# VS-Net: Variable splitting network for accelerated parallel MRI reconstruction

arXiv: https://arxiv.org/abs/1907.10033 · HTML: https://mlsanity.com/paper/1907.10033/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2019.

## Built on by (most cited first, 30 of 31)

- [Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging](https://mlsanity.com/paper/2203.12215/) (2022)
- [Learning a Model-Driven Variational Network for Deformable Image Registration](https://mlsanity.com/paper/2105.12227/) (2021)
- [One-shot Generative Prior in Hankel-k-space for Parallel Imaging Reconstruction](https://mlsanity.com/paper/2208.07181/) (2022)
- [Transfer Learning Enhanced Generative Adversarial Networks for Multi-Channel MRI Reconstruction](https://mlsanity.com/paper/2105.08175/) (2021)
- [WKGM: Weight-K-space Generative Model for Parallel Imaging Reconstruction](https://mlsanity.com/paper/2205.03883/) (2022)
- [An End-To-End-Trainable Iterative Network Architecture for Accelerated Radial Multi-Coil 2D Cine MR Image Reconstruction](https://mlsanity.com/paper/2102.00783/) (2021)
- [A deep cascade of ensemble of dual domain networks with gradient-based T1 assistance and perceptual refinement for fast MRI reconstruction](https://mlsanity.com/paper/2207.01791/) (2022)
- [An Interpretable MRI Reconstruction Network with Two-grid-cycle Correction and Geometric Prior Distillation](https://mlsanity.com/paper/2205.07062/) (2022)
- [Invertible Sharpening Network for MRI Reconstruction Enhancement](https://mlsanity.com/paper/2206.02838/) (2022)
- [Nest-DGIL: Nesterov-optimized Deep Geometric Incremental Learning for CS Image Reconstruction](https://mlsanity.com/paper/2308.03807/) (2023)
- [Assessment of Data Consistency through Cascades of Independently Recurrent Inference Machines for fast and robust accelerated MRI reconstruction](https://mlsanity.com/paper/2111.15498/) (2021)
- [Fast MRI Reconstruction via Edge Attention](https://mlsanity.com/paper/2304.11400/) (2023)
- [Recovering high-quality FODs from a reduced number of diffusion-weighted images using a model-driven deep learning architecture](https://mlsanity.com/paper/2307.15273/) (2023)
- [PRISTA-Net: Deep Iterative Shrinkage Thresholding Network for Coded Diffraction Patterns Phase Retrieval](https://mlsanity.com/paper/2309.04171/) (2023)
- [Score-based Generative Priors Guided Model-driven Network for MRI Reconstruction](https://mlsanity.com/paper/2405.02958/) (2024)
- [A Densely Interconnected Network for Deep Learning Accelerated MRI](https://mlsanity.com/paper/2207.02073/) (2022)
- [HyperCoil-Recon: A Hypernetwork-based Adaptive Coil Configuration Task Switching Network for MRI Reconstruction](https://mlsanity.com/paper/2308.04821/) (2023)
- [MCU-Net: A Multi-prior Collaborative Deep Unfolding Network with Gates-controlled Spatial Attention for Accelerated MR Image Reconstruction](https://mlsanity.com/paper/2402.03383/) (2024)
- [Joint Edge Optimization Deep Unfolding Network for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2405.05564/) (2024)
- [Time-Embedded Algorithm Unrolling for Computational MRI](https://mlsanity.com/paper/2510.16321/) (2025)
- [SDLFormer: A Sparse and Dense Locality-enhanced Transformer for Accelerated MR Image Reconstruction](https://mlsanity.com/paper/2308.04262/) (2023)
- [NoSENSE: Learned unrolled cardiac MRI reconstruction without explicit sensitivity maps](https://mlsanity.com/paper/2309.15608/) (2023)
- [ATOMMIC: An Advanced Toolbox for Multitask Medical Imaging Consistency to facilitate Artificial Intelligence applications from acquisition to analysis in Magnetic Resonance Imaging](https://mlsanity.com/paper/2404.19665/) (2024)
- [Attention Incorporated Network for Sharing Low-rank, Image and K-space Information during MR Image Reconstruction to Achieve Single Breath-hold Cardiac Cine Imaging](https://mlsanity.com/paper/2407.03034/) (2024)
- [SDUM: A Scalable Deep Unrolled Model for Universal MRI Reconstruction](https://mlsanity.com/paper/2512.17137/) (2025)
- [Accelerated First Order Methods for Variational Imaging](https://mlsanity.com/paper/2110.02813/) (2021)
- [Artificial Intelligence-Based Image Reconstruction in Cardiac Magnetic Resonance](https://mlsanity.com/paper/2209.10298/) (2022)
- [Efficient MRI Parallel Imaging Reconstruction by K-Space Rendering via Generalized Implicit Neural Representation](https://mlsanity.com/paper/2309.06067/) (2023)
- [Exploring Siamese Networks in Self-Supervised Fast MRI Reconstruction](https://mlsanity.com/paper/2501.10851/) (2025)
- [Advancing MRI Reconstruction: A Systematic Review of Deep Learning and Compressed Sensing Integration](https://mlsanity.com/paper/2501.14158/) (2025)

## Abstract (arXiv)

In this work, we propose a deep learning approach for parallel magnetic resonance imaging (MRI) reconstruction, termed a variable splitting network (VS-Net), for an efficient, high-quality reconstruction of undersampled multi-coil MR data. We formulate the generalized parallel compressed sensing reconstruction as an energy minimization problem, for which a variable splitting optimization method is derived. Based on this formulation we propose a novel, end-to-end trainable deep neural network architecture by unrolling the resulting iterative process of such variable splitting scheme. VS-Net is evaluated on complex valued multi-coil knee images for 4-fold and 6-fold acceleration factors. We show that VS-Net outperforms state-of-the-art deep learning reconstruction algorithms, in terms of reconstruction accuracy and perceptual quality. Our code is publicly available at https://github.com/j-duan/VS-Net.