  Magnetic resonance (MR)-$T_2^*$ mapping is widely used to study hemorrhage,
calcification and iron deposition in various clinical applications, it provides
a direct and precise mapping of desired contrast in the tissue. However, the
long acquisition time required by conventional 3D high-resolution $T_2^*$
mapping method causes discomfort to patients and introduces motion artifacts to
reconstructed images, which limits its wider applicability. In this paper we
address this issue by performing $T_2^*$ mapping from undersampled data using
compressive sensing (CS). We formulate the reconstruction as a nonconvex
problem that can be decomposed into two subproblems. They can be solved either
separately via the standard approach or jointly via the alternating direction
method of multipliers (ADMM). Compared to previous CS-based approaches that
only apply sparse regularization on the spin density $\boldsymbol X_0$ and the
relaxation rate $\boldsymbol R_2^*$, our formulation enforces additional sparse
priors on the $T_2^*$-weighted images at multiple echoes to improve the
reconstruction performance. We performed convergence analysis of the proposed
algorithm, evaluated its performance on in vivo data, and studied the effects
of different sampling schemes. Experimental results showed that the proposed
joint-recovery approach generally outperforms the state-of-the-art method,
especially in the low-sampling rate regime, making it a preferred choice to
perform fast 3D $T_2^*$ mapping in practice. The framework adopted in this work
can be easily extended to other problems arising from MR or other imaging
modalities with non-linearly coupled variables.
