The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-414

“Low-Rank Modeling of Local k-Space Neighborhoods (LORAKS): Implementation and Examples for Reproducible Research”

by Justin P. Haldar

April 2014

Our recent work has introduced a new signal processing framework that enables the recovery of magnetic resonance (MR) images from undersampled and/or noisy Fourier data, based on the assumptions that the true image has smooth phase and/or limited spatial support. This framework, named LORAKS, relies on embedding the Fourier data from such images into high-dimensional Hankel-like matrices, which we showed would have approximately low rank when the phase/support assumptions were satisfied. The fact that these matrices have low rank implies that they have relatively few degrees of freedom relative to their number of entries, and hence can potentially be recovered from subsampled data. In contrast to conventional reconstruction methods that rely on assumptions of smooth image phase and/or limited spatial support, LORAKS does not place strict requirements on the sampling scheme used for data acquisition, and high-quality reconstructions can be obtained from both highly-structured and highly-unstructured Fourier sampling schemes. In contrast to other recent MR image reconstruction methods based on low-rank modeling, LORAKS can be used with relatively low-dimensional single-channel, single-contrast, single-timepoint MR imaging data. We have also previously demonstrated that LORAKS has distinct features and advantages relative to sparsity-based MR image reconstruction methods (which have been receiving substantial attention over the past several years), which suggests that phase and support constraints can be used in combination with sparsity constraints for even better performance.

In the spirit of reproducible research, this technical report (and corresponding supplementary material available for download at provides a MATLAB implementation of a LORAKS-based reconstruction algorithm proposed in our previous work, and presents several application examples that are not found in the existing literature (including an example outside the context of MR imaging).

To download the report in PDF format click here: USC-SIPI-414.pdf (2.0Mb)