Fast TVL1-L2 MR image reconstruction using variable splitting and accelerated alternating direction method with adaptive restart

This paper presents a fast algorithm for magnetic resonance (MR) image reconstruction from undersampled k-space measurements. The underlying MR image reconstruction is formulated as solving a TVL1-L2 minimization problem whose objective function consists of total variation (TV) regularizer, wavelet-based ℓ
₁
-norm regularizer and ℓ
₂
data fidelity. Our approach is based on a variable splitting strategy and an accelerated alternating direction method of multiplier (ADMM) with restart. This paper shows that our proposed algorithm is fast and efficient for solving the TVL1-L2 MR image reconstruction problem. More precisely, a variable splitting method is used to split the variable into three variables and obtain an equivalent constrained optimization formulation, which is then addressed with an accelerated ADMM with adaptive restart. This ADMM algorithm is acceleration because the next iterate is computed by employing two previous computed iterates, and the restart rule is employed to enforce monotonicity and convergence in solving weakly convex TVL1-L2 optimization. Moreover thanks to intrinsic spatial-frequency encoding in MRI data, the inverse of regularized Hessian matrix can perform efficiently by exploiting fast Fourier transform (FFT) and fast wavelet transform (or tight frame). Experimental examples also demonstrate that the proposed algorithm is fast and efficient compared to the classical ADMM in TVL1-L2 MR image reconstruction.