Multi-coil MRI by analytic continuation

TL;DR

This paper introduces a novel MRI reconstruction method using analytic continuation, transforming a 2-D limited-data problem into solvable 1-D inverse problems, with demonstrated superior performance and stability analysis.

Contribution

The paper develops a new 2-D multi-coil MRI reconstruction technique based on analytic continuation and Fredholm equations, with comprehensive stability analysis and comparison of sampling schemes.

Findings

Achieves high structural similarity in reconstructed images.

Performs well under sub-sampling and motion corruption.

Provides stability insights for different sampling strategies.

Abstract

We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of $\textbf{k}$-space (Fourier space) are lines parallel to either $k_1$ or $k_2$ (i.e., the $\textbf{k}$-space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (``slice-by-slice" imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the $\textbf{k}$-space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different $\textbf{k}$-space sub-sampling schemes (e.g., random vs uniform accelerated sampling).