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REVIEW 4 major objections 6 minor 48 references

Score-based Self-supervised MRI Denoising

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A self-supervised score-matching loss lets MRI denoisers learn the clean-image posterior mean from noisy data alone.

desk verdict C2S is a credible, useful self-supervised MRI denoiser, but the theory does not cover the t=tdata inference point or the sigma_ttarget>0 variant, and a few claims outrun the tables. read the letter →

arxiv 2505.05631 v1 pith:OJFQGTHC submitted 2025-05-08 eess.IV cs.CV

Jiachen Tu , Yaokun Shi , Fan Lam This is my paper · ORCID
classification eess.IVcs.CV
keywords MRIdenoisingself-supervisedscorematchingconditionalexpectationmulti-contrastGaussiannoisecorruption2self
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

Corruption2Self (C2S) sets out to prove that MRI denoising can be learned entirely from noisy acquisitions, without clean or high-SNR labels. The paper's core theoretical claim is that minimizing a generalized denoising score matching (GDSM) loss on pairs of noisy images—the observed image and a further corrupted version—drives the network to the conditional expectation of the cleaner image given the noisier one; with the target noise set to zero, that conditional expectation is the posterior mean of the clean image. On M4Raw and fastMRI, C2S outperforms prior self-supervised denoisers and matches supervised networks trained on higher-SNR references, and with multi-contrast inputs it beats the supervised baselines on M4Raw. The practical payoff is that low-field and accelerated MRI scans, where high-SNR ground truth is expensive or unavailable, could be denoised using only the diagnostic images themselves.

What carries the argument

The load-bearing object is the GDSM loss $$J'(\$\theta$)=\mathbb{E}\left[\left\|\gamma'(\tau,\sigma_{t_{\mathrm{target}}})h_\$\theta$(X_t,t)+\delta'(\tau,\sigma_{t_{\mathrm{target}}})X_t - X_{t_{\mathrm{data}}}\right\|^2\right],$$ with $\gamma'(\tau,\sigma_{t_{\mathrm{target}}}) = \sigma_\tau^2 / (\sigma_\tau^2 + \sigma_{t_{\mathrm{data}}}^2 - \sigma_{t_{\mathrm{target}}}^2)$ and $\delta' = (\sigma_{t_{\mathrm{data}}}^2-\sigma_{t_{\mathrm{target}}}^2)/(\sigma_\tau^2+\sigma_{t_{\mathrm{data}}}^2-\sigma_{t_{\mathrm{target}}}^2)$. Re-expressing the doubly-noisy image from the perspective of both the observed image and the target image yields two score identities whose equality gives $\mathbb{E}[X_{t_{\mathrm{data}}}\mid X_t] = \gamma' \mathbb{E}[X_{t_{\mathrm{target}}}\mid X_t] + \delta' X_t$, so minimizing the loss against $X_{t_{\mathrm{data}}}$ forces $h_\theta$ to output $\mathbb{E}[X_{t_{\mathrm{target}}}\mid X_t]$. This coefficient identity, together with the skip connection $D_\theta = \lambda_{\mathrm{out}} h_\theta + \lambda_{\mathrm{skip}} X_t$ and the $\tau$-reparameterization, carries the whole method.

What would settle it

On a synthetic dataset with a known Gaussian prior for $X_0$ and known noise level, compute the posterior mean $\mathbb{E}[X_0\mid X_{t_{\mathrm{data}}}]$ by Monte Carlo, train C2S from noisy samples only, and compare the network output at $(X_{t_{\mathrm{data}}},t_{\mathrm{data}})$ with that posterior mean; a systematic error that persists when the network is trained only on $t>t_{\mathrm{data}}$ would show the inference step relies on extrapolation outside the theorem's range.

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Extended reading notes

Core claim

The central discovery is that denoising score matching does not need clean training images. By corrupting the observed noisy image $X_{t_{\mathrm{data}}}$ with additional independent Gaussian noise to form $X_t$, and training the network to predict $X_{t_{\mathrm{data}}}$ from $X_t$ through a specially weighted sum of the network output and the input, the minimizer of the GDSM loss satisfies $h_{\theta^*}(X_t,t) = \mathbb{E}[X_{t_{\mathrm{target}}} \mid X_t]$. When $t_{\mathrm{target}} = 0$, the right-hand side is $\mathbb{E}[X_0 \mid X_t]$, the minimum mean-square-error estimate of the clean image, so the network has effectively learned to denoise despite never seeing a clean image. A reparameterized noise scale $\tau$ with uniform sampling and a skip-blended output stabilizes training, and a detail-refinement variant with $\sigma_{t_{\mathrm{target}}} > 0$ retains fine textures. On the M4Raw and fastMRI datasets, the method attains the best PSNR and SSIM among self-supervised methods and competitive numbers against supervised ones, with multi-contrast C2S exceeding all baselines on M4Raw.

Load-bearing premise

The inference step evaluates the trained network at $t=t_{\mathrm{data}}$, but the theorem characterizes the loss minimizer only for $t>t_{\mathrm{data}}$; the method's clean-image estimate therefore rests on the network extrapolating the conditional expectation to the original noise level, and on the added noise being independent Gaussian so the score identity that defines GDSM actually holds.

Editorial extensions

If this is right

  • A single noisy acquisition per image is enough to train a denoiser; no clean labels or paired noisy replicates are required, as long as the noise level is known or estimated.
  • Because the target is the MMSE posterior mean rather than a label-mapped output, C2S generalizes to test data whose SNR is higher than the training labels, a regime where supervised networks trained on averaged labels lose accuracy.
  • The same loss with $\sigma_{t_{\mathrm{target}}}>0$ yields a detail-preserving estimator, giving practitioners a direct trade-off between noise removal and texture retention.
  • Feeding auxiliary MRI contrasts as conditioning inputs improves denoising of the target contrast, and on M4Raw the multi-contrast variant outperforms every supervised baseline tested.
  • Noise-level misestimation within $\pm 50\%$ changes PSNR by at most about 0.1 dB, so the method works with standard noise estimators as a blind denoiser.

Reading between the lines

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

  • If the same Gaussian score identity is validated outside MRI, the GDSM construction transfers directly to other imaging modalities with additive white noise, or to Rician data after a variance-stabilizing transform; the authors point to VST but do not test it.
  • The theory's proof only covers $t > t_{\mathrm{data}}$, while inference evaluates the network at $t = t_{\mathrm{data}}$; a Monte Carlo check of $h_\theta(x,t_{\mathrm{data}})$ against the true posterior mean under a known Gaussian prior would quantify how much of the method's success depends on extrapolation.
  • Because $\sigma_{t_{\mathrm{target}}}$ is sampled uniformly in detail refinement, treating it as a tunable per-contrast hyperparameter might improve texture retention beyond the reported default; this is a testable extension the paper does not explore.
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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. This manuscript introduces Corruption2Self (C2S), a self-supervised MRI denoising method. The authors formulate a generalized denoising score matching (GDSM) loss whose minimizer is the conditional expectation E[X_ttarget | X_t] under stated assumptions, and train a U-Net with time conditioning and NVC-MSA using only noisy observations. They reparameterize noise levels, add a detail refinement stage, and extend the framework to multi-contrast inputs. Experiments on M4Raw and fastMRI compare against classical, supervised, and self-supervised baselines, reporting state-of-the-art self-supervised PSNR/SSIM and competitive supervised performance, with appendices on robustness to noise-level misestimation and additional ablations.

Significance. The core idea--learning conditional expectations of cleaner images from further corrupted noisy observations--is a natural and potentially useful extension of ADSM and Noisier2Noise, and the paper provides a broad experimental comparison on real and simulated MRI data, including low-field M4Raw, multiple contrasts, and multi-contrast fusion. Strengths include the explicit robustness study in Appendix H, the clear positioning relative to prior self-supervised methods, and the reproducibility-oriented details on datasets and training. If the theoretical gap at the inference point is fixed, the method would be a solid contribution to self-supervised MRI denoising; as written, the headline theoretical claim is incomplete in a way that affects the central inference operation.

major comments (4)
  1. [Section 3.1, Theorem 1/Corollary 2 vs. Section 3.2, Eq. (11)] The theorem is stated and proved for t sampled uniformly from (tdata, T], and the proof divides by gamma(t, sigma_ttarget), which is positive only for t > tdata. At t = tdata, gamma = 0 and delta = 1 in Eq. (3), so J(theta) = E[||Xtdata - Xtdata||^2] = 0 for every h_theta, and h_theta*(Xtdata, tdata) is unconstrained. Equation (11) nevertheless evaluates the network exactly at t = tdata and asserts it approximates E[X0 | Xtdata]. The paper needs an explicit continuity/limit argument as t decreases to tdata, or a modified training loss that also constrains t = tdata; as written, the central theoretical claim does not cover the inference operation.
  2. [Appendix B, Theorem 4 proof] For sigma_ttarget > 0, the proof uses the identity Xt = Xttarget + sqrt(sigma_t^2 - sigma_ttarget^2) Z2 with Z2 independent of Xttarget to write the score as (E[Xttarget | Xt] - Xt)/(sigma_t^2 - sigma_ttarget^2). This identity is a coupling assumption that is not stated in Assumptions 3-5: under the natural model Xttarget = X0 + sigma_ttarget Z_ttarget and Xt = X0 + sigma_t Z_t with Z_ttarget and Z_t independent, the conditional variance of Xt - Xttarget given Xttarget is sigma_t^2 + sigma_ttarget^2, not sigma_t^2 - sigma_ttarget^2. Consequently the detail-refinement extension, which relies on sigma_ttarget > 0, is not justified by the stated theorem.
  3. [Appendix G and Section 3.2, Eq. (11)] The detail-refinement stage trains the network to satisfy h_theta*(Xt, t) = E[Xttarget | Xt] with sigma_ttarget sampled from (0, sigma_tdata], while Eq. (11) evaluates h_theta*(Xtdata, tdata) as a clean estimate E[X0 | Xtdata]. These are different quantities, and the paper does not specify how the nonzero-target predictor is converted into the clean estimate whose PSNR/SSIM is reported in Tables 1-3. Please clarify the inference protocol for the refined model or add the missing derivation.
  4. [Appendix E, Table 6 and Section 4] The fastMRI reparameterization comparison reports a 'Without Reparam.' row described as 'estimated baseline results.' Since the main text presents reparameterization as an empirical improvement (Table 4a), the provenance of the estimated values should be disclosed in detail, and ideally the baseline should be run under identical conditions; otherwise the fastMRI reparameterization claim is not directly supported.
minor comments (6)
  1. [Section 3.1] The text says 'proof provided in Appendix 4' for Theorem 1 and 'proof provided in Appendix 5' for Corollary 2, but the appendices are lettered; update the cross-references.
  2. [Section 3.2, Eq. (10) and Algorithm 1] Equation (2) uses Xt for t > tdata, while Eq. (10) and Algorithm 1 construct X_tau from Xtdata + sigma_tau Z; the relationship between t and tau should be made explicit in the main text rather than only in Appendix B.
  3. [Table 1] The paired t-tests are reported without the number of validation subjects/slices or any correction for multiple contrasts; please provide these details.
  4. [Section 4 and Table 5] The Introduction claims multi-contrast C2S achieves state-of-the-art among supervised methods, but Table 5 only compares BM3D, Noise2Noise, and single-contrast C2S; supervised transformers from Table 2 are not included, so the claim should be qualified.
  5. [References and text] There are several typos, including 'Noise2V oid' in the text, 'F ouguier' in the references, and 't target' in Section 3 where 'sigma_ttarget' is meant.
  6. [Appendix J.1] The suggestion that T can be chosen as the maximum pairwise distance between training points is imported from score-based generative modeling; please discuss whether this bound is needed for the denoising objective rather than for generative score estimation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: C2S's clean-image estimate follows from Tweedie/score-matching identities applied to noisy targets, not from reusing a fitted quantity as a prediction.

full rationale

The claimed derivation is not circular. The GDSM loss (Eq. 3 / Eq. 12) is a mean-square regression of the combination gamma(t) h_theta(X_t,t) + delta(t) X_t onto the observed noisy image X_tdata. Lemma 3 (Appendix B) gives the usual conditional-expectation minimizer E[X_tdata | X_t]; Theorem 1 then algebraically rewrites this as E[X_ttarget | X_t] through two Tweedie score identities. The training target throughout is the noisy observation X_tdata, and the clean-image estimate is obtained by removing the added corruption through score matching, not by fitting a parameter to the target and relabeling it as a prediction. The noise level sigma_tdata is an estimated input, not a fitted prediction, and the paper reports a +/-50% robustness study plus held-out evaluation on M4Raw and fastMRI. Citations to ADSM and Noisier2Noise are external prior work; there are no load-bearing self-citations by Tu/Shi/Lam, and the paper explicitly acknowledges that sigma_ttarget = 0 recovers ADSM and the Noisier2Noise special case. The serious caveats are correctness gaps rather than circularity. Theorem 4 states h_theta*(X_t,t) = E[X_ttarget | X_t] for all t >= t_data, but at t = t_data the coefficient gamma in Eq. 12 is zero, so the loss is identically zero for every h_theta and the proof divides by gamma; Eq. 11 evaluates the network at exactly this unconstrained point. In addition, the score identity used for sigma_ttarget > 0 requires a coupling X_t = X_ttarget + sqrt(sigma_t^2 - sigma_ttarget^2) Z with Z independent of X_ttarget, which is not implied by Eqs. 1-2 when X_ttarget is defined using an independent noise N_0. These gaps weaken the stated guarantee of the headline estimate but do not make the derivation equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on the Gaussian noise assumption, the expressiveness of the network, the variance-exploding forward process, and an unstated coupling for target noise levels; no new physical entities are introduced. Free parameters are the estimated noise level, the maximum corruption level, and the loss weighting exponent.

free parameters (3)
  • Data noise level sigma_tdata = estimated per image via skimage noise estimation
    The method requires an estimate of the noise level of the input scan; the paper shows robustness to +/-50% error but the value is not derived from the data-free theory.
  • Maximum corruption level T = T=10 for M4Raw T1, T=5 for fastMRI PD25; selected by validation
    T controls the range of noise levels and is tuned per dataset; performance peaks at different T for different datasets (Appendix I).
  • Weighting exponent alpha in w(tau) = not specified; w(tau)=(sigma_tau^2+sigma_tdata^2)^alpha
    The weighting function is a design choice following prior diffusion models; the value of alpha used in experiments is not reported.
assumptions (4)
  • domain assumption Gaussian noise model: Xtdata = X0 + sigma_tdata N with N ~ N(0,I)
    Used to derive the GDSM loss and Tweedie relations; acknowledged to hold for SNR>2 in MRI, with VST suggested otherwise (Section 3).
  • standard math Sufficient network expressiveness (universal approximation)
    Theorem 4 assumes the function class can realize the conditional expectation; standard in learning theory but not verified in practice.
  • domain assumption Variance-exploding forward process with independent increments
    Equation 2 defines Xt by adding Gaussian noise to Xtdata; this is the operational definition used in training.
  • ad hoc to paper Coupling of Xt and Xttarget along a common noise path for sigma_ttarget > 0
    The proof of Theorem 4 writes Xt = Xttarget + sqrt(sigma_t^2 - sigma_ttarget^2) Z2, which requires a specific coupling that is never stated for the target noise level; this underpins the detail refinement extension.

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Pith. "Pith review of Score-based Self-supervised MRI Denoising." pith.science (2026). https://pith.science/paper/OJFQGTHC

@misc{pith2026250505631,
  author       = {Pith},
  title        = {Pith review of: Score-based Self-supervised MRI Denoising},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJFQGTHC}},
  note         = {Machine review of arXiv:2505.05631}
}
read the original abstract

Magnetic resonance imaging (MRI) is a powerful noninvasive diagnostic imaging tool that provides unparalleled soft tissue contrast and anatomical detail. Noise contamination, especially in accelerated and/or low-field acquisitions, can significantly degrade image quality and diagnostic accuracy. Supervised learning based denoising approaches have achieved impressive performance but require high signal-to-noise ratio (SNR) labels, which are often unavailable. Self-supervised learning holds promise to address the label scarcity issue, but existing self-supervised denoising methods tend to oversmooth fine spatial features and often yield inferior performance than supervised methods. We introduce Corruption2Self (C2S), a novel score-based self-supervised framework for MRI denoising. At the core of C2S is a generalized denoising score matching (GDSM) loss, which extends denoising score matching to work directly with noisy observations by modeling the conditional expectation of higher-SNR images given further corrupted observations. This allows the model to effectively learn denoising across multiple noise levels directly from noisy data. Additionally, we incorporate a reparameterization of noise levels to stabilize training and enhance convergence, and introduce a detail refinement extension to balance noise reduction with the preservation of fine spatial features. Moreover, C2S can be extended to multi-contrast denoising by leveraging complementary information across different MRI contrasts. We demonstrate that our method achieves state-of-the-art performance among self-supervised methods and competitive results compared to supervised counterparts across varying noise conditions and MRI contrasts on the M4Raw and fastMRI dataset.

Figures

Figures reproduced from arXiv: 2505.05631 by the authors.

Figure 1
Figure 1. Overview of the Corruption2Self (C2S) workflow for MRI denoising. Starting from a noisy MRI [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Comparison of different denoising methods for T1 contrast from the M4Raw dataset. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Comparison of denoising methods for the PD contrast ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Comparison of different denoising methods for T1 contrast in the M4Raw dataset. The figure [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Comparison of different denoising methods for PD contrast (noise level 13/255) in fastMRI. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Comparison of different denoising methods for PDFS contrast (noise level 25/255) in fastMRI. [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Comparison of different denoising methods for T1 contrast in M4Raw. The top row shows the [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Top: Comparison of different denoising methods for T1 contrast in M4Raw. Bottom: Comparison [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: Comparison of different denoising methods for FLAIR contrast in M4Raw. [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Comparison of different denoising methods for FLAIR contrast in M4Raw. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Visualization of model performance under varying noise level estimations. The plots demonstrate [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: Effectiveness of Reparameterization in Noise Level Adjustment on the M4Raw FLAIR validation Dataset. Comparison of PSNR and SSIM metrics on the validation set for different model configurations over the first 125 training epochs (T = 5, σtarget = 0). The combination o…

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    \@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should ...

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    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.