Accelerated Weighted ℓ1-Minimization for MRI Reconstruction Under Tight Frames in Complex Domain

Abstract

We propose an improvement of the projected fast iterative soft-thresholding algorithm (pFISTA) and smoothing FISTA (SFISTA) to achieve faster convergence and improved reconstruction accuracy. The pFISTA addresses the problem of compressed sensing magnetic resonance imaging (CS-MRI) reconstruction under tight frames and considers standard $\ell_{1}$ norm minimization. The $\ell_{1} -$norm weighs each component in a sparse vector equally. However, this is restrictive. We employ the weighted $\ell_{1} -$regularizer, defined over a complex-domain as the sparsity-promoting function in CS-MRI reconstruction. The weighted $\ell_{1} -$regularizer assigns different weights to the components in a sparse vector to improve upon reconstruction accuracy. The optimization objective in CS-MRI is a real-valued function defined over a complex-domain and is therefore not holomorphic. We derive an algorithm, namely, projected weighted iterative soft-thresholding algorithm (pWISTA) based on Wirtinger calculus to solve the weighted $\ell_{1} -$regularized CS-MRI reconstruction under tight frames. We show that the proximal operator for the weighted $\ell_{1}$ regularizer over a complex-domain is the soft-thresholding operator, but with a different threshold for each component. We also incorporate Nesterov’s momentum into the pWISTA update to obtain the projected weighted fast iterative soft-thresholding algorithm (pWFISTA), which result in accelerated optimization as shown by the experimental results.