A Probabilistic Bayesian Approach to Recover $R_2^*$ map and Phase Images for Quantitative Susceptibility Mapping

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.