  Magnetorelaxometry imaging is a novel tool for quantitative determination of
the spatial distribution of magnetic nanoparticle inside an organism. The use
of multiple excitation patterns has been demonstrated to significantly improve
spatial resolution. However, increasing the number of excitation patterns is
considerably more time consuming, because several sequential measurements have
to be performed. In this paper, we use compressed sensing in combination with
sparse recovery to reduce the total measurement time and to improve spatial
resolution. For image reconstruction, we propose using the Douglas-Rachford
splitting algorithm applied to the sparse Tikhonov functional including a
positivity constraint. Our numerical experiments demonstrate that the resulting
algorithm is capable to accurately recover the magnetic nanoparticle
distribution from a small number of activation patterns. For example, our
algorithm applied with 10 activations yields half the reconstruction error of
quadratic Tikhonov regularization applied with 50 activations, for a tumor-like
phantom.
