A probabilistic Bayesian approach to recover map and phase images for quantitative susceptibility mapping

Undersampling is used to reduce the scan time for high-resolution three-dimensional magnetic resonance imaging. In order to achieve better image quality and avoid manual parameter tuning, we propose a probabilistic Bayesian approach to recover $$ map and phase images for quantitative susceptibility mapping (QSM), while allowing automatic parameter estimation from undersampled data.

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.

Undersampling takes place in the y-z plane of k-space according to the Poisson-disk pattern. Retrospective undersampling is performed to evaluate the performances of different reconstruction approaches, prospective undersampling is performed to demonstrate the feasibility of undersampling in practice.

The proposed AMP with parameter estimation (AMP-PE) approach successfully recovers $$ maps and phase images for QSM across various undersampling rates. It is more computationally efficient, and performs better than the state-of-the-art $$ -norm regularization (L1) approach in general, except a few cases where the L1 approach performs as well as AMP-PE.

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.