Dictionary learning based regularization in quantitative MRI: a nested alternating optimization framework

Abstract In this article we propose a novel regularization method for a class of nonlinear inverse problems that is inspired by an application in quantitative magnetic resonance imaging. The latter is a special instance of a general dynamical image reconstruction technique, wherein a radio-frequency pulse sequence gives rise to a time discrete physics-based mathematical model which acts as a side constraint in our inverse problem. To enhance reconstruction quality, we employ dictionary learning as a data-adaptive regularizer, capturing complex tissue structures beyond handcrafted priors. For computing a solution of the resulting non-convex and non-smooth optimization problem, we alternate between updating the physical parameters of interest via a Levenberg–Marquardt approach and performing several iterations of a dictionary learning algorithm. This process falls under the category of nested alternating optimization schemes. We develop a general overall algorithmic framework whose convergence theory is not directly available in the literature. Global sub-linear and local strong linear convergence in infinite dimensions under certain regularity conditions for the sub-differentials are investigated based on the Kurdyka–Łojasiewicz inequality. Eventually, numerical experiments demonstrate the practical potential and unresolved challenges of the method.