Dynamic MRI Reconstruction via Weighted Tensor Nuclear Norm Regularizer

Abstract

In this paper, we propose a novel multi-dimensional reconstruction method based on the low-rank plus sparse tensor ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ) decomposition model to reconstruct dynamic magnetic resonance imaging (dMRI). The multi-dimensional reconstruction method is formulated using a non-convex alternating direction method of multipliers (ADMM), where the weighted tensor nuclear norm (WTNN) and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm are used to enforce the low-rank in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> and the sparsity in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> , respectively. In particular, the weights used in the WTNN are sorted in a non-descending order, and we obtain a closed-form optimal solution of the WTNN minimization problem. The theoretical properties provided guarantee the weak convergence of our reconstruction method. In addition, a fast inexact reconstruction method is proposed to increase imaging speed and efficiency. Experimental results demonstrate that both of our reconstruction methods can achieve higher reconstruction quality than the state-of-the-art reconstruction methods.