pith. sign in

arxiv: 2503.09559 · v3 · submitted 2025-03-12 · 📡 eess.IV · cs.CV· cs.LG· eess.SP

Interlaced R2D2 DNN Series for Scalable Non-Cartesian MRI with Sensitivity Self-calibration

Shijie Chen , Yiwei Chen , Amir Aghabiglou , Motahare Torki , Chao Tang , Ruud B. van Heeswijk , Yves Wiaux This is my paper

Pith reviewed 2026-05-23 00:14 UTC · model grok-4.3

classification 📡 eess.IV cs.CVcs.LGeess.SP
keywords MRI reconstructionnon-Cartesian MRIsensitivity self-calibrationdeep learningradial samplingundersampled k-spaceR2D2iterative reconstruction
0
0 comments X

The pith

An interlaced pair of R2D2 DNN series jointly reconstructs MR images and self-calibrates coil sensitivity maps from undersampled non-Cartesian k-space.

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

The paper proposes iR2D2 to overcome scalability limits in deep learning MRI reconstruction by treating the problem as an iterative residual estimation process inspired by matching pursuit. It introduces an interlaced architecture where two R2D2 series alternate updates to refine both the image and the sensitivity maps, avoiding the need for precomputed maps that can be inaccurate with undersampling. This adaptive method with error-controlled stopping allows high-fidelity imaging at scale for radial sampling trajectories. Readers would care because accurate self-calibration could improve diagnostic quality in accelerated MRI scans without additional calibration time.

Core claim

iR2D2 extends the R2D2 paradigm by introducing a bespoke interlaced architecture that alternates between two R2D2 DNN series to jointly self-calibrate sensitivity maps and form the MR image, operating as an adaptive solver governed by an error-controlled update condition that enforces sufficient residual energy descent.

What carries the argument

The interlaced architecture of two alternating R2D2 DNN series for joint image formation and sensitivity map self-calibration.

If this is right

  • iR2D2 significantly improves upon R2D2 in reconstruction quality.
  • It outperforms state-of-the-art benchmarks on both simulated and real undersampled radial k-space data.
  • The method scales to large numbers of coils or higher-dimensional imaging where embedding NUFFT in backprop is impractical.
  • It delivers high-fidelity images with corrected sensitivity profiles.
  • The adaptive error-controlled updates enable dynamic iteration without fixed forward passes.

Where Pith is reading between the lines

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

  • This joint optimization approach might reduce the need for separate sensitivity calibration scans in clinical MRI workflows.
  • The residual-based iterative nature could be applied to other non-Cartesian sampling patterns or modalities with similar operator inaccuracies.
  • Future extensions might incorporate the interlacing into other iterative algorithms beyond R2D2.
  • It raises the possibility of fully data-driven operator calibration in inverse problems where the measurement model is partially unknown.

Load-bearing premise

The two interlaced R2D2 series remain consistent with each other during joint optimization rather than diverging into incompatible image and sensitivity estimates.

What would settle it

A test case in which the self-calibrated sensitivity maps from iR2D2 produce higher reconstruction error than maps estimated independently from a fully sampled calibration region would falsify the joint optimization benefit.

Figures

Figures reproduced from arXiv: 2503.09559 by Amir Aghabiglou, Chao Tang, Motahare Torki, Ruud B. van Heeswijk, Shijie Chen, Yiwei Chen, Yves Wiaux.

Figure 1
Figure 1. Figure 1: Illustration of the R2D2 algorithm. The image iterates [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the U-WDSR model. (a) illustrates the U-WDSR architecture, and (b) depicts its WDSR layer. The WDSR residual body (in green [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the non-Cartesian MRI problem. Panel (a) displays [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantitative reconstruction results from simulated experiments. The PSNR and SSIM of R2D2(U-Net) and R2D2(U-WDSR) (solid lines) are reported [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visual reconstruction results for one of test inverse problems of the simulated experiments, with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visual reconstruction results for one of test inverse problems of the simulated experiments, with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Training scalability comparison. Training times are reported against [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DR-specific quantitative reconstruction results from simulated exper [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Visual reconstruction results at AF = 4 and AF = 8 for the 15-coil real data experiment. The back-projected image with 192 sampling spokes (denoted as AF = 1) is reported for reference on the top left panel. The following panels show the back-projected images as well as reconstructions with R2D2(U-Net) at iteration i = 5, R2D2(U-WDSR) at iteration i = 3, and benchmark algorithms, both at AF = 4 (48 spokes)… view at source ↗
Figure 10
Figure 10. Figure 10: Quantitative evaluation of data fidelity for the 15-coil real data [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

We introduce interlaced R2D2 (iR2D2), a DNN series paradigm for scalable image reconstruction from accelerated non-Cartesian k-space acquisitions in MRI with sensitivity map self-calibration. While unrolled DNN architectures provide robust image formation, embedding non-uniform fast Fourier transform operators within the backpropagation graph becomes impractical to train at large scale, e.g., in 2D MRI with a large number of coils, or for higher-dimensional imaging. To address this scalability challenge, we leverage the R2D2 paradigm as a learned version of the Matching Pursuit algorithm that was recently introduced in radio astronomy for fast large-scale Fourier imaging. R2D2's reconstruction is formed as a series of residual images iteratively estimated as outputs of DNN modules taking the previous iteration's data residual as input. Specific to MRI, precomputed sensitivity maps derived from undersampled data can yield an inaccurate measurement operator, which may adversely affect the performance of iterative algorithms such as R2D2. Thus, we extend the R2D2 framework to iR2D2 by introducing a bespoke interlaced architecture that alternates between two R2D2 DNN series to jointly self-calibrate sensitivity maps and form the MR image. We further enhance iR2D2 to operate as an adaptive solver governed by an error-controlled update condition that enforces a sufficient residual energy descent, a dynamic capability fundamentally incompatible with the predefined forward passes of unrolled architectures. Extensive experiments in simulation and on real data, targeting undersampled radial k-space sampling, demonstrate that iR2D2 significantly improves upon R2D2 and outperforms state-of-the-art benchmarks, delivering scalable, high-fidelity imaging with corrected sensitivity profiles.

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

1 major / 2 minor

Summary. The manuscript introduces interlaced R2D2 (iR2D2), extending the R2D2 DNN series paradigm (originally from radio astronomy) to scalable non-Cartesian MRI reconstruction. It proposes a bespoke interlaced architecture that alternates between two independent R2D2 DNN series to jointly self-calibrate sensitivity maps and reconstruct the image from undersampled radial k-space data, together with an adaptive solver using an error-controlled update condition for sufficient residual descent. Experiments in simulation and on real data are reported to show significant improvements over R2D2 and state-of-the-art benchmarks, with corrected sensitivity profiles.

Significance. If the central claims hold, the work offers a practical scalable alternative to unrolled networks for large-coil or higher-dimensional MRI by avoiding NUFFT operators inside the backpropagation graph. The adaptive error-controlled solver is a genuine strength, as it is fundamentally incompatible with fixed unrolled architectures. Self-calibration of sensitivities from undersampled data addresses a common practical limitation, and the reported gains on radial sampling could have impact if the joint optimization is shown to be robust.

major comments (1)
  1. The section introducing the interlaced architecture (described in the abstract and methods as alternating between two R2D2 DNN series for joint self-calibration): no explicit coupling mechanism—such as a joint loss term, shared parameters across series, or consistency penalty on the forward operator using the estimated sensitivities—is specified. Without this, alternation alone does not prevent the two series from producing inconsistent image and sensitivity estimates, which is load-bearing for the claims of 'corrected sensitivity profiles' and improved fidelity over R2D2.
minor comments (2)
  1. Abstract: the claim of 'significantly improves upon R2D2 and outperforms state-of-the-art benchmarks' should be supported by at least one quantitative metric (e.g., SSIM or NRMSE with error bars) rather than qualitative language alone.
  2. The adaptive solver is presented as a dynamic capability; the precise form of the error-controlled update condition (e.g., the threshold or descent criterion) should be given explicitly, preferably as an equation or pseudocode.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance. We address the single major comment below.

read point-by-point responses

  1. Referee: The section introducing the interlaced architecture (described in the abstract and methods as alternating between two R2D2 DNN series for joint self-calibration): no explicit coupling mechanism—such as a joint loss term, shared parameters across series, or consistency penalty on the forward operator using the estimated sensitivities—is specified. Without this, alternation alone does not prevent the two series from producing inconsistent image and sensitivity estimates, which is load-bearing for the claims of 'corrected sensitivity profiles' and improved fidelity over R2D2.

    Authors: We agree that the current description of the interlaced architecture would benefit from a more explicit statement of the coupling mechanism. In the revised manuscript we will expand the methods section to include the precise mathematical formulation: the output sensitivity maps from the first R2D2 series are immediately inserted into the forward operator that computes the data residual for the second series, and vice versa at each alternation step; this residual is then passed as input to the next DNN module. We will also add a short paragraph clarifying that no joint loss, shared weights, or explicit consistency penalty is employed—the coupling is realized solely through this residual-update pathway. These additions will directly address the concern about potential inconsistency between the two series. revision: yes

Circularity Check

0 steps flagged

No significant circularity in architectural extension or claims

full rationale

The paper presents iR2D2 as a novel interlaced DNN series architecture that alternates two R2D2 modules for joint image reconstruction and sensitivity self-calibration in non-Cartesian MRI. This is described as an engineering extension of the prior R2D2 paradigm (cited from radio astronomy) combined with an error-controlled adaptive solver. No derivation chain, equations, or load-bearing steps reduce a claimed prediction or result to an input quantity by construction, self-definition, or fitted-parameter renaming. The central claims rest on the bespoke architecture design and empirical results on radial sampling data rather than any mathematical reduction to prior outputs. Minor self-citation to the R2D2 source is present but not load-bearing for the new interlacing mechanism or performance claims. The method is self-contained as an independent architectural proposal with external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the R2D2 residual-learning structure transfers from radio astronomy to MRI and that alternating two such series can stably calibrate sensitivities; no free parameters, axioms, or invented entities are explicitly listed in the abstract.

pith-pipeline@v0.9.0 · 5885 in / 1246 out tokens · 32196 ms · 2026-05-23T00:14:37.576955+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 4 internal anchors

  1. [1]

    Compressed sensing imaging techniques for radio interferometry,

    Y . Wiaux, L. Jacques, G. Puy, A. M. Scaife, and P. Vandergheynst, “Compressed sensing imaging techniques for radio interferometry,”Mon. Not. R. Astron. Soc. , vol. 395, no. 3, pp. 1733–1742, 2009

    work page 2009

  2. [2]

    Overview of functional magnetic resonance imaging,

    G. H. Glover, “Overview of functional magnetic resonance imaging,” Neurosurg. Clin., vol. 22, no. 2, pp. 133–139, 2011

    work page 2011

  3. [3]

    Liang and P

    Z.-P. Liang and P. C. Lauterbur, Principles of magnetic resonance imaging. SPIE Optical Engineering Press Bellingham, 2000

    work page 2000

  4. [4]

    Un- dersampled MRI reconstruction with patch-based directional wavelets,

    X. Qu, D. Guo, B. Ning, Y . Hou, Y . Lin, S. Cai, and Z. Chen, “Un- dersampled MRI reconstruction with patch-based directional wavelets,” Magn. Reson. Imaging , vol. 30, no. 7, pp. 964–977, 2012

    work page 2012

  5. [5]

    Sparse MRI: The application of compressed sensing for rapid MR imaging,

    M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med., vol. 58, no. 6, pp. 1182–1195, 2007

    work page 2007

  6. [6]

    Dynamic MRI using smoothness regularization on manifolds (SToRM),

    S. Poddar and M. Jacob, “Dynamic MRI using smoothness regularization on manifolds (SToRM),” IEEE Trans. Med. Imaging , vol. 35, no. 4, pp. 1106–1115, 2015

    work page 2015

  7. [7]

    4D flow imaging with MRI,

    Z. Stankovic, B. D. Allen, J. Garcia, K. B. Jarvis, and M. Markl, “4D flow imaging with MRI,” Cardiovasc. Diagn. Ther ., vol. 4, no. 2, p. 173, 2014

    work page 2014

  8. [8]

    U-net: Convolutional networks for biomedical image segmentation,

    O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in Proc. MICCAI. Springer, Oct. 2015, pp. Part III 18, 234–241

    work page 2015

  9. [9]

    Deep learning for undersampled MRI reconstruction,

    C. M. Hyun, H. P. Kim, S. M. Lee, S. Lee, and J. K. Seo, “Deep learning for undersampled MRI reconstruction,” Phys. Med. Biol., vol. 63, no. 13, p. 135007, 2018

    work page 2018

  10. [10]

    k-space deep learning for accelerated MRI,

    Y . Han, L. Sunwoo, and J. C. Ye, “k-space deep learning for accelerated MRI,” IEEE Trans. Med. Imaging , vol. 39, no. 2, pp. 377–386, 2019

    work page 2019

  11. [11]

    KIKI-net: cross-domain convolutional neural networks for reconstructing under- sampled magnetic resonance images,

    T. Eo, Y . Jun, T. Kim, J. Jang, H.-J. Lee, and D. Hwang, “KIKI-net: cross-domain convolutional neural networks for reconstructing under- sampled magnetic resonance images,” Magn. Reson. Med., vol. 80, no. 5, pp. 2188–2201, 2018

    work page 2018

  12. [12]

    Learned primal-dual reconstruction,

    J. Adler and O. ¨Oktem, “Learned primal-dual reconstruction,” IEEE Trans. Med. Imaging , vol. 37, no. 6, pp. 1322–1332, 2018

    work page 2018

  13. [13]

    Plug-and-play methods for magnetic res- onance imaging: Using denoisers for image recovery,

    R. Ahmad, C. A. Bouman, G. T. Buzzard, S. Chan, S. Liu, E. T. Reehorst, and P. Schniter, “Plug-and-play methods for magnetic res- onance imaging: Using denoisers for image recovery,” IEEE Signal Process. Mag., vol. 37, no. 1, pp. 105–116, 2020

    work page 2020

  14. [14]

    NC-PDNet: A density-compensated unrolled network for 2D and 3D non-Cartesian MRI reconstruction,

    Z. Ramzi, G. Chaithya, J.-L. Starck, and P. Ciuciu, “NC-PDNet: A density-compensated unrolled network for 2D and 3D non-Cartesian MRI reconstruction,” IEEE Trans. Med. Imaging , vol. 41, no. 7, pp. 1625–1638, 2022

    work page 2022

  15. [15]

    Decomposed diffusion sampler for accelerating large-scale inverse problems,

    H. Chung, S. Lee, and J. C. Ye, “Decomposed diffusion sampler for accelerating large-scale inverse problems,” in Proc. ICLR, May 2024

    work page 2024

  16. [16]

    Matching pursuits with time-frequency dictionaries,

    S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. , vol. 41, no. 12, pp. 3397– 3415, 1993

    work page 1993

  17. [17]

    Deep network series for large-scale high-dynamic range imaging,

    A. Aghabiglou, M. Terris, A. Jackson, and Y . Wiaux, “Deep network series for large-scale high-dynamic range imaging,” in Proc. IEEE ICASSP. IEEE, Jun. 2023, pp. 1–5

    work page 2023

  18. [18]

    The R2D2 deep neural network series paradigm for fast precision imaging in radio astronomy,

    A. Aghabiglou, C. San Chu, A. Dabbech, and Y . Wiaux, “The R2D2 deep neural network series paradigm for fast precision imaging in radio astronomy,” Astrophys. J. Suppl. Ser ., vol. 273, no. 1, p. 3, 2024

    work page 2024

  19. [19]

    Scal- able non-cartesian magnetic resonance imaging with R2D2,

    Y . Chen, C. Tang, A. Aghabiglou, C. S. Chu, and Y . Wiaux, “Scal- able non-cartesian magnetic resonance imaging with R2D2,” in Proc. EUSIPCO, Aug. 2024, pp. 1511–1515

    work page 2024

  20. [20]

    To- wards a robust R2D2 paradigm for radio-interferometric imaging: revis- iting DNN training and architecture,

    A. Aghabiglou, C. S. Chu, C. Tang, A. Dabbech, and Y . Wiaux, “To- wards a robust R2D2 paradigm for radio-interferometric imaging: revis- iting DNN training and architecture,” arXiv preprint arXiv:2503.02554 , 2025

    work page arXiv 2025

  21. [21]

    Non-cartesian parallel imaging reconstruction,

    K. L. Wright, J. I. Hamilton, M. A. Griswold, V . Gulani, and N. Seiber- lich, “Non-cartesian parallel imaging reconstruction,” J. Magn. Reson. Imaging, vol. 40, no. 5, pp. 1022–1040, 2014

    work page 2014

  22. [22]

    Nonuniform fast Fourier transforms using min-max interpolation,

    J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. , vol. 51, no. 2, pp. 560–574, 2003

    work page 2003

  23. [23]

    Sampling density compensation in MRI: rationale and an iterative numerical solution,

    J. G. Pipe and P. Menon, “Sampling density compensation in MRI: rationale and an iterative numerical solution,” Magn. Reson. Med. , vol. 41, no. 1, pp. 179–186, 1999

    work page 1999

  24. [24]

    SENSE: sensitivity encoding for fast MRI,

    K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, “SENSE: sensitivity encoding for fast MRI,” Magn. Reson. Med. , vol. 42, no. 5, pp. 952–962, 1999

    work page 1999

  25. [25]

    Generalized autocalibrating partially parallel acquisitions (GRAPPA),

    M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V . Jellus, J. Wang, B. Kiefer, and A. Haase, “Generalized autocalibrating partially parallel acquisitions (GRAPPA),” Magn. Reson. Med. , vol. 47, no. 6, pp. 1202–1210, 2002

    work page 2002

  26. [26]

    ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA,

    M. Uecker, P. Lai, M. J. Murphy, P. Virtue, M. Elad, J. M. Pauly, S. S. Vasanawala, and M. Lustig, “ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA,” Magn. Reson. Med. , vol. 71, no. 3, pp. 990–1001, 2014

    work page 2014

  27. [27]

    Deep-learning methods for parallel magnetic resonance imaging reconstruction: A survey of the current approaches, trends, and issues,

    F. Knoll, K. Hammernik, C. Zhang, S. Moeller, T. Pock, D. K. Sodick- son, and M. Akcakaya, “Deep-learning methods for parallel magnetic resonance imaging reconstruction: A survey of the current approaches, trends, and issues,” IEEE Signal Process. Mag. , vol. 37, no. 1, pp. 128– 140, 2020

    work page 2020

  28. [28]

    A fast wavelet-based reconstruction method for magnetic resonance imaging,

    M. Guerquin-Kern, M. Haberlin, K. P. Pruessmann, and M. Unser, “A fast wavelet-based reconstruction method for magnetic resonance imaging,” IEEE Trans. Med. Imaging , vol. 30, no. 9, pp. 1649–1660, 2011

    work page 2011

  29. [29]

    Undersampled radial MRI with multiple coils. iterative image reconstruction using a total variation constraint,

    K. T. Block, M. Uecker, and J. Frahm, “Undersampled radial MRI with multiple coils. iterative image reconstruction using a total variation constraint,” Magn. Reson. Med. , vol. 57, no. 6, pp. 1086–1098, 2007

    work page 2007

  30. [30]

    fastMRI: An Open Dataset and Benchmarks for Accelerated MRI

    J. Zbontar, F. Knoll, A. Sriram, T. Murrell, Z. Huang, M. J. Muckley, A. Defazio, R. Stern, P. Johnson, M. Bruno et al. , “fastMRI: An open dataset and benchmarks for accelerated MRI,” arXiv preprint arXiv:1811.08839, 2018

    work page internal anchor Pith review arXiv 2018

  31. [31]

    End-to-end variational networks for accelerated MRI reconstruction,

    A. Sriram, J. Zbontar, T. Murrell, A. Defazio, C. L. Zitnick, N. Yakubova, F. Knoll, and P. Johnson, “End-to-end variational networks for accelerated MRI reconstruction,” in Proc. MICCAI. Springer, Oct. 2020, pp. Part II 23, 64–73

    work page 2020

  32. [32]

    Wide Activation for Efficient and Accurate Image Super-Resolution

    J. Yu, Y . Fan, J. Yang, N. Xu, Z. Wang, X. Wang, and T. Huang, “Wide activation for efficient and accurate image super-resolution,” arXiv preprint arXiv:1808.08718 , 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  33. [33]

    Practical blind image denoising via Swin- Conv-UNet and data synthesis,

    K. Zhang, Y . Li, J. Liang, J. Cao, Y . Zhang, H. Tang, D.-P. Fan, R. Timofte, and L. V . Gool, “Practical blind image denoising via Swin- Conv-UNet and data synthesis,” Mach. Intell. Res. , vol. 20, no. 6, pp. 822–836, 2023

    work page 2023

  34. [34]

    Scalable precision wide-field imaging in radio interferometry: I. uSARA validated on ASKAP data,

    A. G. Wilber, A. Dabbech, A. Jackson, and Y . Wiaux, “Scalable precision wide-field imaging in radio interferometry: I. uSARA validated on ASKAP data,” Mon. Not. R. Astron. Soc. , vol. 522, no. 4, pp. 5558– 5575, 2023

    work page 2023

  35. [35]

    Image quality metrics: PSNR vs. SSIM,

    A. Hore and D. Ziou, “Image quality metrics: PSNR vs. SSIM,” in Proc. ICPR. IEEE, Aug. 2010, pp. 2366–2369

    work page 2010

  36. [36]

    Denoising diffusion implicit models,

    J. Song, C. Meng, and S. Ermon, “Denoising diffusion implicit models,” in Proc. ICLR, May 2021

    work page 2021

  37. [37]

    Pytorch: An imperative style, high-performance deep learning library,

    A. Paszke, S. Gross, F. Massa, Lerer et al. , “Pytorch: An imperative style, high-performance deep learning library,” Proc. NIPS , pp. 8024– 8035, Dec. 2019

    work page 2019

  38. [38]

    Adam: A Method for Stochastic Optimization

    D. P. 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

  39. [39]

    Diffusion models beat GANs on image synthesis,

    P. Dhariwal and A. Nichol, “Diffusion models beat GANs on image synthesis,” Proc. NIPS, vol. 34, pp. 8780–8794, Dec. 2021

    work page 2021

  40. [40]

    Decoupled Weight Decay Regularization

    I. Loshchilov, F. Hutter et al. , “Fixing weight decay regularization in adam,” arXiv preprint arXiv:1711.05101 , vol. 5, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  41. [41]

    TorchKbNufft: A high-level, hardware-agnostic non-uniform fast Fourier transform,

    M. J. Muckley, R. Stern, T. Murrell, and F. Knoll, “TorchKbNufft: A high-level, hardware-agnostic non-uniform fast Fourier transform,” in ISMRM Workshop, Jan. 2020

    work page 2020

  42. [42]

    Domain gener- alization: A survey,

    K. Zhou, Z. Liu, Y . Qiao, T. Xiang, and C. C. Loy, “Domain gener- alization: A survey,” IEEE Trans. Pattern Anal. Mach. Intell. , vol. 45, no. 4, pp. 4396–4415, 2022

    work page 2022