Matlab implementation of the algorithm for reconstructing images from highly undersampled magnetic resonance imaging (MRI) data as described in detail in this publication.
To reproduce the figures from the above publication, simply run the script MAIN_script.m.
The code uses Jeffrey Fessler’s image reconstruction toolbox (http://web.eecs.umich.edu/~fessler/code/) for the nonuniform FFT computation (NUFFT).
The image reconstruction problem in MRI can be described as
A (X) = Y
where Y is the measured undersampled k-space data, X is the (unknown) full image series (in real space) to be reconstructed, and A is a linear operator consisting of the calculation of the Fourier transform (FT) of X and the subsequent evaluation at the k-space trajectory’s positions. Since the data is sub-sampled, Y is usually smaller than X, making the problem ill-posed and not solvable by simply inverting the reconstruction problem. Instead, we try to estimate X in an iterative fashion by applying the following prior knowledge:
- We assume that the CEST image series X can be approximated by a low-rank matrix (see e.g. this paper)
- We drive the solution X towards zero in areas where no sample is present (limited object support (LS), see for example this paper)
The following pseudo-code outlines the algorithm.
