An efficient auxiliary variable method for quantification of spin density, R2* decay and field inhomogeneity maps in magnetic resonance imaging
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Quantification of spin density, $R_2^*$ decay and off-resonance frequency maps is very important in some applications of magnetic resonance imaging (MRI). To reconstruct these parameter maps, a time-varying model such as mono-exponentials must be used to represent the signal from each voxel. When only a single-shot trajectory is adopted, the underlying reconstruction problem is significantly nonlinear and therefore requires an iterative algorithm. The regularized trust region method previously proposed to address this problem is stable but lacks speed. In this paper, we propose a novel auxiliary variable method that is very efficient in solving the underlying optimization problem. This method introduces an auxiliary variable in the spatial-temporal domain that separates the data fidelity term and the structure fidelity term. The algorithm then alternately optimizes the data fidelity and the structure fidelity to reach the solution. The data fidelity optimization has a closed-form solution and can be solved very efficiently. The structure fidelity optimization fits the exponential model with the auxiliary variable and can also be rapidly computed. Some preliminary comparisons between the auxiliary variable method and the trust region method show that the new method is 10 times faster than the trust region method at a reasonable reconstruction precision.