phi^{-1} and G_theta_y = E_omega^{-1}, but the architecture in Section II-C only shares parameters between F_phi and G_theta_z; it does not enforce exact inversion, so the cycle-consistency terms in Eq. (8) need not vanish in the implemented model. The reduction to Eq. (9) should be restated as an approximation or proved under conditions satisfied by the network.
[Tables I, V, II, VI] Tables I and V report single-run performance without error bars, confidence intervals, or the number of runs. The p-values in Tables II and VI are computed with paired tests over image patches, not over independent model trainings, so they conflate per-patch variation with model-instance variation and do not establish that the observed differences are due to the method rather than to random initialization or training seed. The experimental section should report mean plus/minus standard deviation over multiple random seeds and should apply significance tests at the level of repeated runs.
[Abstract, Tables II and VI] The abstract's claim that the model 'performs significantly better than the state-of-the-art methods in 87.5% cases' overstates the evidence in Tables II and VI. The 87.5% figure aggregates 63 of 72 model-metric pairs, but on TCGA the differences against BoNuS are not significant for Dice (p=1.42e-01), Jaccard (p=1.05e-01), or Recall (p=9.43e-01), and the differences against Asym-CycleGAN are not significant for Dice (p=6.49e-02), Jaccard (p=5.42e-02), or Recall (p=8.31e-02). The 'significantly better' claim should be restricted to the cases where the paired tests give p<0.05, or a global multiple-comparison procedure should be used.

minor comments (8)

[Eq. (3), Section III-A] Please specify how V_O is implemented (Otsu threshold plus Voronoi labeling details, including any morphological post-processing); as written, the H-channel prior is not fully reproducible.
[Section II-C] The statement that Z is assumed to follow a standard normal distribution is not reconciled with the measure eta introduced in Section II-B; state explicitly whether eta is fixed to N(0,I) or learned.
[Fig. 2 caption] The caption says 'x[:,:,0:2] denotes 3 channel input image', but the slice 0:2 gives only two channels; this is likely a typo for x[:,:,0:3].
[Table II] BoNuS is labeled '(2022)' in the first block but '(2024)' in the second block; align the year with reference [18].
[Algorithm 1 and Section III-A-3] The text states that lambda_1 decreases gradually during training, but Algorithm 1 shows a constant lambda_1; please specify the decay schedule.
[Section III-B-5] The sentence '29 out of 36 cases, better but not significantly in 6 cases' accounts for only 35 cases; check whether a third category is needed or whether the numbers are misreported.
[Appendix, Eq. (37)] The notation x = G_theta(y,z) = G_theta_y(z) = G_theta_z(y) conflates the two sections of G_theta; these are generally different and should be written separately with the intended equality stated explicitly.
[Eq. (6) and Eq. (10)] The cycle-consistency term with weight lambda_2 appears in Eq. (6), but Theorem 1 eliminates it and Eq. (10) has no lambda_2 term; clarify whether any cycle-consistency weight is used in the final objective.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the OT/invertible-generator derivation is conditional and self-contained, and the H-channel regularization is transparently an auxiliary self-supervised term rather than a hidden fitted input.

full rationale

The paper's central derivation (Theorem 1) explicitly assumes Gθ_z = Fφ^{-1} and Gθ_y = Eω^{-1}; under that assumption the cycle-consistency terms vanish by composition, so the reduction to Eq. (9) is a valid conditional equivalence rather than a circular inference. No self-citations are load-bearing: the cited invertible-generator and normalizing-flow building blocks [22,28,29] are external prior work and are not used to smuggle in the paper's own conclusions. The l_ssim term in Eq. (3) uses a deterministic transform VO(H-channel of x) as a pseudo-label; while this is a strong inductive bias and arguably confounds the comparison with supervised baselines that do not receive the same prior, it is openly defined, explicitly ablated (Table I shows Ostrich Dice 0.788 vs Ostrich\R 0.762), and it is not a parameter fitted to a subset of the evaluation data, so it does not make the reported prediction equivalent to its input by construction. The main legitimate concerns are experimental fairness (baselines without l_ssim) and the gap between the exact invertibility assumed in Theorem 1 and the parameter-sharing implementation, but these are correctness and benchmarking risks, not circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central derivation relies on standard mathematical tools, but the key invertibility assumption is an ad hoc design premise that is not validated. The embedding space is a known construct, and the SSIM pseudo-label is a hand-crafted component. The free parameters are under-specified, which limits independent reproduction.

free parameters (3)

lambda_1 = not specified; annealed during training
Weights the SSIM regularization term l_ssim in the objective (Eq. 6). The ablation shows this term is essential for full performance, but the schedule is not reported.
lambda_2 = not used; cycle loss vanishes by assumption
Cycle-consistency weight in Eq. 6. The paper claims it becomes unnecessary under exact invertibility, but since that condition is not enforced, λ2 is effectively absent from the training objective.
embedding dimension of Z = not specified
The latent code dimensionality is a free design choice. The paper states Z follows a standard normal but does not give its dimension.

assumptions (6)

standard math Kantorovich duality for optimal transport
Used to derive the dual formulation in Eq. 37 and the GAN-like objective; standard in OT theory.
standard math Fubini's theorem for product measures
Invoked in the proof of Theorem 1 (Eq. 20) to interchange integrals over Y and Z.
standard math 1-Lipschitz continuity of Kantorovich potentials
Used to bound the discriminator gradient and to relate the primal and dual costs.
ad hoc to paper Gθ_z = Fφ^{-1} and Gθ_y = Eω^{-1} (exact invertibility)
Core assumption of Theorem 1; the paper claims the architecture shares parameters to approximate this, but exact invertibility is not proven and is likely unattainable when mapping between 3-channel images and 1-channel masks plus a latent.
domain assumption The embedding space Z follows a standard normal distribution
Used for sampling in the GAN objective; a common but unverified assumption about the latent structure.
domain assumption The Voronoi-labeled H-channel (via Otsu) provides a useful pseudo-segmentation for the SSIM regularization
The regularization term in Eq. 3 relies on this hand-crafted transformation being a reasonable proxy for nuclei locations. Its validity is not evaluated independently.

invented entities (1)

Embedding space Z
purpose: Captures information lost when compressing the image domain X to the segmentation domain Y, enabling reconstruction through the invertible generator.
The latent code is a standard technique in multimodal I2I (e.g., MUNIT, BicycleGAN), not a new physical entity. Its effectiveness is only demonstrated within the proposed model, not via external falsifiable predictions.

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Cite this review

Pith. "Pith review of Optimal Transport Driven Asymmetric Image-to-Image Translation for Nuclei Segmentation of Histological Images." pith.science (2026). https://pith.science/paper/KJK2M6JN

@misc{pith2026250607023,
  author       = {Pith},
  title        = {Pith review of: Optimal Transport Driven Asymmetric Image-to-Image Translation for Nuclei Segmentation of Histological Images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJK2M6JN}},
  note         = {Machine review of arXiv:2506.07023}
}

read the original abstract

Segmentation of nuclei regions from histological images enables morphometric analysis of nuclei structures, which in turn helps in the detection and diagnosis of diseases under consideration. To develop a nuclei segmentation algorithm, applicable to different types of target domain representations, image-to-image translation networks can be considered as they are invariant to target domain image representations. One of the important issues with image-to-image translation models is that they fail miserably when the information content between two image domains are asymmetric in nature. In this regard, the paper introduces a new deep generative model for segmenting nuclei structures from histological images. The proposed model considers an embedding space for handling information-disparity between information-rich histological image space and information-poor segmentation map domain. Integrating judiciously the concepts of optimal transport and measure theory, the model develops an invertible generator, which provides an efficient optimization framework with lower network complexity. The concept of invertible generator automatically eliminates the need of any explicit cycle-consistency loss. The proposed model also introduces a spatially-constrained squeeze operation within the framework of invertible generator to maintain spatial continuity within the image patches. The model provides a better trade-off between network complexity and model performance compared to other existing models having complex network architectures. The performance of the proposed deep generative model, along with a comparison with state-of-the-art nuclei segmentation methods, is demonstrated on publicly available histological image data sets.

Figures

Figures reproduced from arXiv: 2506.07023 by the authors.

**Figure 2.** Block diagram of the squeeze-unsqueeze operation. H [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Qualitative performance analysis in nuclei instanc [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Performance analysis in nuclei segmentation: (a) Or [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Performance analysis of nuclei segmentation: (a) Or [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 8.** Figure 8: The qualitative comparison through zoomed-in regio [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 7.** Figure 7: (a) Original images of TCGA data, (b) ground-truth se [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: (a) Original images of CoNIC data, (b) ground-truth s [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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