  Objective: We propose a method for the reconstruction of parameter-maps in
Quantitative Magnetic Resonance Imaging (QMRI).
  Methods: Because different quantitative parameter-maps differ from each other
in terms of local features, we propose a method where the employed dictionary
learning (DL) and sparse coding (SC) algorithms automatically estimate the
optimal dictionary-size and sparsity level separately for each parameter-map.
We evaluated the method on a $T_1$-mapping QMRI problem in the brain using the
BrainWeb data as well as in-vivo brain images acquired on an ultra-high field
7T scanner. We compared it to a model-based acceleration for parameter mapping
(MAP) approach, other sparsity-based methods using total variation (TV),
Wavelets (Wl) and Shearlets (Sh), and to a method which uses DL and SC to
reconstruct qualitative images, followed by a non-linear (DL+Fit).
  Results: Our algorithm surpasses MAP, TV, Wl and Sh in terms of RMSE and
PSNR. It yields better or comparable results to DL+Fit by additionally
significantly accelerating the reconstruction by a factor of approximately
seven.
  Conclusion: The proposed method outperforms the reported methods of
comparison and yields accurate $T_1$-maps. Although presented for $T_1$-mapping
in the brain, our method's structure is general and thus most probably also
applicable for the the reconstruction of other quantitative parameters in other
organs.
  Significance: From a clinical perspective, the obtained $T_1$-maps could be
utilized to differentiate between healthy subjects and patients with
Alzheimer's disease. From a technical perspective, the proposed unsupervised
method could be employed to obtain ground-truth data for the development of
data-driven methods based on supervised learning.+
