vSHARP: variable Splitting Half-quadratic Admm algorithm for Reconstruction of inverse-Problems

TL;DR

vSHARP tackles ill-posed inverse problems in medical imaging by unrolling a Half-Quadratic Variable Splitting (HQVS) formulation within an ADMM framework. It combines a differentiable data-consistency gradient descent with DL-based denoisers (e.g., U-Net) and a learned Lagrange-multiplier initializer, with trainable penalties and step sizes to adapt optimization dynamics. The approach is rigorously evaluated on static and dynamic accelerated MRI across multiple datasets, showing superior reconstruction quality (high SSIM and pSNR, low NMSE) and robust performance under high acceleration factors, with an ablation study highlighting the method’s adaptability and trade-offs. Overall, vSHARP establishes a principled, adaptable pipeline that blends optimization theory with DL components to deliver high-fidelity reconstructions in clinically relevant MRI inverse problems, while offering practical guidance for deployment and future extensions to other imaging modalities.

Abstract

Medical Imaging (MI) tasks, such as accelerated parallel Magnetic Resonance Imaging (MRI), often involve reconstructing an image from noisy or incomplete measurements. This amounts to solving ill-posed inverse problems, where a satisfactory closed-form analytical solution is not available. Traditional methods such as Compressed Sensing (CS) in MRI reconstruction can be time-consuming or prone to obtaining low-fidelity images. Recently, a plethora of Deep Learning (DL) approaches have demonstrated superior performance in inverse-problem solving, surpassing conventional methods. In this study, we propose vSHARP (variable Splitting Half-quadratic ADMM algorithm for Reconstruction of inverse Problems), a novel DL-based method for solving ill-posed inverse problems arising in MI. vSHARP utilizes the Half-Quadratic Variable Splitting method and employs the Alternating Direction Method of Multipliers (ADMM) to unroll the optimization process. For data consistency, vSHARP unrolls a differentiable gradient descent process in the image domain, while a DL-based denoiser, such as a U-Net architecture, is applied to enhance image quality. vSHARP also employs a dilated-convolution DL-based model to predict the Lagrange multipliers for the ADMM initialization. We evaluate vSHARP on tasks of accelerated parallel MRI Reconstruction using two distinct datasets and on accelerated parallel dynamic MRI Reconstruction using another dataset. Our comparative analysis with state-of-the-art methods demonstrates the superior performance of vSHARP in these applications.

Figures (9)

• Figure 1: Graphical Illustration of vSHARP. Utilizing acquired measurements $\Tilde{\mathbf{y}}$ from the acquisition domain $\mathcal{Y}$, initial estimations are made for the image $\mathbf{x}_0$ and the auxiliary variable $\mathbf{z}_0$ (in the image domain $\mathcal{X}$. The Lagrange Multiplier Initializer Network $\mathcal{G}_{\boldsymbol{\psi}}$Yiasemis_2022_CVPR generates an initialization $\mathbf{u}_0$ for the Lagrange multipliers, using $\mathbf{x}_0$ as input. These images, namely $\mathbf{z}_0$, $\mathbf{x}_0$, and $\mathbf{u}_0$, serve as the initial quantities in the unrolled ADMM algorithm spanning $T$ iterations. During each iteration, a DL-based denoiser $\mathcal{R}_{\boldsymbol{\theta}_t}$ (in this case, a U-Net) is employed to refine $\mathbf{z}_t$, while $\mathbf{x}_t$ undergoes optimization through a differentiable Data Consistency via Gradient Descent (DCGD) algorithm. The DCGD algorithm itself is further unrolled over $T_{\mathbf{x}}$ iterations. As the result of these operations, vSHARP produces $\mathbf{x}_T$ as the predicted reconstructed image.
• Figure 2: Sample reconstructions from the Calgary-Campinas test set.
• Figure 3: Sample reconstructions from the fastMRI prostate test set.
• Figure 4: Sample reconstructions (cropped region of interest) from the CMRxRecon cardiac cine test set in the dynamic MRI reconstruction experiments. Only time-steps $t=1, 4, 7, 10$ (out of 12) are shown.
• Figure S1: Evaluation metrics distribution on the Calgary-Campinas test set (box plots).