  Purpose: Undersampling is used to reduce the scan time for high-resolution 3D
magnetic resonance imaging. In order to achieve better image quality and avoid
manual parameter tuning, we propose a probabilistic Bayesian approach to
recover $R_2^*$ map and phase images for quantitative susceptibility mapping
(QSM), while allowing automatic parameter estimation from undersampled data.
  Theory: Sparse prior on the wavelet coefficients of images is interpreted
from a Bayesian perspective as sparsity-promoting distribution. A novel
nonlinear approximate message passing (AMP) framework that incorporates a
mono-exponential decay model is proposed. The parameters are treated as unknown
variables and jointly estimated with image wavelet coefficients.
  Results: The proposed AMP with parameter estimation (AMP-PE) approach
successfully recovers $R_2^*$ maps and phase images for QSM across various
undersampling rates. It is more computationally efficient, and performs better
than the state-of-the-art $l_1$-norm regularization (L1) approach in general,
except a few cases where the L1 approach performs as well as AMP-PE.
  Conclusion: AMP-PE achieves better performance by drawing information from
both the sparse prior and the mono-exponential decay model. It does not require
parameter tuning, and works with a clinical, prospective undersampling scheme
where parameter tuning is often impossible or difficult due to the lack of
ground-truth image.
