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arxiv: 2605.18709 · v1 · pith:OAXWTFCFnew · submitted 2026-05-18 · 📡 eess.IV · eess.SP

Dynamic MRI Reconstruction Via Dual Deep Priors and Low-Rank Plus Sparse Modeling

Yongliang Sun , Siddhant Gautam , Chaoyan Huang , Nicole Seiberlich , Ismail Alkhouri , Saiprasad Ravishankar This is my paper

Pith reviewed 2026-05-20 07:39 UTC · model grok-4.3

classification 📡 eess.IV eess.SP
keywords dynamic MRI reconstructiondeep image priorlow-rank plus sparseunsupervised learningADMM optimizationcine MRIundersampled k-spacetraining-free reconstruction
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The pith

Dynamic MRI from undersampled data can be reconstructed by fitting two separate untrained neural networks to its low-rank background and sparse dynamic components.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a training-data-free method for dynamic MRI reconstruction that models both spatial quality and temporal consistency in cine series. It decomposes each scan into a low-rank background component and a sparse dynamic component, with each part represented by its own untrained convolutional network. These two networks are optimized jointly through an accelerated extrapolated ADMM algorithm that includes a convergence proof for the nonconvex case. The approach merges the scan-specific adaptation of deep image priors with the structure of classical low-rank plus sparse regularization. Experiments at multiple acceleration factors show the method surpasses both traditional reconstruction techniques and other supervised or unsupervised learning methods.

Core claim

Instead of reconstructing the cine image series directly, the low-rank background and sparse dynamic components are each parameterized by a dedicated deep image prior network. These networks are jointly optimized using accelerated extrapolated ADMM, for which a sufficient descent property is shown and every cluster point is proved to be a critical point of the associated Lyapunov function. This formulation delivers reconstructions that preserve both spatial detail and temporal dynamics better than classical low-rank plus sparse methods or existing supervised and unsupervised deep learning approaches across a range of acceleration factors.

What carries the argument

Dual deep image prior networks for low-rank plus sparse decomposition, jointly optimized via accelerated extrapolated ADMM (eADMM).

If this is right

  • The method adapts directly to each individual scan without any need for a large collection of fully sampled training examples.
  • Convergence of the iterates to a critical point is guaranteed even though the DIP networks introduce nonconvexity.
  • Explicit separation into low-rank and sparse parts improves exploitation of spatiotemporal structure compared with single-network DIP approaches.
  • The retained L+S interpretability allows inspection of which components the reconstruction attributes to background versus motion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dual-prior structure could transfer to other dynamic imaging modalities such as CT perfusion or ultrasound where low-rank plus sparse models are already used.
  • Adding explicit physics-based constraints on k-space consistency inside the eADMM loop might further reduce residual aliasing at very high accelerations.
  • The same dual-DIP idea could be tested on static MRI tasks by replacing the temporal low-rank model with a spatial one.
  • Replacing the convolutional DIP networks with transformer-based priors might capture longer-range temporal dependencies if overfitting remains controlled.

Load-bearing premise

That jointly optimizing two DIP networks for the low-rank and sparse components via eADMM will reliably capture spatiotemporal correlations on individual scans without overfitting while the supplied convergence analysis for nonconvex DIP parameterizations remains valid.

What would settle it

On a standard public dynamic cardiac MRI dataset at acceleration factor 8, compare the proposed method's PSNR, SSIM, and temporal gradient error against the strongest competing unsupervised baseline; failure to show improvement would contradict the central performance claim.

Figures

Figures reproduced from arXiv: 2605.18709 by Chaoyan Huang, Ismail Alkhouri, Nicole Seiberlich, Saiprasad Ravishankar, Siddhant Gautam, Yongliang Sun.

Figure 1
Figure 1. Figure 1: Overview of the proposed L+S DIP reconstruction framework. The dynamic MR image [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of the extrapolation parameters α and β in the proposed eADMM algorithm on the reconstruction convergence. The curves show PSNR as a function of optimization iteration for different choices of α and β, demonstrating that moderate extrapolation improves convergence speed. and surrounding dynamic regions. Multi-dynamic DIP produces smoother reconstructions, but still loses some fine structural details… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the proposed L+S-DIP framework with other dynamic MRI reconstruction [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effect of random network initialization on the proposed L+S-DIP reconstruction. From [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of the learned low-rank and sparse components for the OCMR dataset. [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
read the original abstract

Dynamic MRI reconstruction from undersampled measurements is a challenging inverse problem that requires preserving both spatial reconstruction quality and temporal consistency across the frames of the cine series. While recent learning-based approaches achieve strong performance, they heavily rely on large training, mostly fully sampled, datasets, and may otherwise generalize poorly. In contrast, training-data-free methods such as deep image prior (DIP) adapt directly to individual scans but often fail to fully exploit temporal structure and are prone to overfitting. They are particularly attractive for dynamic MRI due to the limited large, public, high-quality datasets. In this work, we propose a structured DIP framework for dynamic MRI reconstruction that explicitly models spatiotemporal correlations through a low-rank plus sparse (L+S) decomposition. Instead of directly reconstructing the cine image series, we parameterize the low-rank background and sparse dynamic components using two DIP untrained convolutional neural networks, jointly optimized using accelerated extrapolated ADMM (eADMM). This formulation combines the implicit regularization of DIP with the interpretability of classical L+S regularization. We provide a convergence analysis for the proposed eADMM algorithm in the presence of DIP-based nonconvex parameterizations. In particular, we establish a sufficient descent property and show that every cluster point of the generated sequence is a critical point of the associated Lyapunov function. Across various acceleration factors, our numerical results demonstrate that the proposed method consistently outperforms classical reconstruction and existing supervised and unsupervised MRI reconstruction techniques.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a training-data-free method for dynamic MRI reconstruction from undersampled k-space data. It decomposes the cine series into low-rank background and sparse dynamic components, each parameterized by a separate untrained convolutional neural network (DIP). These networks are jointly optimized using an accelerated extrapolated ADMM (eADMM) algorithm. A convergence analysis is supplied establishing sufficient descent and that every cluster point of the iterates is a critical point of an associated Lyapunov function. Numerical experiments across acceleration factors are reported to show consistent outperformance relative to classical compressed-sensing methods and both supervised and unsupervised learning-based baselines.

Significance. If the empirical outperformance and practical behavior of the nonconvex optimization hold, the work would provide a useful bridge between implicit DIP regularization and interpretable L+S modeling for dynamic MRI, where large fully-sampled training sets are scarce. The convergence result for DIP-parameterized nonconvex ADMM is a modest but positive technical contribution that could be referenced in related inverse-problem papers.

major comments (2)
  1. Convergence Analysis section: the proof establishes that cluster points are critical points of the Lyapunov function and that the algorithm exhibits sufficient descent, but this is a standard result for nonconvex ADMM and supplies no guarantee that the attained critical points avoid fitting noise or aliasing artifacts rather than recovering true spatiotemporal structure. Because the central claim is reliable outperformance on real undersampled scans without overfitting, this gap between theory and generalization is load-bearing and requires either additional analysis or explicit discussion of how the implicit DIP regularization plus explicit L+S penalty prevents such fitting.
  2. Experimental results (numerical results paragraph and associated tables/figures): the abstract asserts consistent outperformance across acceleration factors, yet the manuscript does not appear to include ablations on iteration count, early-stopping criteria, network depth, or initialization sensitivity. Without these, it is difficult to confirm that the practical behavior matches the theoretical critical-point guarantee or that the reported gains are robust rather than dependent on particular hyper-parameter choices.
minor comments (2)
  1. Notation in the method section: the distinction between the two DIP networks (one for low-rank, one for sparse) and the explicit L+S penalty terms could be clarified with a single diagram or table summarizing all variables and their roles in the eADMM updates.
  2. References: several classical L+S MRI papers and recent DIP-MRI works are cited, but the manuscript should explicitly state how the dual-DIP parameterization differs from prior single-network DIP or L+S combinations to better highlight novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight important aspects of the convergence analysis and experimental validation. We address each major comment below and propose targeted revisions to clarify the connection between theory and practical generalization while strengthening the empirical robustness claims.

read point-by-point responses

  1. Referee: Convergence Analysis section: the proof establishes that cluster points are critical points of the Lyapunov function and that the algorithm exhibits sufficient descent, but this is a standard result for nonconvex ADMM and supplies no guarantee that the attained critical points avoid fitting noise or aliasing artifacts rather than recovering true spatiotemporal structure. Because the central claim is reliable outperformance on real undersampled scans without overfitting, this gap between theory and generalization is load-bearing and requires either additional analysis or explicit discussion of how the implicit DIP regularization plus explicit L+S penalty prevents such fitting.

    Authors: We agree that the convergence result follows standard arguments for nonconvex ADMM and does not by itself guarantee avoidance of noise fitting. However, the dual-DIP parameterization is not arbitrary: the low-rank network is constrained to produce a temporally coherent background while the sparse network is encouraged to capture only residual temporal variations, and both are coupled through the explicit nuclear-norm and sparsity penalties. This structural bias, combined with the data-fidelity term, limits the capacity to fit unstructured aliasing. We will add a dedicated paragraph in the convergence section that explicitly discusses this mechanism and its relation to generalization, drawing on known properties of DIP in inverse problems. No new theoretical guarantees will be claimed, but the discussion will better bridge the theory-practice gap. revision: partial

  2. Referee: Experimental results (numerical results paragraph and associated tables/figures): the abstract asserts consistent outperformance across acceleration factors, yet the manuscript does not appear to include ablations on iteration count, early-stopping criteria, network depth, or initialization sensitivity. Without these, it is difficult to confirm that the practical behavior matches the theoretical critical-point guarantee or that the reported gains are robust rather than dependent on particular hyper-parameter choices.

    Authors: The referee correctly notes the absence of these ablations. In the revised manuscript we will add a new subsection (or supplementary section) reporting: (i) reconstruction quality versus iteration count with early-stopping based on the Lyapunov function and data consistency; (ii) results for two different network depths; and (iii) statistics over five random initializations. These experiments will be performed on the same datasets used in the main paper and will be summarized in a table. We believe the added material will confirm that the observed gains are stable within the operating regime used in the original experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation combines established DIP and L+S components with independent convergence analysis and experimental validation.

full rationale

The paper proposes a dual-DIP parameterization of the L+S decomposition for dynamic MRI, jointly optimized via eADMM, and supplies a convergence proof establishing sufficient descent and critical-point convergence for the nonconvex case. These elements are presented as a synthesis of prior DIP and L+S ideas with a new dual-network formulation; the performance claims rest on numerical experiments across acceleration factors rather than any fitted parameter or self-citation that is redefined as a prediction. No equation or step reduces by construction to its own inputs, and the convergence result is a standard nonconvex ADMM argument applied to the proposed objective. The approach is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific free parameters, axioms, or invented entities; the method appears to rest on standard assumptions of DIP implicit regularization and L+S decomposition from prior literature.

pith-pipeline@v0.9.0 · 5806 in / 1212 out tokens · 45549 ms · 2026-05-20T07:39:20.177340+00:00 · methodology

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    Alkhouri, E

    I. Alkhouri, E. Bell, A. Ghosh, S. Liang, R. Wang, and S. Ravishankar. Understanding untrained deep models for inverse problems [from the guest editors]: Algorithms and theory.IEEE Signal Processing Magazine, 43(2):77–92, 2026

    work page 2026

  2. [2]

    E. Bell, S. Liang, I. Alkhouri, and S. Ravishankar. Tada-dip: Input-adaptive deep image prior for one-shot 3d image reconstruction.arXiv preprint arXiv:2512.03962, 2025

    work page arXiv 2025

  3. [3]

    Bernstein, K

    M. Bernstein, K. King, and X. Zhou.Handbook of MRI pulse sequences. Elsevier, 2004

    work page 2004

  4. [4]

    Bolte, S

    J. Bolte, S. Sabach, and M. Teboulle. Proximal alternating linearized minimization for nonconvex and nonsmooth problems.Mathematical Programming, 146(1):459–494, 2014

    work page 2014

  5. [5]

    J.-F. Cai, E. J. Candès, and Z. Shen. A singular value thresholding algorithm for matrix completion.SIAM Journal on optimization, 20(4):1956–1982, 2010

    work page 1956

  6. [6]

    Cascarano, A

    P. Cascarano, A. Sebastiani, M. C. Comes, G. Franchini, and F. Porta. Combining weighted total variation and deep image prior for natural and medical image restoration via admm. In 2021 21st international conference on computational science and its applications (ICCSA), pages 39–46. IEEE, 2021

    work page 2021

  7. [7]

    C. Chen, Y. Liu, P. Schniter, M. Tong, K. Zareba, O. Simonetti, L. Potter, and R. Ahmad. Ocmr (v1.0)–open-access multi-coil k-space dataset for cardiovascular magnetic resonance imaging. arXiv preprint arXiv:2008.03410, 2020

    work page arXiv 2008

  8. [8]

    Cheng and Q

    C. Cheng and Q. Zhou. Mcdip-admm: Overcoming overfitting in dip-based ct reconstruction. Expert Systems, 40(10):e13440, 2023

    work page 2023

  9. [9]

    Gamper, P

    U. Gamper, P. Boesiger, and S. Kozerke. Compressed sensing in dynamic MRI.Magnetic Resonance in Medicine, 59(2):365–373, 2008. 12

    work page 2008

  10. [10]

    Gautam, A

    S. Gautam, A. Li, P. P. Agarwal, A. K. Attili, J. A. Fessler, N. Seiberlich, and S. Ravishankar. Scan-adaptive dynamic mri undersampling using a dictionary of efficiently learned patterns. arXiv preprint arXiv:2602.13984, 2026

    work page arXiv 2026

  11. [11]

    Y. Guan, K. Zhang, Q. Qi, D. Wang, Z. Ke, S. Wang, D. Liang, and Q. Liu. Zero-shot dynamic mri reconstruction with global-to-local diffusion model.NMR in Biomedicine, 38(10):e70128, 2025

    work page 2025

  12. [12]

    J. I. Hamilton, W. Truesdell, M. Galizia, N. Burris, P. Agarwal, and N. Seiberlich. A low-rank deep image prior reconstruction for free-breathing ungated spiral functional cmr at 0.55 t and 1.5 t.Magnetic Resonance Materials in Physics, Biology and Medicine, 36(3):451–464, 2023

    work page 2023

  13. [13]

    Huang, Z

    C. Huang, Z. Wu, and T. Zeng. Edge-guided low-light image enhancement with inertial bregman alternating linearized minimization.arXiv preprint arXiv:2403.01142, 2024

    work page arXiv 2024

  14. [14]

    Huang, Z

    W. Huang, Z. Ke, Z.-X. Cui, J. Cheng, Z. Qiu, S. Jia, L. Ying, Y. Zhu, and D. Liang. Deep low-rank plus sparse network for dynamic mr imaging.Medical Image Analysis, 73:102190, 2021

    work page 2021

  15. [15]

    Huang, V

    W. Huang, V. Spieker, S. Xu, G. Cruz, C. Prieto, J. A. Schnabel, K. Hammernik, T. Kuestner, and D. Rueckert. Subspace implicit neural representations for real-time cardiac cine mr imaging. InInternational Conference on Information Processing in Medical Imaging, pages 168–183. Springer, 2025

    work page 2025

  16. [16]

    H. Jung, K. Sung, K. S. Nayak, E. Y. Kim, and J. C. Ye. k-t focuss: a general compressed sensing framework for high resolution dynamic mri.Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 61(1):103–116, 2009

    work page 2009

  17. [17]

    Z. Ke, W. Huang, Z.-X. Cui, J. Cheng, S. Jia, H. Wang, X. Liu, H. Zheng, L. Ying, Y. Zhu, et al. Learned low-rank priors in dynamic MR imaging.IEEE Transactions on Medical Imaging, 40(12):3698–3710, 2021

    work page 2021

  18. [18]

    Adam: A Method for Stochastic Optimization

    D. Kingma and J. Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  19. [19]

    Knoll, K

    F. Knoll, K. Hammernik, C. Zhang, S. Moeller, T. Pock, D. K. Sodickson, and M. Akcakaya. Deep-learning methods for parallel magnetic resonance imaging reconstruction: A survey of the current approaches, trends, and issues.IEEE signal processing magazine, 37(1):128–140, 2020

    work page 2020

  20. [20]

    J. F. Kunz, S. Ruschke, and R. Heckel. Implicit neural networks with fourier-feature inputs for free-breathing cardiac mri reconstruction.IEEE Transactions on Computational Imaging, 10:1280–1289, 2024

    work page 2024

  21. [21]

    Liang, E

    S. Liang, E. Bell, Q. Qu, R. Wang, and S. Ravishankar. Analysis of deep image prior and exploiting self-guidance for image reconstruction.IEEE Transactions on Computational Imaging, 11:435–451, 2025

    work page 2025

  22. [22]

    Liang and P

    Z. Liang and P. Lauterbur.Principles of magnetic resonance Imaging. SPIE Optical Engineering Press Belllingham, WA, 2000

    work page 2000

  23. [23]

    Lingala and M

    S. Lingala and M. Jacob. Blind compressive sensing dynamic MRI.IEEE Transactions on Medicine Imaging, 32(6):1132–1145, 2013. 13

    work page 2013

  24. [24]

    S. G. Lingala, Y. Hu, E. DiBella, and M. Jacob. Accelerated dynamic mri exploiting sparsity and low-rank structure: kt slr.IEEE transactions on medical imaging, 30(5):1042–1054, 2011

    work page 2011

  25. [25]

    Lustig, D

    M. Lustig, D. Donoho, and J. Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging.Magnetic Resonance in Medicine, 58(6):1182–1195, 2007

    work page 2007

  26. [26]

    Lustig, D

    M. Lustig, D. Donoho, J. Santos, and J. Pauly. Compressed sensing MRI.IEEE signal processing magazine, 25(2):72–82, 2008

    work page 2008

  27. [27]

    L. C. Maximon. Differential and difference equations.Comparison of Methods of Solution, Washington, 2016

    work page 2016

  28. [28]

    Otazo, E

    R. Otazo, E. Candes, and D. K. Sodickson. Low-rank plus sparse matrix decomposition for accelerated dynamic mri with separation of background and dynamic components.Magnetic resonance in medicine, 73(3):1125–1136, 2015

    work page 2015

  29. [29]

    C. Qin, J. Duan, K. Hammernik, J. Schlemper, T. Küstner, R. Botnar, C. Prieto, A. N. Price, J. V. Hajnal, and D. Rueckert. Complementary time-frequency domain networks for dynamic parallel MR image reconstruction.Magnetic Resonance in Medicine, 86(6):3274–3291, 2021

    work page 2021

  30. [30]

    C. Qin, J. Schlemper, J. Caballero, A. N. Price, J. V. Hajnal, and D. Rueckert. Convolutional recurrent neural networks for dynamic mr image reconstruction.IEEE transactions on medical imaging, 38(1):280–290, 2018

    work page 2018

  31. [31]

    Ronneberger, P

    O. Ronneberger, P. Fischer, and T. Brox. U-Net: Convolutional networks for biomedical image segmentation. InInternational Conference on Medicine image computing and computer-assisted intervention, pages 234–241. Springer, 2015

    work page 2015

  32. [32]

    Schlemper, J

    J. Schlemper, J. Caballero, J. V. Hajnal, A. N. Price, and D. Rueckert. A deep cascade of convolutional neural networks for dynamic MR image reconstruction.IEEE transactions on Medical Imaging, 37(2):491–503, 2017

    work page 2017

  33. [33]

    Taylor, B

    A. Taylor, B. Van Scoy, and L. Lessard. Lyapunov functions for first-order methods: Tight automated convergence guarantees. InInternational Conference on Machine Learning, pages 4897–4906. PMLR, 2018

    work page 2018

  34. [34]

    J. Tsao. Ultrafast imaging: principles, pitfalls, solutions, and applications.Journal of Magnetic Resonance Imaging, 32(2):252–266, 2010

    work page 2010

  35. [35]

    J. Tsao, P. Boesiger, and K. P. Pruessmann. k-t BLAST and k-t SENSE: dynamic MRI with high frame rate exploiting spatiotemporal correlations.Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 50(5):1031–1042, 2003

    work page 2003

  36. [36]

    Uecker, P

    M. Uecker, P. Lai, M. Murphy, P. Virtue, M. Elad, J. Pauly, S. Vasanawala, and M. Lustig. ESPIRiT — an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA.Magnetic Resonance in Medicine, 71(3):990–1001, 2014

    work page 2014

  37. [37]

    Ulyanov, A

    D. Ulyanov, A. Vedaldi, and V. Lempitsky. Deep image prior. arxiv.Preprint]. doi, 10, 2017

    work page 2017

  38. [38]

    Ulyanov, A

    D. Ulyanov, A. Vedaldi, and V. Lempitsky. Deep image prior. InProceedings of the IEEE conference on computer vision and pattern recognition, pages 9446–9454, 2018. 14

    work page 2018

  39. [39]

    Vornehm, C

    M. Vornehm, C. Chen, M. A. Sultan, S. M. Arshad, Y. Han, F. Knoll, and R. Ahmad. Multi- dynamic deep image prior for cardiac mri.Magnetic Resonance in Medicine, 94(6):2668–2679, 2025

    work page 2025

  40. [40]

    T. Wu, Z. Du, Z. Li, F.-L. Fan, and T. Zeng. Vdip-tgv: Blind image deconvolution via variational deep image prior empowered by total generalized variation.arXiv preprint arXiv:2310.19477, 2023

    work page arXiv 2023

  41. [41]

    Z. Wu, C. Huang, and T. Zeng. Extrapolated plug-and-play three-operator splitting methods for nonconvex optimization with applications to image restoration.SIAM Journal on Imaging Sciences, 17(2):1145–1181, 2024

    work page 2024

  42. [42]

    J. Yoo, K. H. Jin, H. Gupta, J. Yerly, M. Stuber, and M. Unser. Time-dependent deep image prior for dynamic mri.IEEE Transactions on Medical Imaging, 40(12):3337–3348, 2021. 15 Appendix A Theoretical convergence analysis To ensure the completeness and readability of the proof, we will restate the problem and the assumptions. We analyze the convergence of ...

    work page arXiv 2021