FOREX Press I. J. of Electrical & Electronics Research
Support Open Access
  • Home/
  • IJEER-140223

Research Article |

A Learned Three Operator Splitting Network for Accelerated MRI Reconstruction Using Complex-Valued Deep Unrolling

Author(s): David Muigai1*, Elijah Mwangi2, Henry Kiragu3

Published In : International Journal of Electrical and Electronics Research (IJEER)        Volume 14, Issue 2

Publisher : FOREX Publication

Published : 30 June 2026

e-ISSN : 2347-470X

Page(s) : 467-478

DOI: https://doi.org/10.37391/IJEER.140223

Abstract

Magnetic Resonance Imaging (MRI) is the cornerstone of modern medical diagnosis and research, providing high spatial resolution visualization of the anatomical and functional information of the body’s internal structure in a non-ionizing, non-carcinogenic and non-invasive manner. Despite these superior properties, the MRI data acquisition process is inherently slow, limiting this technique in scenarios where time is crucial. Consequently, accelerating MRI through k-space under-sampling at sub-Nyquist rates and reconstructing high-quality images from incomplete measurements has emerged as an active area of research in the past few decades. This paper proposes a learned three-operator splitting algorithm implemented in an unrolled complex-valued deep neural network architecture for accelerated Magnetic Resonance (MR) image reconstruction. Objective results evaluated using Peak Signal–to–Noise Ratio (PSNR), Structural Similarity Index (SSIM) and Normalized Root–Mean–Squared Error (NRMSE) at ×2 to ×5 acceleration show superior reconstruction performance of the proposed network compared to other state–of–the–art learned algorithms at comparable reconstruction time. Subjective results show that the proposed network reconstructed images visually similar to ground truth images. The proposed network has the potential to enable real-time MRI applications with high-quality images.

Keywords: Accelerated MRI Reconstruction, Compressed Sensing, Complex-Valued Deep Learning, Learned Three Three-Operator Splitting Network.




David Muigai, Department of Electrical Engineering, Pan African University Institute of Basic Sciences, Technology and Innovation, Nairobi, Kenya; Email: david.waweru@egerton.ac.ke

Elijah Mwangi, Faculty of Engineering, University of Nairobi, Kenya; Email: elijah.mwangi@uonbi.ac.ke

Henry Kiragu,Department of Electrical and Telecommunication Engineering, Multimedia University of Kenya, Nairobi, Kenya; Email: hkiragu@mmu.ac.ke

    [1] Y. Lian, Z. Liu, J. Wang, and S. Lu, “Spatial-frequency aware zero-centric residual unfolding network for MRI reconstruction,” Magnetic Resonance Imaging, vol. 117, pp. 110334, Jan. 2025, doi:10.1016/j.mri.2025.110334.
    [2] W. Bian and Y. K. Tamilselvam, “A review of Optimization-Based Deep Learning Models for MRI Reconstruction,” AppliedMath, vol. 4, no. 3, pp. 1098–1127, Sep. 2024, doi: 10.3390/appliedmath4030059.
    [3] Y. Zhang, H. Gui, N. Yang, and Y. Hu, “JotlasNet: Joint tensor low-rank and attention-based sparse unrolling network for accelerating dynamic MRI,” Magnetic Resonance Imaging, vol. 118, pp. 110337, Jan. 2025, doi: 10.1016/j.mri.2025.110337.
    [4] B. Vasudeva, P. Deora, S. Bhattacharya, and P. M. Pradhan, “Compressed Sensing MRI Reconstruction with Co-VeGAN: Complex-Valued Generative Adversarial Network,” 2022 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pp. 1779–1788, Jan. 2022, doi: 10.1109/wacv51458.2022.00184.
    [5] M. I. Doneva, M. Akcakaya, and C. Prieto, Magnetic resonance image reconstruction: Theory, Methods and Applications. Academic Press,2022.
    [6] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, pp. 1289–1306, Apr. 2006, doi: 10.1109/tit.2006.871582.
    [7] M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, pp. 1182–1195, Oct. 2007, doi: 10.1002/mrm.21391.
    [8] S. Ma, W. Yin, Y. Zhang and A. Chakraborty, "An efficient algorithm for compressed MR imaging using total variation and wavelets," 2008 IEEE Conference on Computer Vision and Pattern Recognition, Anchorage, AK, USA, 2008, pp. 1-8, doi: 10.1109/CVPR.2008.4587391.
    [9] J. C. Ye, “Compressed sensing MRI: a review from signal processing perspective,” BMC Biomedical Engineering, vol. 1, no. 1, pp. 8, Mar. 2019, doi: 10.1186/s42490-019-0006-z.
    [10] N. Fuin, A Bustin, T Küstner et al., “A multi-scale variational neural network for accelerating motion-compensated whole-heart 3D coronary MR angiography,” Magnetic Resonance Imaging, vol. 70, pp. 155–167, Apr. 2020, doi: 10.1016/j.mri.2020.04.007.
    [11] Md. B. Hossain, R. K. Shinde, S. Oh, K.-C. Kwon, and N. Kim, “A systematic review and identification of the challenges of deep learning techniques for undersampled magnetic resonance image reconstruction,” Sensors, vol. 24, no. 3, pp. 753, Jan. 2024, doi: 10.3390/s24030753.
    [12] S. Kiryu, H. Akai, K. Yasaka et al., “Clinical impact of deep learning reconstruction in MRI,” Radiographics, vol. 43, no. 6, pp. e220133, May 2023, doi: 10.1148/rg.220133.
    [13] P. Guo, Y. Mei, J. Zhou, S. Jiang, and V. M. Patel, “ReconFormer: Accelerated MRI Reconstruction using Recurrent Transformer,” IEEE Transactions on Medical Imaging, vol. 43, no. 1, pp. 582–593, Sep. 2023, doi: 10.1109/tmi.2023.3314747.
    [14] X. Zhao, T. Yang, B. Li, and X. Zhang, “SwinGAN: A dual-domain Swin Transformer-based generative adversarial network for MRI reconstruction,” Computers in Biology and Medicine, vol. 153, Dec. 2022, doi: 10.1016/j.compbiomed.2022.106513.
    [15] G. Ongie, A. Jalal, C. A. Metzler, R. G. Baraniuk, A. G. Dimakis, and R. Willett, “Deep learning techniques for inverse problems in imaging,” IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 1, pp. 39–56, May 2020, doi: 10.1109/jsait.2020.2991563.
    [16] G. Yang, S. Yu, H. Dong et al., “DAGAN: Deep De-Aliasing Generative Adversarial Networks for Fast Compressed Sensing MRI Reconstruction,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 1310–1321, Dec. 2017, doi: 10.1109/tmi.2017.2785879.
    [17] Y. Han, L. Sunwoo, and J. C. Ye, K -Space Deep Learning for Accelerated MRI,” IEEE Transactions on Medical Imaging, vol. 39, no. 2, pp. 377–386, Jul. 2019, doi: 10.1109/tmi.2019.2927101.
    [18] Y. Yang, J. Sun, H. Li and Z. Xu. ADMM-Net: a deep learning approach for compressive sensing MRI. In: Advances in neural information processing systems; 2017. pp. 10–8.
    [19] 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, vol. 38, no. 1, pp. 280–290, Aug. 2018, doi: 10.1109/tmi.2018.2863670.
    [20] H. K. Aggarwal, M. P. Mani, and M. Jacob, “MODL: Model-Based Deep Learning Architecture for Inverse Problems,” IEEE Transactions on Medical Imaging, vol. 38, no. 2, pp. 394–405, Aug. 2018, doi: 10.1109/tmi.2018.2865356.
    [21] C. M. Sandino, P. Lai, SS. Vasanawala, JY Cheng. Accelerating cardiac cine MRI using a deep learning-based ESPIRiT reconstruction. Magnetic Resonance in Medicine, 2020, doi:10.1002/mrm.28420.
    [22] Y. Zhang, X. Li, W. Li, and Y. Hu, “Deep unrolling shrinkage network for dynamic MR imaging,” 2023 IEEE International Conference on Image Processing (ICIP), pp. 1145–1149, Sep. 2023, doi: 10.1109/icip49359.2023.10223077.
    [23] M. Vornehm, J. Wetzl, D. Giese et al., “CineVN: Variational network reconstruction for rapid functional cardiac cine MRI,” Magnetic Resonance in Medicine, vol. 93, no. 1, pp. 138–150, Aug. 2024, doi: 10.1002/mrm.30260.
    [24] 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, vol. 37, no. 2, pp. 491–503, Feb. 2018, doi: 10.1109/tmi.2017.2760978.
    [25] J. Duan, J. Schlemper, C. Qin et al., “VS-Net: Variable splitting network for accelerated parallel MRI reconstruction,” arXiv (Cornell University), Jan. 2019, doi: 10.48550/arxiv.1907.10033.
    [26] B. Xin, TS Phan, L Axel, et al. Learned half-quadratic splitting network for magnetic resonance image reconstruction. arXiv 2021.
    [27] Z. Fabian, B Tinaz, M Soltanolkotabi, “HUMUS-net: hybrid unrolled multi-scale network architecture for accelerated MRI reconstruction” Adv Neural Inf Process Syst, 2022;35:25306–19.
    [28] Y. Zhang, P. Li, and Y. Hu, “T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstruction,” Computers in Biology and Medicine, vol. 170, pp. 108034, Jan. 2024, doi: 10.1016/j.compbiomed.2024.108034.
    [29] D. Davis and W. Yin, “A Three-Operator Splitting Scheme and its Optimization Applications,” Set-Valued and Variational Analysis, vol. 25, no. 4, pp. 829–858, Jun. 2017, doi: 10.1007/s11228-017-0421-z
    [30] H. Wang, M. Fazlyab, S. Chen, and V. M. Preciado, Eds. Robust Convergence Analysis of Three-Operator Splitting, 2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, 2019. doi: 10.1109/ALLERTON.2019.8919695
    [31] H. Zhao, O. Gallo, I. Frosio, and J. Kautz, “Loss functions for image restoration with neural networks,” IEEE Transactions on Computational Imaging, vol. 3, pp. 47–57, Dec. 2016, doi: 10.1109/tci.2016.2644865.
    [32] D.P. Kingma and J.L. Ba, “Adam: A Method for Stochastic Optimization,” International Conference on Learning Representations (ICLR), Dec 2014.
    [33] J. Zbontar, F. Knoll, A. Sriram et al., “FastMRI: an open dataset and benchmarks for Accelerated MRI,” arXiv (Cornell University), Jan. 2018, doi: 10.48550/arxiv.1811.08839.
    [34] C. Chen, Y. Liu, P. Schniter et al., “OCMR (V1.0) --Open-Access Multi-Coil K-Space Dataset for Cardiovascular Magnetic Resonance Imaging,” arXiv (Cornell University), Jan. 2020, doi: 10.48550/arxiv.2008.03410.
    [35] 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, vol. 37, no. 2, pp. 491–503, Oct. 2017.
    [36] J. Zhang and B. Ghanem, "ISTA-Net: Interpretable Optimization-Inspired Deep Network for Image Compressive Sensing," 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Salt Lake City, UT, USA, 2018, pp. 1828-1837, doi: 10.1109/CVPR.2018.00196.
    [37] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, and F. Knoll, “Learning a variational network for reconstruction of accelerated MRI data,” Magnetic Resonance in Medicine, vol. 79, no. 6, pp. 3055–3071, Nov. 2017.
    [38] J. Adler and O. Oktem, “Learned Primal-Dual reconstruction,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 1322–1332, Jan. 2018, doi: 10.1109/tmi.2018.2799231.

David Muigai, Elijah Mwangi and Henry Kiragu (2026), A Learned Three Operator Splitting Network for Accelerated MRI Reconstruction Using Complex-Valued Deep Unrolling . IJEER 14(2), 467-478. DOI: 10.37391/IJEER.140223.