  Quantitative magnetic resonance imaging (qMRI) is concerned with estimating
(in physical units) values of magnetic and tissue parameters e.g., relaxation
times $T_1$, $T_2$, or proton density $\rho$. Recently in [Ma et al., Nature,
2013], Magnetic Resonance Fingerprinting (MRF) was introduced as a technique
being capable of simultaneously recovering such quantitative parameters by
using a two step procedure: (i) given a probe, a series of magnetization maps
are computed and then (ii) matched to (quantitative) parameters with the help
of a pre-computed dictionary which is related to the Bloch manifold. In this
paper, we first put MRF and its variants into a perspective with optimization
and inverse problems to gain mathematical insights concerning identifiability
of parameters under noise and interpretation in terms of optimizers. Motivated
by the fact that the Bloch manifold is non-convex and that the accuracy of the
MRF-type algorithms is limited by the "discretization size" of the dictionary,
a novel physics-based method for qMRI is proposed. In contrast to the
conventional two step method, our model is dictionary-free and is rather
governed by a single non-linear equation, which is studied analytically. This
non-linear equation is efficiently solved via robustified Newton-type methods.
The effectiveness of the new method for noisy and undersampled data is shown
both analytically and via extensive numerical examples for which also
improvement over MRF and its variants is documented.
