Robust Quantitative Susceptibility Mapping via Approximate Message Passing

Purpose: It is challenging to recover magnetic susceptibility in the presence of phase errors, which may be caused by noise or strong local-susceptibility shifts in cases of brain hemorrhage and calcification. We propose a Bayesian formulation for quantitative susceptibility mapping (QSM) where a customized Gaussian-mixture distribution is used to model the long-tailed noise distribution. Theory: Complex exponential functions of the phase are used as nonlinear measurements. Wavelet coefficients of the susceptibility map are modeled by the Laplace distribution. Measurement noise is modeled by a two-component Gaussian-mixture distribution, where the second component is reserved to model the noise outliers. The susceptibility map and distribution parameters are jointly recovered using approximate message passing (AMP). Methods: The proposed AMP with built-in parameter estimation (AMP-PE) is compared with the state-of-the-art nonlinear L1-QSM and MEDI approaches that adopt the L1-norm and L2-norm data-fidelity terms respectively. They are tested on the simulated and in vivo datasets. Results: On the simulated Sim2Snr1 dataset, AMP-PE achieved the lowest NRMSE and SSIM, MEDI achieved the lowest HFEN. On the in vivo datasets, AMP-PE is more robust and better at preserving structural details and removing streaking artifacts in the hemorrhage cases than L1-QSM and MEDI. Conclusion: By leveraging a customized Gaussian-mixture noise prior, AMP-PE achieves better performance in challenging cases of brain hemorrhage and calcification. It is equipped with built-in parameter estimation, which avoids subjective bias from the usual visual-tuning step of in vivo reconstruction.