Structured Low-Rank Algorithms: Theory, MR Applications, and Links to Machine Learning

TL;DR

This survey reviews recent structured low-rank matrix completion techniques for continuous domain multidimensional signal recovery, highlighting their theoretical foundations, algorithmic developments, and applications in accelerated magnetic resonance imaging.

Contribution

It introduces a unified low-rank structured matrix framework linking signal sparsity to matrix rank, with performance guarantees and scalable algorithms for MR applications.

Findings

Effective in highly accelerated MRI imaging

Enables calibration-free acquisition methods

Improves artifact correction in MRI

Abstract

In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as a low-rank structured matrix completion, which comes with performance guarantees. We will also review fast algorithms that are comparable in complexity to current compressed sensing methods, which enables the application of the framework to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.