Diffusion-Based Limited-Angle CT Reconstruction under Noisy Conditions

Jiaqi Guo; Santiago L\'opez-Tapia

REVIEW 3 major objections 5 minor 25 references

Diffusion-Based Limited-Angle CT Reconstruction under Noisy Conditions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A noise-aware rectification step makes diffusion-based limited-angle CT reconstruction robust to measurement noise, with roughly 3 dB gains in sinogram-domain PSNR.

desk verdict A narrow noise-aware extension of RNSD with promising experiments, but the central schedule as printed has a unit error that makes it undefined; needs major revision before the numbers can be trusted. read the letter →

arxiv 2507.05647 v1 pith:2MRY32TI submitted 2025-07-08 eess.IV cs.CV

Jiaqi Guo , Santiago López-Tapia This is my paper · ORCID

classification eess.IVcs.CV

keywords limited-angleCTsinograminpaintingmean-revertingstochasticdifferentialequationdiffusionmodelnoise-awarerectificationrange-nullspacedecompositiondataconsistency

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

Limited-angle CT scans are hard because many projection angles are missing; adding real measurement noise makes the problem harder still. The paper argues that a diffusion model can complete the missing sinogram angles accurately under noise, provided the rectification step is redesigned to fold the measurement noise into the diffusion process instead of pretending it is absent. The proposed RNSD+ correction scales the range-space data-consistency term and adjusts the injected noise at every reverse step, so the total noise level stays inside what the score network was trained to remove. On synthesized noisy chromatin electron-tomography data, the paper reports that RNSD+ beats the non-rectified version of the same model and NafNet in both sinogram and final image metrics, while a damping parameter gives robustness to noise-level mismatch.

What carries the argument

The machinery is the mean-reverting stochastic differential equation, whose forward process pulls the state toward the partially observed sinogram as terminal mean $\mu$, paired with RNSD+ rectification at each reverse step. RNSD+ splits the current estimate into the range space of the measurement operator (what the measurements tell us) and the null space (the missing angles), corrects the range-space part against $y$ with the time-dependent scale $\lambda_t$, and injects the adjusted noise $n'_t$ with scale $\gamma_t$ so that the effective residual noise stays within the maximum level the score model can denoise. A damping factor $\beta \in (0,1]$ scales down the correction when the noise estimate is uncertain.

What would settle it

Measure the variance of the residual $x'_{0|t} - x_0$ at each reverse step under the proposed schedule; if it exceeds the training-time $\sigma_t$ when the noise level is known exactly, or if the reconstruction quality peaks away from the $\gamma_t$ values in Eq. (7), the rectification schedule is not doing the work the paper claims.

Watch

Extended reading notes

Core claim

The central claim is that the RNSD+ rectification, replacing Eq. (5) with $x'_{0|t} = \hat{x}_{0|t} - \lambda_t A^\dagger(A\hat{x}_{0|t} - y)$ and adding the adjusted noise $n'_t = [A\gamma_t + (I - A)\sigma_t]\,d\bar{w}$, lets a mean-reverting diffusion model complete missing angular views even when the observed sinogram is noisy. In this design the measurement noise is not a nuisance to be filtered out; it is absorbed into the Gaussian noise budget and removed by the score network as the reverse process evolves. The paper's experiments support this by showing that, on the ChromSTET2025 dataset, RNSD+ gives the best PSNR, SSIM and LPIPS among the tested methods in both the sinogram domain and after filtered back projection, including a roughly 3 dB gain over the same model without rectification.

Load-bearing premise

The load-bearing premise is that the score network trained on the MR-SDE noise schedule can also remove the residual noise left after the measurement-consistent rectification, so the inference-time noise never exceeds the maximum level the model has seen during training.

Editorial extensions

If this is right

On the tested dataset, adding RNSD+ to the MR-SDE sinogram-completion pipeline improves sinogram-domain PSNR from 32.03 to 35.91 dB, with SSIM rising from 0.911 to 0.950 and LPIPS falling from 0.0300 to 0.0222.
After filtered back projection, the method improves image-domain PSNR to 20.43 dB and LPIPS to 0.1053, ahead of NafNet and the non-rectified baseline, meaning the gains survive the reconstruction step.
A damping factor $\beta$ lets users trade data consistency for stability when the noise standard deviation is not known exactly, with $\beta=1$ applying the full correction and smaller values protecting against overestimated noise.
Because the pipeline completes sinograms first and then applies ordinary FBP, it can be dropped into existing limited-angle CT workflows without changing the reconstruction backend.
The framework is stated for Gaussian measurement noise, so it carries over across the tested noise intensities (5–15 dB SNR) and acquisition scenarios used in the experiments.

Reading between the lines

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

Beyond the paper's experiments, the same noise-aware rectification should transfer to other linear inverse problems with Gaussian measurement noise, such as magnetic-resonance reconstruction or deblurring, because the algebra only uses the measurement operator and the noise variance.
A natural testable extension is to replace the known-noise assumption with a lightweight noise estimator and see whether the $\beta$ damping recovers most of the reported PSNR gains when the estimate is intentionally biased.
The paper's time-travel default suggests the rectification may leave mild artifacts in the early reverse steps; measuring per-step error would show whether the benefit comes from the later stages of denoising.

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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

RNSD+ is a real but narrow extension: it takes the range-null-space rectification from the authors' earlier RNSD/MR-SDE work and adds an explicit noise-aware cap on the correction plus a rebalanced noise injection, with the sigma_y dependence appearing in lambda_t, gamma_t, and h_t. That specific schedule is not in [10], [15], [16], or [17], so it is a new result. The experimental direction is plausible and consistent with DDNM-style noise-aware correction results; the robustness sweep over beta and noise-scale mismatch is a useful addition. If the numbers in Table 1 are reproducible, the ~3 dB gain over the no-rectification baseline matters for electron tomography.

That said, the printed math does not hold up. Eq (6) claims x'_{0|t} = (I - lambda_t A) x_hat - lambda_t y, but the correct expansion is (I - lambda_t A^dagger A) x_hat - lambda_t A^dagger y; as printed the dimensions do not match. That is likely a typo, but it makes the method ambiguous. More seriously, Eq (7) sets lambda_t = 1 when sigma_t >= h_t sigma_y, yet gamma_t^2 = sigma_t^2 dt - (h_t lambda_t sigma_y)^2 requires sigma_t sqrt(dt) >= h_t sigma_y. With dt = 0.01, the radicand goes negative whenever h_t sigma_y > sigma_t/10, so the update is undefined as written. The stress-test note is right: sigma_t is a per-square-root-time SDE coefficient and sigma_y is a per-sample noise level; Eq (7) mixes them. The text asserts the coefficients keep the added noise within the model-trained range, but that is a claim, not a derivation, and the printed condition does not guarantee it.

The empirical work is also thinner than the abstract implies: the main baseline is the authors' own unpublished MR-SDE [15], no external noisy LACT method is run, there are no error bars, and no code or data are released. The experiment is one synthetic dataset (ChromSTET2025). None of this disproves the results, but the paper currently overclaims generalization.

Who should read this: people working on diffusion-based inverse problems in limited-angle CT. It is worth a serious referee, but only if the referee demands corrected equations, a real derivation of the schedule, external baselines, and code/data. I would send it to review with that expectation, not desk-reject it and not accept it as-is.

Referee Report

3 major / 5 minor

Summary. The paper addresses noisy limited-angle CT by formulating it as sinogram inpainting with a Mean-Reverting Stochastic Differential Equation (MR-SDE). It introduces RNSD+, an inference-time rectification that scales the range-space data-consistency correction by a coefficient λ_t and adjusts the injected diffusion noise by a coefficient γ_t, with an additional damping parameter β for robustness to noise-level misestimation. Experiments on the ChromSTET2025 dataset compare FBP, NafNet, MR-SDE without RNSD+, and the proposed method, reporting best sinogram-domain PSNR/SSIM/LPIPS and best image-domain PSNR/LPIPS for RNSD+. A robustness study examines the effect of varying σ_y estimation error and β.

Significance. If the rectification schedule is correct, the paper makes a useful contribution: explicitly accounting for measurement noise at inference time in diffusion-based sinogram completion is a practical need, and the reported gains (e.g., sinogram-domain PSNR 35.91 vs 32.03) are encouraging. The paper is also honest about the role of the estimated noise level and provides a tunable damping factor. However, the central mathematical derivation is not clean as printed: the key equations contain dimensional inconsistencies and an unverified nonnegativity condition. The empirical comparison also leans heavily on the authors' own prior work as the main diffusion baseline. The core idea is plausible and likely fixable, but the paper as printed does not yet provide a verified description of what was implemented.

major comments (3)

[Section 2.1, Eq. (6)] The second equality in Eq. (6) is algebraically and dimensionally inconsistent as printed. From the first expression, x'_{0|t} = \hat{x}_{0|t} - λ_t A^†(A \hat{x}_{0|t} - y) expands to (I - λ_t A^† A) \hat{x}_{0|t} + λ_t A^† y, not (I - λ_t A) \hat{x}_{0|t} - λ_t y. Because A ∈ R^{N×M}, the matrix I - λ_t A is not defined, and the printed form drops the pseudo-inverse that defines the range-space projection. Since this rectification is the paper's central contribution and is used in all experiments, the equation must be corrected and the implemented form must be stated unambiguously; the same issue affects Eq. (5), where the symbol x on the right-hand side is undefined.
[Section 2.1, Eq. (7)] The nonnegativity condition for γ_t² is dimensionally mismatched. With d \bar{w} = √dt σ_t ε_t, the per-step variance of the injected SDE noise is σ_t² dt, whereas the measurement-noise contribution in the range space has variance on the order of (h_t λ_t σ_y)². The condition for γ_t² ≥ 0 is therefore σ_t² dt ≥ (h_t λ_t σ_y)², not σ_t ≥ h_t σ_y as printed. With T = 100, dt = 0.01, whenever h_t σ_y exceeds σ_t/10 the radicand is negative and the update cannot be executed as written. Please report the effective dt convention and the exact implemented schedule, or revise the definitions so that the radicand is guaranteed nonnegative in the reported experiments.
[Section 2.1, Eqs. (6)-(7)] The claim that the rectification keeps the effective inference-time noise within the score model's trained range is asserted without derivation. In particular, σ_t is a per-√t SDE coefficient while σ_y is a per-sample noise level, so comparing them requires an explicit discrete-time conversion; the distribution of ε_t is written as N(μ, λ²), which is incompatible with d \bar{w} being a noise increment; and h_t is defined using ζ'_t, which appears identical to \bar ζ_t introduced in Eq. (3), making the formula ambiguous. A clean derivation of λ_t, γ_t, and h_t, with all distributions and the discrete-time variance of each term, is necessary to verify that the score network is not fed out-of-distribution residual noise.

minor comments (5)

[Table 1] The table reports 'ten-run average' results without standard deviations or significance tests; given that some image-domain gaps are modest (e.g., PSNR 20.43 vs 19.19), variance information is needed to assess the claim of consistent improvement.
[Section 3.2] The text refers to 'Figure 2.1'; this should be 'Figure 2'. The heading of Section 3.3, 'Robusteness', should be 'Robustness'.
[Eq. (7)] The expression for h_t is ambiguous: ζ'_t appears to be the same as \bar ζ_t, and the exponent e^{-\bar ζ_t - 1} is unclear. Please define all quantities explicitly.
[Abstract] The abstract claims generalization 'across varying noise intensity and acquisition scenarios', but the experiments vary noise level and σ_y estimation error only; the missing-angle range and noise type are fixed, so the claim is stronger than what is demonstrated.
[References] Reference [15], which serves as the main baseline, is cited only as '2025' without a venue or arXiv identifier; it should be made identifiable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; RNSD+ is an explicit construction with measured gains, and the self-cited baseline is not load-bearing.

full rationale

The paper's load-bearing component is the RNSD+ rectification in Eqs. (5)-(7). This is presented as an explicit construction, not as an empirical prediction: λ_t, γ_t, and h_t are defined (with h_t imported from the MR-SDE transition kernel) so that the measurement-induced noise is folded into the reverse-diffusion noise budget. The performance gains in Table 1 are measured outcomes of running this construction against implemented baselines, including the authors' prior MR-SDE without RNSD+ [15]. That baseline is self-cited, and the narrative frames the contribution as 'our new method beats our old method,' but the comparison is a direct experiment, not a logical reduction; the cited [15] is not used to prove the RNSD+ result. No fitted parameter is renamed as a prediction: σ_y is simulated and β is swept post hoc in the robustness analysis, with default β=1. The algebraic and dimensional concerns about Eq. (6) (for example, the apparent dropping of A† in the printed simplification, or the possibility that γ_t² becomes negative under the stated condition) are correctness risks, not circularity: an incorrect or unverified schedule does not make the derivation equivalent to its inputs. No uniqueness theorem or ansatz is smuggled in via a self-citation; Eq. (7)'s schedule is asserted, not derived from [15]. Therefore no step reduces by construction to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on an asserted noise-matching schedule (Eq 7), a Gaussian-and-known noise model, transfer of the trained score network to rectified states, a mask interpretation of the forward operator, and one synthetic dataset. Three free parameters (beta, sigma_y, inherited training schedule) plus five assumptions carry the load; no independent external validation or formal verification is provided.

free parameters (3)

beta (damping factor) = No principled value; swept over (0,1] in Fig 3
Scales the range-space correction so smaller beta trades data consistency for robustness to sigma_y misestimation. The paper gives no rule for choosing beta in practice.
sigma_y (measurement noise standard deviation) = Assumed known in main experiments; simulated at 5-15 dB SNR
Both lambda_t and gamma_t are functions of sigma_y. The main experiments assume it is known exactly; Fig 3 only tests relative misestimation of it.
Inherited MR-SDE training configuration (variance lambda^2, T=100, time-travel settings) = T=100, lambda^2 fixed, per [10]-[16]
The rectification noise budget depends on the sigma_t schedule, and the paper defers the training configuration to its own preprint [10] rather than deriving or re-justifying it.

assumptions (5)

domain assumption The score network trained on the MR-SDE noise schedule remains accurate when measurement noise is folded in via lambda_t and gamma_t, because the post-rectification noise stays within sigma_t, the maximum level the model was trained to denoise.
This is the stated design criterion for Eq (7) in Sec 2.1. If the residual noise after rectification is out-of-distribution for the denoiser, the iterative inference would diverge.
domain assumption The unknown measurement noise n is zero-mean Gaussian with variance sigma_y^2 I, in the same family as the diffusion noise.
Sec 2.1 argues that 'since n ~ N(0, sigma_y^2 I)... it can be treated as part of the noise and subsequently removed through diffusion'. Real CT noise is only approximately Gaussian.
domain assumption The operator A with adjoint A-dagger behaves as a sinogram mask, so that sinogram-domain inpainting with a mask exactly represents the limited-angle Radon forward model of Eq (1).
The paper overloads A (Radon matrix in Eq 1, masking operator in Eqs 5-7). The rectification assumes y already lives in sinogram coordinates and that A-dagger A projects onto the observed views.
ad hoc to paper The lambda_t, gamma_t, and h_t formulas of Eq (7) are the correct noise-matching solution for RNSD+.
The text says the coefficients 'are set to ensure' the noise bound, but no derivation is shown, and h_t mixes zeta-bar_t and zeta-prime_t in a way that cannot be verified from the printed equations.
domain assumption The synthetic ChromSTET2025 volumes generated by SR-EV [18] are representative of real electron-tomography limited-angle data.
All conclusions rest on this single simulated dataset; no real experimental tomograms are used for validation.

invented entities (1)

RNSD+ rectification schedule (lambda_t / gamma_t / h_t noise-adjusted correction)
purpose: Enforces data consistency during reverse diffusion under Gaussian measurement noise by bounding the injected noise below the trained noise level sigma_t.
The schedule's only support is Table 1 and Fig 3 of this paper. It has not been reproduced by an independent group, and no formal verification or external benchmark is offered.

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Cite this review

Pith. "Pith review of Diffusion-Based Limited-Angle CT Reconstruction under Noisy Conditions." pith.science (2026). https://pith.science/paper/2MRY32TI

@misc{pith2026250705647,
  author       = {Pith},
  title        = {Pith review of: Diffusion-Based Limited-Angle CT Reconstruction under Noisy Conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2MRY32TI}},
  note         = {Machine review of arXiv:2507.05647}
}

abstract

Limited-Angle Computed Tomography (LACT) is a challenging inverse problem where missing angular projections lead to incomplete sinograms and severe artifacts in the reconstructed images. While recent learning-based methods have demonstrated effectiveness, most of them assume ideal, noise-free measurements and fail to address the impact of measurement noise. To overcome this limitation, we treat LACT as a sinogram inpainting task and propose a diffusion-based framework that completes missing angular views using a Mean-Reverting Stochastic Differential Equation (MR-SDE) formulation. To improve robustness under realistic noise, we propose RNSD$^+$, a novel noise-aware rectification mechanism that explicitly models inference-time uncertainty, enabling reliable and robust reconstruction. Extensive experiments demonstrate that our method consistently surpasses baseline models in data consistency and perceptual quality, and generalizes well across varying noise intensity and acquisition scenarios.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [10]

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Jiaming Liu, Rushil Anirudh, Jayaraman J Thiagarajan, Stewart He, K Aditya Mohan, Ulugbek S Kamilov, and Hyojin Kim, “Dolce: A model-based probabilistic diffu- sion framework for limited-angle ct reconstruction,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2023, pp. 10498–10508

work page 2023

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Rn- sdes: Limited-angle ct reconstruction with residual null- space diffusion stochastic differential equations,

Jiaqi Guo, Santiago Lopez-Tapia, Wing Shun Li, Yunnan Wu, Marcelo Carignano, Vadim Backman, Vinayak P Dravid, and Aggelos K Katsaggelos, “Rn- sdes: Limited-angle ct reconstruction with residual null- space diffusion stochastic differential equations,” arXiv preprint arXiv:2409.13930, 2024

work page arXiv 2024

[5] [1]

Diffusion-Based Limited-Angle CT Reconstruction under Noisy Conditions

INTRODUCTION Tomographic reconstruction aims to recover high-dimensional structures from their lower-dimensional projections. This process is governed by the Radon Transform [1], which mod- els each projection as a line integral along specific scanning paths. Under ideal full-angle acquisition, these projections sufficiently sample the Fourier space, allo...

work page Pith review arXiv 2025

[6] [2]

METHOD 2.1. Noisy Sinogram Inpainting with Rectified Diffusion In this work, we leverage the Mean-Reverting Stochastic Dif- ferential Equation (MR-SDE) framework proposed by [16] to recover incomplete sinograms by modeling the missing an- gular views. The MR-SDE paradigm captures the forward and reverse evolution of a diffusion process through stochas- ti...

work page

[7] [3]

General Experiment Setting 3.1.1

EXPERIMENT 3.1. General Experiment Setting 3.1.1. Dataset and Preprocessing ChromSTET2025 Dataset: The Chromatin Scanning Trans- mission Electron Tomography (ChromSTET) dataset is cre- ated using the SR-EV [18] model, developed by the Center for Chromatin NanoImaging in Cancer at Northwestern Uni- versity. This model simulates 3D chromatin structures at n...

work page

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CONCLUSION & DISCUSSION In this paper, we proposed a diffusion-based framework for robust limited-angle computed tomography (LACT) recon- struction under noisy conditions. Our method formulates LACT as a sinogram inpainting task, leveraging a Mean- Reverting Stochastic Differential Equation (MR-SDE) to recover missing angular views by progressively refini...

work page