  In the field of quantitative imaging, the image information at a pixel or
voxel in an underlying domain entails crucial information about the imaged
matter. This is particularly important in medical imaging applications, such as
quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of
biophysical parameters can characterize the imaged tissue and thus lead to more
accurate diagnoses. Such quantitative values can also be useful in subsequent,
automatized classification tasks in order to discriminate normal from abnormal
tissue, for instance. The accurate reconstruction of these quantitative maps is
typically achieved by solving two coupled inverse problems which involve a
(forward) measurement operator, typically ill-posed, and a physical process
that links the wanted quantitative parameters to the reconstructed qualitative
image, given some underlying measurement data. In this review, by considering
qMRI as a prototypical application, we provide a mathematically-oriented
overview on how data-driven approaches can be employed in these inverse
problems eventually improving the reconstruction of the associated quantitative
maps.
