  We propose a novel numerical approach to separate multiple tissue
compartments in image voxels and to estimate quantitatively their nuclear
magnetic resonance (NMR) properties and mixture fractions, given magnetic
resonance fingerprinting (MRF) measurements. The number of tissues, their types
or quantitative properties are not a-priori known, but the image is assumed to
be composed of sparse compartments with linearly mixed Bloch magnetisation
responses within voxels. Fine-grid discretisation of the multi-dimensional NMR
properties creates large and highly coherent MRF dictionaries that can
challenge scalability and precision of the numerical methods for (discrete)
sparse approximation. To overcome these issues, we propose an off-the-grid
approach equipped with an extended notion of the sparse group lasso
regularisation for sparse approximation using continuous (non-discretised)
Bloch response models. Further, the nonlinear and non-analytical Bloch
responses are approximated by a neural network, enabling efficient
back-propagation of the gradients through the proposed algorithm. Tested on
simulated and in-vivo healthy brain MRF data, we demonstrate effectiveness of
the proposed scheme compared to the baseline multicompartment MRF methods.
