Dictionary Learning Based Regularization in Quantitative MRI: A Nested Alternating Optimization Framework

TL;DR

This paper introduces a novel dictionary learning-based regularization framework for nonlinear inverse problems in quantitative MRI, employing nested alternating optimization with convergence analysis and demonstrating promising numerical results.

Contribution

It proposes a new nested alternating optimization method using dictionary learning for regularization in qMRI inverse problems, with convergence guarantees in infinite dimensions.

Findings

Demonstrates improved reconstruction quality in qMRI.

Provides convergence analysis for the proposed optimization scheme.

Shows practical potential through numerical experiments.

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 (qMRI). 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-Lojasiewicz inequality. Eventually, numerical experiments demonstrate the practical potential and unresolved challenges of the method.