Model Learning: Primal Dual Networks for Fast MR imaging

Dong Liang; HaiFeng Wang; Jing Cheng; Leslie Ying

REVIEW 4 major objections 4 minor 15 references

Model Learning: Primal Dual Networks for Fast MR imaging

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Unrolling the primal-dual hybrid gradient algorithm into learned networks, with progressively relaxed constraints, reconstructs high-quality MR images from 6×-undersampled k-space, outperforming state-of-the-art baselines.

desk verdict A competent, honest incremental unrolling of Chambolle–Pock for MRI, but the headline claim over D5C5 is undercut by the authors' own admission that the baseline may have been undertrained. read the letter →

arxiv 1908.02426 v1 pith:RTWKJNR2 submitted 2019-08-07 eess.IV cs.CVcs.LGphysics.med-phstat.ML

Jing Cheng , Haifeng Wang , Leslie Ying , Dong Liang This is my paper · ORCID

classification eess.IVcs.CVcs.LGphysics.med-phstat.ML

keywords MRIreconstructionprimal-dualhybridgradientChambolle-Pockalgorithmdeeplearningk-spaceundersamplingcompressedsensingdataconsistencymodel-driven

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 tries to show that an MRI reconstruction method built by unrolling the primal-dual hybrid gradient (Chambolle–Pock) algorithm into a learned network can reconstruct high-quality images from k-space data sampled at 6× undersampling. The authors propose a ladder of three networks—PDHG-CSnet, CP-net, and PD-net—in which constraints of the original compressed-sensing model are progressively relaxed and replaced by learned operations. They report that the most relaxed network, PD-net, outperforms the data-driven D5C5 cascade and the model-driven ADMM-CSNet on in-vivo brain data. A sympathetic reader would take the contribution to be evidence that optimization-derived network architectures with learned data consistency beat both pure end-to-end learning and strictly constrained unrolled methods.

What carries the argument

The load-bearing object is the unrolled Chambolle–Pock (primal-dual hybrid gradient) iteration, whose three update equations—dual ascent on the data-consistency term, primal proximal step on the image prior, and an acceleration extrapolation—are turned into network layers. The learned primal proximal operator $\Lambda$ and dual proximal operator $\Gamma$ are small convolutional networks that take the current variable and either the image-domain gradient or the k-space mismatch as input; the step sizes $\sigma$, $\tau$ and the extrapolation weight $\theta$ are learnable. Cascading ten iterations with learned operators $\Lambda$ and $\Gamma$ lets the network alternate between k-space and image-domain updates while learning both the regularization and the data-consistency similarity measure.

What would settle it

Run PD-net and D5C5 on prospectively undersampled scans with the same acceleration but real noise and coil behavior instead of retrospective Poisson-disk masks; if PD-net's advantage in PSNR or SSIM disappears or reverses, the paper's central superiority claim fails.

Watch

Extended reading notes

Core claim

The central claim is that gradually relaxing the fixed structure of an unrolled primal-dual optimization algorithm improves reconstruction quality while retaining the alternating k-space and image-domain updates of the original method. Starting from the Chambolle–Pock iterations for the compressed-sensing objective, the authors first replace the proximal operator of the image prior by a CNN (PDHG-CSnet), then replace the data-consistency dual proximal also by a learned operator (CP-net), and finally let the network choose how to combine primal and dual variables (PD-net). At 6× Poisson-disk undersampling, PD-net is reported to achieve better quantitative and visual results than D5C5, ADMM-CSNet, and a traditional TV-based compressed-sensing method, with the gap growing as the training set grows.

Load-bearing premise

The results rest on the assumption that retrospectively undersampling fully sampled single-coil brain data with a Poisson disk mask behaves like real accelerated MRI; if prospective multi-coil acquisition differs (noise correlation, motion, coil sensitivities), the reported superiority may not transfer to clinical scans.

Editorial extensions

If this is right

PD-net is a competitive single-coil reconstruction method at 6× undersampling; the paper reports it outperforming D5C5 and ADMM-CSNet on retrospective Poisson-disk data.
Relaxing model constraints shifts the burden to training data: with 100 training samples CP-net matches PD-net, while PD-net only pulls ahead when trained on 1400 samples.
Learned data consistency can beat explicit $\ell_2$ k-space correction, because the dual network learns the similarity to acquired data instead of forcing an Euclidean-distance match.
The unrolled primal-dual formulation offers a route to architectures that are neither purely data-driven nor purely model-driven; they inherit the alternating-update skeleton of an optimization algorithm.

Reading between the lines

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

The same constraint-relaxation ladder could be applied to multi-coil reconstruction: replacing the single-coil encoding matrix by SENSE or GRAPPA-style operators would test whether learned dual updates still help when coil sensitivities are known.
If the training-set dependence reported here is real, practitioners should expect PD-net-type networks to need proportionally more data as constraints are relaxed; a scaling-law measurement over training-set size would make this explicit.
The learned dual variable can be read as a k-space correction field; one could inspect it directly to see whether it isolates alias artifacts, providing an interpretability check the paper does not report.
The comparison to D5C5 suggests the gain comes from learned similarity rather than network capacity; an ablation that swaps PD-net's dual block for an explicit $\ell_2$ data-consistency layer would isolate that factor.

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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

Two things to know. First, this is a competent and honest incremental paper: it unrolls the Chambolle-Pock primal-dual algorithm into three progressively more relaxed learned networks for undersampled MRI, and the comparison among PDHG-CSnet, CP-net, and PD-net is a useful pedagogical demonstration of the model-driven/data-driven spectrum. Second, the paper's central empirical claim — that PD-net beats D5C5 and other state-of-the-art methods — is not adequately supported as written. The authors themselves concede that D5C5 may not have had enough training data, which is a direct confound; without details of D5C5's training setup, the comparison cannot decide whether PD-net's learned data consistency is genuinely better or whether the baseline was simply undertrained. This is the load-bearing weakness, and it is real, not manufactured.

What is genuinely new is modest: applying the learned primal-dual architecture from Adler and Oktem (CT) to MRI, and much of the unrolling structure is already in ADMM-CSNet and variational networks. The paper does a good job of clearly laying out the three relaxations and the network architectures, and the k-space/image-domain alternating updates with learned data consistency are a reasonable design. The figures appear to show convincing visual improvements, but the quantitative values live only in captions, with no standard deviations or statistical tests. That is a reportability problem, not a fatal one.

The convergence-guarantee language in the abstract is overstated: the learned network no longer satisfies the assumptions of Chambolle-Pock, so the guarantee does not transfer. The stress-test note is correct on this point as well.

The synthetic Poisson-disk undersampling of single-coil data is standard in the field, so I would not count that against the paper much, but it does mean the clinical transfer claim is prospective at best.

Who is this paper for? Anyone working on unrolled optimization for inverse problems in MRI would want to cite it as an early example of learned primal-dual applied to MR. It deserves a serious referee — the architecture is sound and the experimental approach is redeemable. I would send it out with a request for controlled baseline training, quantitative tables in the main text, and a more modest statement of what the convergence theory actually guarantees.

Referee Report

4 major / 4 minor

Summary. The paper proposes three progressively relaxed unrolled networks based on the Chambolle-Pock primal-dual hybrid gradient algorithm for single-coil MR reconstruction from undersampled k-space: PDHG-CSnet (learned proximal in the primal update), CP-net (learned data consistency in the dual update), and PD-net (free-form learned primal/dual updates with no explicit variable structure). The architecture is described by Eqs. (2)-(7), training uses an MSE loss with 10 cascaded iterations of 3x3 convolutional blocks, and experiments compare against zero-filling, TV-based CS (Rec_PF), ADMM-CSNet, and D5C5 on 7 in vivo brain datasets acquired from three scanner vendors at 6x Poisson-disk undersampling. The abstract and Section 3 claim that PD-net achieves superior reconstructions over state-of-the-art methods and that the approach combines the theoretical convergence guarantee of optimization methods with deep learning capacity.

Significance. If the empirical claim were fully supported, PD-net would be a useful contribution bridging model-driven and data-driven MR reconstruction: the gradual relaxation taxonomy from Eq. (2) through Eq. (7) is pedagogically clear, the derivation of the unrolled updates is standard and correctly presented, and the use of in vivo test data from three scanner vendors is a genuine strength. The paper also honestly flags the possible undertraining of D5C5. However, the central 'superior' claim currently rests on quantitative metrics that appear only inside figure captions and on an uncontrolled baseline comparison, so the significance is conditional on a substantially strengthened evaluation.

major comments (4)

[Section 3, Fig. 4] The central empirical claim in the Abstract and Section 3 that PD-net 'achieves better performance than D5C5' and provides 'superior MR reconstructions' is not verifiable from the text: no PSNR, SSIM, or other quantitative metrics are stated outside the figure captions, no standard deviations are reported, and no statistical test across the 7 test datasets is provided. Please add a results table with per-method metrics on each test dataset, summary statistics across datasets, and an indication of how many independent runs or scans support each number.
[Section 3 (D5C5 baseline)] The comparison with D5C5 is uncontrolled: the training setup for D5C5, including number of training samples, epochs, learning rate, augmentation, and hyperparameter tuning, is not reported, and the text itself concedes that 'the training data may be not enough for D5C5.' If D5C5 was undertrained, the claimed architectural advantage of PD-net is confounded. Please report D5C5's training details, ensure it receives the same training data and protocol as PD-net, and verify that it has converged (for example, by showing training loss curves) before drawing conclusions about relative performance.
[Section 2.5 and Abstract] The statement that the method 'combines the theoretical convergence guarantee of optimization methods' is not justified: after replacing the proximal operators with learned CNN operators in Eqs. (4) and (6), and after breaking the explicit variable structure in Eq. (7), the Chambolle-Pock convergence theorem no longer applies to the resulting network. Please state explicitly which properties of the original algorithm are preserved and which are heuristic, or remove the convergence-guarantee claim from the abstract and Section 2.5.
[Section 3 (evaluation data)] All experiments use retrospectively undersampled single-coil k-space data obtained from fully sampled acquisitions with a Poisson disk mask. The paper does not test or discuss prospective multi-coil acquisitions, noise correlation, motion, or coil sensitivity effects, so the claimed superiority may not transfer to clinical scans. Please add this as an explicit limitation and, if possible, validate the method on prospectively undersampled or multi-coil data.

minor comments (4)

[Section 2.4] There are several typographical errors: 'Rectified Linear Unites' should be 'Rectified Linear Units', 'Ubunt u' should be 'Ubuntu', and 'Erlgen' should be 'Erlangen'.
[Section 2.4] The number of iterations (10), the number of channels (32), and the filter size (3x3) are fixed without any ablation or sensitivity analysis; a brief justification or ablation study would strengthen the architecture choices.
[References] Reference [14] is the authors' own ISMRM abstract; the manuscript should explicitly state how this journal/workshop paper extends that preliminary version.
[Figures] Figures 2-4 are referenced only by figure number, not by panel; please specify which panels correspond to which methods and which training sizes, since the captions currently carry a large portion of the experimental results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PD-net derivation is a standard unrolling of Chambolle-Pock with learned proximal operators, and the empirical claims rest on held-out in-vivo tests rather than on fitted constants or self-citation chains.

full rationale

The derivation chain is input-to-model, not model-to-input. Starting from the CS objective in Eq. (1), the paper writes the Chambolle-Pock iterations in Eq. (2), replaces the primal proximal operator with a learned CNN in Eq. (4), relaxes the data-consistency term and replaces the dual proximal operator with a learned CNN in Eq. (6), and finally relaxes the variable structure in Eq. (7). Each step is an explicit modification of the update equations, and the network parameters are trained end-to-end by minimizing the MSE loss in Eq. (8) on paired undersampled/fully sampled data. No fitted parameter is renamed as a prediction, and no evaluated metric is used as a training target. The claim that 'PD-net achieves better performance than D5C5' is an empirical comparison; the paper's own admission that 'the training data may be not enough for D5C5' is a possible confound in the baseline comparison, but it is an experimental-control issue, not a circular derivation. The statement that the method 'combines the theoretical convergence guarantee of optimization methods' is an overstatement because the learned replacements void the Chambolle-Pock assumptions, but that is a correctness/accuracy concern rather than circularity. The only self-citation is reference [14], the authors' own ISMRM abstract, cited for the design choice of learning where to evaluate the forward operator; however, Eq. (7) explicitly defines the PD-net updates and the inspiration is also attributed to the external learned-primal-dual work [13], so this self-citation is not load-bearing. The 'gradual relaxation' narrative is an explanatory ordering of architectures, not a derivation that assumes the superiority conclusion. Therefore no circular step is exhibited, and the paper receives a score of 0.

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

The method does not introduce new physical entities. The free parameters are network architecture choices fixed by hand; the learned weights are trained from data. The key assumptions are standard for model-driven deep learning MRI reconstruction, with the convergence-guarantee claim being the main ad hoc element.

free parameters (3)

iteration_count = 10
Number of unrolled iterations fixed to 10 in all three networks (Section 2.4).
convolution_channels = 32
Channel width of the CNN blocks in primal and dual updates (Section 2.4).
filter_size = 3x3
Convolution kernel size used throughout the networks (Section 2.4).

assumptions (4)

domain assumption The encoding operator A and its adjoint A* are exactly known and linear.
All updates in Eqs. (2), (4), (6), and (7) multiply by A or A*; the forward model in Eq. (1) is taken as given.
domain assumption Fully sampled k-space data, after coil combination, provides a valid ground-truth image for supervised training.
Section 3 uses retrospectively undersampled data from fully sampled acquisitions and computes MSE against reference reconstructions.
domain assumption CNN blocks with 3x3 convolutions and ReLU can approximate the proximal operators of the unknown prior and data-consistency terms.
Sections 2.1 to 2.3 replace proximal operators with learned operators Lambda and Gamma.
ad hoc to paper The convergence guarantee of Chambolle-Pock remains a meaningful property of the unrolled network.
The abstract claims the method 'combines the theoretical convergence guarantee' but once proximal operators are replaced with learned networks and the iteration count is fixed, the CP convergence theorem no longer applies.

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Cite this review

Pith. "Pith review of Model Learning: Primal Dual Networks for Fast MR imaging." pith.science (2026). https://pith.science/paper/RTWKJNR2

@misc{pith2026190802426,
  author       = {Pith},
  title        = {Pith review of: Model Learning: Primal Dual Networks for Fast MR imaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTWKJNR2}},
  note         = {Machine review of arXiv:1908.02426}
}

read the original abstract

Magnetic resonance imaging (MRI) is known to be a slow imaging modality and undersampling in k-space has been used to increase the imaging speed. However, image reconstruction from undersampled k-space data is an ill-posed inverse problem. Iterative algorithms based on compressed sensing have been used to address the issue. In this work, we unroll the iterations of the primal-dual hybrid gradient algorithm to a learnable deep network architecture, and gradually relax the constraints to reconstruct MR images from highly undersampled k-space data. The proposed method combines the theoretical convergence guarantee of optimi-zation methods with the powerful learning capability of deep networks. As the constraints are gradually relaxed, the reconstruction model is finally learned from the training data by updating in k-space and image domain alternatively. Experi-ments on in vivo MR data demonstrate that the proposed method achieves supe-rior MR reconstructions from highly undersampled k-space data over other state-of-the-art image reconstruction methods.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Lustig, M., Donoho, D., Pauly, J.: Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182-1195 (2007)

work page 2007

[2] [2]

Science, 349, 255-60 (2015)

Jordan, M., Mitchell, T., Machine learning: Trends, perspectives, and prospects . Science, 349, 255-60 (2015)

work page 2015

[3] [3]

In: 13th International Symposium on Biomedical Imag- ing, pp

Wang, S., Su, Z., Ying, L., Peng, X., Zhu, S., Liang, F., et al.: Accelerating Magnetic Reso- nance Imaging via Deep Learning. In: 13th International Symposium on Biomedical Imag- ing, pp. 514-517, IEEE, Prague, Czech Republic (2016)

work page 2016

[4] [4]

Nature 555, 487-492 (2018)

Zhu, B., Liu, J., Cauley, S., Rosen, B., Rosen, M.: Image reconstruction by domain-trans- form manifold learning. Nature 555, 487-492 (2018)

work page 2018

[5] [5]

Magn Reson Med 80, 2188-2201 (2018)

Eo, T., Jun, Y., Kim, T., Jang, J., Lee, H., Hwang, D.: KIKI-net: cross-domain convolutional neural networks for reconstructing undersampled magnetic resonance images. Magn Reson Med 80, 2188-2201 (2018)

work page 2018

[6] [6]

IEEE Trans Med Imag- ing 37, 491-503 (2018)

Schlemper, J., Caballero, J., Hajnal, J., Price, A., Rueckert, D.: A Deep Cascade of Convo- lutional Neural Networks for Dynamic MR Image Reconstruction . IEEE Trans Med Imag- ing 37, 491-503 (2018)

work page 2018

[7] [7]

IEEE Trans Med Imaging 37, 1488- 1497 (2018)

Quan, T., Nguyen-Duc, T., Jeong, W.: Compressed Sensing MRI Reconstruction Using a Generative Adversarial Network With a Cyclic Loss . IEEE Trans Med Imaging 37, 1488- 1497 (2018)

work page 2018

[8] [8]

IEEE Trans Med Imaging, 38, 394-405 (2019)

Aggarwal, H., Mani, M., Jacob, M.: MoDL: Model-Based Deep Learning Architecture for Inverse Problems. IEEE Trans Med Imaging, 38, 394-405 (2019)

work page 2019