  Purpose: To achieve automatic hyperparameter estimation for the joint
recovery of quantitative MR images, we propose a Bayesian formulation of the
reconstruction problem that incorporates the signal model. Additionally, we
investigate the use of complementary undersampling patterns to determine
optimal undersampling schemes for quantitative MRI.
  Theory: We introduce a novel nonlinear approximate message passing framework,
referred to as ``AMP-PE'', that enables the simultaneous recovery of
distribution parameters and quantitative maps.
  Methods: We employed the variable flip angle multi-echo (VFA-ME) method to
acquire measurements. Both retrospective and prospective undersampling
approaches were utilized to obtain Fourier measurements using variable-density
and Poisson-disk patterns. Furthermore, we extensively explored various
undersampling schemes, incorporating complementary patterns across different
flip angles and/or echo times.
  Results: AMP-PE adopts a model-based joint recovery strategy, it outperforms
the $l_1$-norm minimization approach that follows a decoupled recovery
strategy. A comparison with an existing joint-recovery approach further
demonstrates the advantageous outcomes of AMP-PE. For quantitative $T_1$
mapping using VFA-ME, employing identical k-space sampling patterns across
different echo times produced the best performance. Whereas for $T_2^*$ and
proton density mappings, using complementary sampling patterns across different
flip angles yielded the best performance.
  Conclusion: AMP-PE is equipped with built-in parameter estimation, and works
naturally in clinical settings with varying acquisition protocols and scanners.
It also achieves improved performance by combining information from the MR
signal model and the sparse prior on images.
