A probabilistic Bayesian approach to recover $$$ {R}_{2\ast } $$$ map and phase images for quantitative susceptibility mapping

Figure S1. Retrospective undersampling: sagittal views of recovered QSM using the least squares approach (LSQ), the l1-norm regularization approach with parameter tuning (L1-T) and L-curve method (L1-L), the proposed AMP-PE approach.

Figure S2. Retrospective undersampling: coronal views of recovered QSM using the least squares approach (LSQ), the l1-norm regularization approach with parameter tuning (L1-T) and L-curve method (L1-L), the proposed AMP-PE approach.

Figure S3. Retrospective undersampling: axial views of recovered local field maps using the least squares approach (LSQ), the l1-norm regularization approach with parameter tuning (L1-T) and L-curve method (L1-L), the proposed AMP-PE approach.

Figure S4. Retrospective undersampling: sagittal views of recovered local field maps using the least squares approach (LSQ), the l1-norm regularization approach with parameter tuning (L1-T) and L-curve method (L1-L), the proposed AMP-PE approach.

Figure S5. Retrospective undersampling: coronal views of recovered local field maps using the least squares approach (LSQ), the l1-norm regularization approach with parameter tuning (L1-T) and L-curve method (L1-L), the proposed AMP-PE approach.

Figure S6. Prospective undersampling: recovered initial magnetization $$$ {\hat{\boldsymbol{z}}}_0 $$$ using the least squares approach (LSQ), the l1-norm regularization approach with the L-curve method (L1-L), the proposed AMP-PE approach.

Figure S7. Prospective undersampling: recovered $$$ {R}_{2\ast } $$$ map $$$ {\hat{\boldsymbol{r}}}_{2\ast } $$$ using the least squares approach (LSQ), the l1-norm regularization approach with the L-curve method (L1-L), the proposed AMP-PE approach.

Figure S8. Prospective undersampling: axial view of recovered QSM using the least squares approach (LSQ), the l1-norm regularization approach with the L-curve method (L1-L), the proposed AMP-PE approach.

Figure S9. Prospective undersampling: sagittal view of recovered QSM using the least squares approach (LSQ), the l1-norm regularization approach with the L-curve method (L1-L), the proposed AMP-PE approach.

Figure S10. Prospective undersampling: coronal view of recovered QSM using the least squares approach (LSQ), the l1-norm regularization approach with the L-curve method (L1-L), the proposed AMP-PE approach.

Figure S11. The Poisson-disk sampling pattern produces a more uniform sampling across k-space than the random sampling pattern.

Figure S12. Comparison of the recovered initial magnetizations $$$ {\hat{\boldsymbol{z}}}_0 $$$ using random sampling and Poisson-disk sampling with the proposed AMP-PE approach. The Poisson-disk sampling pattern leads to lower errors.

Figure S13. Comparison of the recovered $$$ {R}_{2\ast } $$$ map $$$ {\hat{\boldsymbol{r}}}_{2\ast } $$$ using random sampling and Poisson-disk sampling with the proposed AMP-PE approach. The Poisson-disk sampling pattern leads to lower errors.

Figure S14. Comparison of the recovered QSM using random sampling and Poisson-disk sampling with the proposed AMP-PE approach. The Poisson-disk sampling pattern leads to lower errors.

Figure S15. Comparison of the recovered $$$ {\hat{\boldsymbol{z}}}_1 $$$ using Bernoulli–Gaussian-mixture prior and the Laplace prior with the proposed AMP-PE approach. The Laplace prior leads to lower errors.

Table S1. Retrospective undersampling (P1-R): pixel-wise absolute errors of recovered images across different subjects.

Table S2. Retrospective undersampling (P2-R): pixel-wise absolute errors of recovered images across different subjects.

Table S3. Retrospective undersampling: HFEN values of recovered QSM $$$ \hat{\chi} $$$.

Table S4. Retrospective undersampling: normalized absolute errors of recovered local fields.

Table S5. Retrospective undersampling: pixel-wise absolute errors of recovered local fields across different subjects.

Table S6. Prospective undersampling (P1-P): normalized absolute errors of recovered images.

Table S7. Prospective undersampling (P2-P): normalized absolute errors of recovered images.

Table S8. Prospective undersampling (P1-P): pixel-wise absolute errors of recovered images across different subjects.

Table S9. Prospective undersampling (P2-P): pixel-wise absolute errors of recovered images across different subjects.

Table S10. Parameters in the l1-norm regularization approach. For retrospective undersampling, the 1st (S1) and 8th (S8) subjects are used as training data, the rest are used as test data.

Table S11. Retrospective undersampling (P1-R): normalized absolute errors of recovered images from L1 with Exhaustive search (L1-E) and AMP.

Table S12. Retrospective undersampling (P2-R): normalized absolute errors of recovered images from L1 with Exhaustive search (L1-E) and AMP.

Algorithm S1. Recovery of the multi-echo image distribution $$$ {p}_{\mathcal{M}} $$$(z_in | y).

Algorithm S2. Recovery of $$$ {R}_{2\ast } $$$ map $$$ {r}_{2\ast } $$$, initial magnetization z₀ and multi-echo image z_i.