# American Institute of Mathematical Sciences

November  2013, 7(4): 1215-1233. doi: 10.3934/ipi.2013.7.1215

## Hybrid regularization for MRI reconstruction with static field inhomogeneity correction

 1 Department of Mathematics, University of California, Los Angeles, CA 90095, United States 2 Department of Chemistry and Biochemistry, University of California, Los Angeles, CA, 90095, United States

Received  June 2012 Revised  January 2013 Published  November 2013

Rapid acquisition of magnetic resonance (MR) images via reconstruction from undersampled $k$-space data has the potential to greatly decrease MRI scan time on existing medical hardware. To this end, iterative image reconstruction based on the technique of compressed sensing has become the method choice for many researchers [1]. However, while conventional compressed sensing relies on random measurements from a discrete Fourier transform, actual MR scans often suffer from off-resonance effects and thus generate data by way of a non-Fourier operator [2]. Correcting for these effects requires that one employs more sophisticated image reconstruction methods and introduces computational bottlenecks that are not encountered in standard compressed sensing.
In this work, we demonstrate how one may accelerate the convergence of algorithms for solving the image reconstruction problem, $$\label{eq:caneq} (1) \underset{\rho}{argmin} J(\rho) subject to A \rho = s$$ by opting for a regularization of the form: $$\label{eq:hybrid} (2) J(\rho) = | \nabla \rho| + \nu | F \rho |$$ when $F$ is a tight frame and $A$ is only approximately a Fourier transform. In our experiments, reconstructing field-corrected MR images with the hybrid regularization of 2 provides a speedup of roughly one order of magnitude when compared with an approach based solely on total-variation and may produce higher quality images than an approach based solely on tight frames.
Citation: Ryan Compton, Stanley Osher, Louis-S. Bouchard. Hybrid regularization for MRI reconstruction with static field inhomogeneity correction. Inverse Problems & Imaging, 2013, 7 (4) : 1215-1233. doi: 10.3934/ipi.2013.7.1215
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