Weakly-Convex Regularization for Magnetic Resonance Image Denoising

Abhishek Shreekant Bhandiwad; Abijith Jagannath Kamath; Akash Prabakar; Chandra Sekhar Seelamantula

REVIEW 3 major objections 6 minor 24 references

Weakly-Convex Regularization for Magnetic Resonance Image Denoising

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A bottom-up design of weakly-convex regularizers yields MRI denoisers that are provably convergent, interpretable, and competitive with Patch2Self.

desk verdict Plausible application of known weakly-convex regularization to diffusion MRI, but the 'on par' and 'fewer artifacts' claims are not backed by any numbers. read the letter →

arxiv 2508.14438 v1 pith:FVTP3BMG submitted 2025-08-20 eess.SP

Akash Prabakar , Abhishek Shreekant Bhandiwad , Abijith Jagannath Kamath , Chandra Sekhar Seelamantula This is my paper · ORCID

classification eess.SP

keywords magneticresonanceimagingdiffusion-weightedimagedenoisingweakly-convexregularizationWelschpenaltynonconvexoptimizationdeepunfoldingtractography

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

The paper proposes a recipe for building MRI denoisers that are both data-driven and provably convergent. Instead of treating deep networks as black boxes, it starts from scalar weakly-convex penalty functions—such as the minimax-concave penalty, SCAD, and the Welsch penalty—and stacks them as activations in a convolutional ridge-regularizer network. Because each component is weakly convex, the regularizer inherits a convergence guarantee for the denoising objective while still being trainable. On diffusion-weighted MR images, the method matches Patch2Self, a state-of-the-art self-supervised denoiser, and the authors argue it removes fewer real structures, using fiber-to-bundle coherence tractography as evidence. If this holds, MRI gains an explainable, stable denoiser with performance comparable to deep learning.

What carries the argument

The engine is the prototype construction: scalar weakly-convex penalties—minimax-concave penalty, SCAD, and the Welsch penalty—are treated as activations and combined in a convolutional ridge-regularizer network. Theorem 1 ties the exact weak-convexity parameter to the maximum concavity of the prototype (kappa(psi)=rho), and Theorem 2 admits smooth functions with Lipschitz derivative as convergent regularizers. The network is trained by unrolling the iterates of subgradient descent, so the same network that is learned is the one whose convergence is guaranteed.

What would settle it

Run the proposed network and Patch2Self on the same diffusion-weighted dataset across several noise realizations, and compute quantitative fiber-to-bundle coherence statistics (or phantom metrics with known ground truth) in the regions where residuals show bright structure; if the numbers do not reliably favor the proposed method, the fewer-artifacts claim fails.

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Extended reading notes

Core claim

The paper claims that weakly-convex regularizers for the denoising problem minimize over x of one-half the squared difference between x and the noisy image plus a weakly-convex penalty g(x), and that such regularizers can be constructed bottom-up from scalar prototype functions—the minimax-concave penalty, SCAD, and the Welsch penalty. These prototypes are symmetrized as g(x)=psi(|x|), and their weak-convexity parameters are tied exactly to the maximum concavity of the prototype. Stacked as activations in a convolutional ridge-regularizer network and trained by unrolling subgradient descent, the resulting denoiser is provably convergent and interpretable while performing on par with Patch2Se

Load-bearing premise

The fewer-artifacts claim rests on the assumption that the fiber-to-bundle coherence density maps, presented only as visuals, capture the denoising artifacts that matter; if those maps are not a sensitive measure, that claim is unsupported.

Editorial extensions

If this is right

Convex denoising is no longer required for guarantees: weakly convex regularizers with rho<1 keep the objective convex, so nonconvex penalties like MCP, SCAD, and Welsch can be used without giving up convergence.
The same bottom-up recipe turns other weakly-convex scalar functions into convergent convolutional regularizers, so interpretability is built in rather than added after training.
On diffusion-weighted MRI, the method matches Patch2Self's denoising while preserving coherent fiber streamlines and removing incoherent ones, which should make downstream tractography more faithful.
Because the network is trained by unrolling, the regularizer is data-driven: users do not have to hand-pick penalty parameters for each scan.
If the artifact claim transfers, clinical MRI can use a denoiser whose behavior is explainable and stable, unlike black-box deep networks.

Reading between the lines

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

Editorial inference: the construction is problem-agnostic; the same prototype-to-ridge-regularizer pipeline could be tried on other linear inverse problems, such as CT or PET reconstruction, where explainability is required.
Editorial inference: the article's few-artifacts evidence is visual fiber-to-bundle coherence maps; quantifying FBC or using phantom ground truth would let the claim be tested directly and is a natural follow-up.
Editorial inference: Theorem 1's exact equality kappa(psi)=rho means the designer can search over prototype families while keeping the convergence guarantee, so the regularizer shape could be optimized for a downstream metric.
Editorial inference: comparing against a wider set of baselines, not only Patch2Self, would show where the weakly-convex construction gains most.

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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

Short version: this is an application note that moves the authors' own weakly-convex regularizer from their ICASSP 2025 paper onto diffusion-weighted MRI. That move is new, and the tractography-based evaluation idea is a good instinct. The problem is that the central claims are supported only by eyeballing two figures. There are no PSNR/SSIM numbers, no FBC summary statistics, no confidence intervals, no multi-subject results. 'On par' and 'fewer artifacts' are simply asserted, not measured.

What the paper does well: the construction is honestly inherited from [1] and [7], and the convergence guarantees carry over. The application to HARDI data with a Patch2Self comparison is a sensible test bed. Showing FBC density maps as evidence of artifact removal is creative, though it stops short of quantification. The writing is compact and clear, with no overreach in the theory sections.

Where it falls short: the experimental section is three paragraphs and two figures. The residual images in Fig. 3 are suggestive but not evidence. The FBC maps in Fig. 4 could be cherry-picked, and without numeric values or a statistical test they prove nothing about systematic improvement. The paper also does not report training details for the unfolded network beyond referring to [1], so the results are not independently reproducible from the text. This is a load-bearing gap because the entire contribution here is empirical; the theory is already published elsewhere.

Is the central idea wrong? Probably not. The method has a theoretical foundation and the qualitative results look plausible. But as written, the paper is an extended abstract, not a complete claim. A serious referee could fix it by requiring quantitative evaluation, error bars, and code release. The paper deserves that referee rather than a desk reject, because the application is real and the claims are testable.

Bottom line: read it if you work on diffusion MRI denoising, but do not cite it for the empirical claims until the numbers appear. I would not put it in my own references right now. If a referee report comes back asking for measurements, that is the right outcome.

Referee Report

3 major / 6 minor

Summary. The paper presents a weakly-convex regularization approach for diffusion-weighted MR image denoising. The authors build on their earlier ICASSP 2025 construction [1]: prototype weakly-convex functions (MCP, SCAD, Welsch) are used in a convolutional ridge regularizer network, and the denoising problem is posed as (1). They claim the technique performs on par with state-of-the-art denoisers and exhibits fewer denoising artifacts, supporting this with visual comparisons against Patch2Self (Fig. 3) and fiber-to-bundle coherence (FBC) density maps from probabilistic tractography (Fig. 4). The theoretical convergence and interpretability claims are inherited from prior work [1], [7].

Significance. If the empirical claims were properly quantified, the paper would offer a useful contribution: a provably convergent, interpretable, data-driven denoiser that is competitive with black-box deep learning in a clinically relevant MRI pipeline. The choice of an external baseline (Patch2Self) and an independent tractography evaluation are appropriate and commendable. However, the current evidence is almost entirely qualitative, and the central claims are not yet supported by measurements. The significance is therefore conditional on the authors providing quantitative validation.

major comments (3)

[Section III, Figs. 3-4] The central claims that the proposed technique 'performs on par with state-of-the-art denoisers' and 'exhibits fewer denoising artifacts' are not supported by any quantitative evaluation. There are no PSNR/SSIM or other image-quality metrics, no numerical FBC statistics, no streamline counts or bundle-overlap measures, no multiple slices/subjects, and no error bars or statistical tests. The entire empirical contribution rests on visual inspection of one slice and one FBC map. This is load-bearing because the theory is inherited from [1] and [7]; the added value of this manuscript is its application to MRI, and that application is currently unquantified.
[Section III, 'Details of training the network are given in [1]'] The experimental setup is not self-contained. The reader cannot reproduce the method from this manuscript alone: the network architecture, training data, noise model, hyperparameter values (e.g., lambda, gamma, Welsch scale), and optimization details are all deferred to [1]. In addition, the only comparison is against Patch2Self, so the abstract's phrase 'state-of-the-art' is overstated. The paper should state exactly what is new relative to [1], and should provide a reproducible experimental protocol, including quantitative baselines.
[Section III, Fig. 4 and FBC analysis] The 'fewer artifacts' claim is further weakened by the use of FBC density maps as the sole evidence. These maps are presented qualitatively, with no numbers, confidence intervals, or statistical tests. It is not shown that FBC is sensitive to the specific artifacts the denoiser removes, nor that the observed differences are not within the variability of the tractography pipeline. If FBC is to be used as a proxy for denoising quality, that proxy must be validated or at least quantified. As it stands, this is an unsupported measurement-modeling assumption.

minor comments (6)

[Abstract] The phrase 'state-of-the-art denoisers' is stronger than the evidence justifies, since only one baseline (Patch2Self) is evaluated.
[Fig. 3 caption] The caption refers to a 'weakly-convex ridge regularizer network with the Welsch function as activation' but the network architecture and the Welsch activation are not defined in the text; please add definitions or reference a specific equation.
[Fig. 4] The colorbar is labeled 'HIGHLOW' with no units or numeric scale; this should be made quantitative or removed.
[Section II] The term 'weakly-monotone derivatives' is used without definition; since the paper is about weakly-convex functions, please define or avoid this term.
[Eq. (1)] The displayed minimization uses 'minimize' instead of the standard '\(\min\)' notation; this is a minor formatting issue.
[Section III] There is no discussion of computational cost, runtime, or scalability. If 'on par' is meant to include practical usability, these are relevant.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the MRI denoising comparison is externally benchmarked, though it is only visually assessed.

full rationale

The paper's construction chain is Eq. (1) with a weakly-convex regularizer; Definition 1 and the MCP/SCAD/Welsch examples are standard mathematical objects. Theorem 1 is stated as a result from the authors' prior ICASSP paper [1], and Theorem 2 is attributed to [7]; neither is a fitted parameter or a quantity defined in terms of the claimed outcome. The denoising experiment uses the external Stanford HARDI dataset [22], the external Patch2Self baseline [24], and the external tractography/FBC pipeline [20]; the reported behavior is not obtained by fitting a value and then renaming that value a prediction. The main weakness is evidentiary, not circular: 'on par' and 'fewer denoising artifacts' are supported only by the visual figures (Fig. 3 and Fig. 4) with no PSNR/SSIM, FBC statistics, or statistical tests. The self-citations [1] and [9]-[11] are used for construction and convergence properties, and the training details are deferred to [1], but these are mathematical/engineering premises that do not make the empirical comparison equivalent to its inputs. No step was found in which an equation reduces to another by construction or a fitted parameter is relabeled as a prediction. Score 2 reflects the presence of self-citation for the construction theorem and training details, while the central empirical claim remains independently benchmarked.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The mathematical core is imported from the authors' earlier ICASSP paper [1] and from [7]; the empirical claims depend on undisclosed trained parameters and a measurement assumption in the tractography validation. No new physical entities are introduced.

free parameters (3)

MCP/SCAD regularization parameters (lambda, gamma) = not reported
Prototype penalties in Section II are parameterized by threshold and concavity; the paper does not state chosen or learned values.
Welsch activation scale = not reported
The Welsch function used as activation has a scale parameter; its value or fitting rule is not given.
Trained network weights = not reported
The deep-unfolded ridge regularizer network is learned, so all weights and biases are fitted to training data; training details are deferred to [1].

assumptions (5)

standard math Definition 1: rho-weak convexity
Foundation of the whole construction; standard definition in convex analysis.
domain assumption Theorem 1 (from [1]): kappa(psi)=rho iff g(x)=psi(|x|) is rho-weakly convex
Used to justify MCP and SCAD; stated but not proved in this paper.
domain assumption Theorem 2 (cf. [7]): L-Lipschitz derivative implies rho-weak convexity with rho<=L
Used to claim Welsch, sinusoidal, Gabor activations are weakly convex; cited from [7].
domain assumption Convergence of the unrolled subgradient descent algorithm
The provably-convergent claim relies on convergence guarantees for weakly-convex objectives, not demonstrated in this paper.
domain assumption FBC density maps as measure of denoising artifacts
The tractography validation assumes the FBC pipeline [20] accurately reflects artifact removal; no independent evidence is given.

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Cite this review

Pith. "Pith review of Weakly-Convex Regularization for Magnetic Resonance Image Denoising." pith.science (2026). https://pith.science/paper/FVTP3BMG

@misc{pith2026250814438,
  author       = {Pith},
  title        = {Pith review of: Weakly-Convex Regularization for Magnetic Resonance Image Denoising},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVTP3BMG}},
  note         = {Machine review of arXiv:2508.14438}
}

read the original abstract

Regularization for denoising in magnetic resonance imaging (MRI) is typically achieved using convex regularization functions. Recently, deep learning techniques have been shown to provide superior denoising performance. However, this comes at the price of lack of explainability, interpretability and stability, which are all crucial to MRI. In this work, we present a constructive approach for designing weakly-convex regularization functions for MR image denoising. We show that our technique performs on par with state-of-the-art denoisers for diffusion-weighted MR image denoising. Our technique can be applied to design weakly-convex convolutional neural networks with prototype activation functions that impart interpretability and are provably convergent. We also show that our technique exhibits fewer denoising artifacts by demonstrating its effect on brain microstructure modelling.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

On the design of weakly-convex regularizers for solving linear inverse problems,

A. J. Kamath, A. S. Bhandiwad, and C. S. Seelamantula, “On the design of weakly-convex regularizers for solving linear inverse problems,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), 2025

work page 2025

[2] [7]

Learning weakly convex regularizers for convergent image-reconstruction algorithms,

A. Goujon, S. Neumayer, and M. Unser, “Learning weakly convex regularizers for convergent image-reconstruction algorithms,”SIAM J. Imag. Sci., vol. 17, no. 1, pp. 91–115, 2024

work page 2024

[3] [2]

Plug-and-play priors for model based reconstruction,

S. V . Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” inProc. IEEE Glob. Conf. Signal Inf. Process. (GlobalSIP), pp. 945–948, 2013

work page 2013

[4] [3]

Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications,

U. S. Kamilov, C. A. Bouman, G. T. Buzzard, and B. Wohlberg, “Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications,”IEEE Signal Process. Mag., vol. 40, pp. 85–97, 2023

work page 2023

[5] [4]

Plug-and- play methods provably converge with properly trained denoisers,

E. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, and W. Yin, “Plug-and- play methods provably converge with properly trained denoisers,” in Proc. PMLR Int. Conf. Mach. Learn., pp. 5546–5557, 2019

work page 2019

[6] [5]

Plug- and-play image restoration with deep denoiser prior,

K. Zhang, Y . Li, W. Zuo, L. Zhang, L. Van Gool, and R. Timofte, “Plug- and-play image restoration with deep denoiser prior,”IEEE Trans. Patt. Anal. Mach. Intell., vol. 44, no. 10, pp. 6360–6376, 2021

work page 2021

[7] [6]

Lanza, S

A. Lanza, S. Morigi, I. W. Selesnick, and F. Sgallari,Convex Non-convex Variational Models. Springer, 2021

work page 2021

[8] [8]

Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation

Z. Shumaylov, J. Budd, S. Mukherjee, and C.-B. Sch ¨onlieb, “Weakly convex regularisers for inverse problems: Convergence of critical points and primal-dual optimisation,”arXiv preprint arXiv:2402.01052, 2024

work page Pith review arXiv 2024