  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.
