vSHARP: variable Splitting Half-quadratic ADMM algorithm for
  Reconstruction of inverse-Problems
George Yiasemis Nikita Moriakov Jan-Jakob Sonke Jonas Teuwen
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 supervised and self-supervised 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 the proposed model by applying it to the task
of accelerated Parallel MRI Reconstruction on two distinct datasets. We present
a comparative analysis of our experimental results with state-of-the-art
approaches, highlighting the superior performance of vSHARP.