Generalized Fast Iteratively Reweighted Soft-Thresholding Algorithm for Sparse Coding Under Tight Frames in the Complex-Domain

We present a new method for fast magnetic resonance image (MRI) reconstruction in the complex-domain under tight frames. We propose a generalized problem formulation that allows for different weight-update strategies for iteratively reweighted ℓ1-minimization under tight frames. Further, we impose sufficient conditions on the function of the weights that leads to the reweighting strategy, which follows the interpretation originally given by Candès et al, but is more efficient than theirs. Since the objective function in complex-domain compressive sensing MRI (CS-MRI) reconstruction problem is nonholomorphic, we resort to Wirtinger calculus for deriving the update strategies. We develop an algorithm called generalized iteratively reweighted soft-thresholding algorithm (GIRSTA) and its fast variant, namely, generalized fast iteratively reweighted soft-thresholding algorithm (GFIRSTA). We provide convergence guarantees for GIRSTA and empirical convergence results for GFIRSTA. Our experiments show a remarkable performance of the proposed algorithms for complex-domain CS-MRI reconstruction considering both random sampling and radial sampling strategies. GFIRSTA outperforms state-of-the-art techniques in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM).