Dictionary learning based regularization in quantitative MRI: A nested alternating optimization framework
Authors
- Dong, Guozhi
ORCID: 0000-0002-9674-6143 - Hintermüller, Michael
ORCID: 0000-0001-9471-2479 - Sirotenko, Clemens
2020 Mathematics Subject Classification
- 35R30 49J52 49J53 92C55 68T05
Keywords
- Quantitative MRI, quantitative image reconstruction, regularization, variational methods, machine learning, non-convex and non-smooth optimization
DOI
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 (MRI). It is a special instance of a general dynamical image reconstruction problem with an underlying time discrete physical model. Our regularization strategy is based on dictionary learning, a method that has been proven to be effective in classical MRI. To address 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 such algorithmic framework, integrated with the Levenberg-Marquardt method, of which the convergence theory is not directly available from 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.
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