  We consider several detection situations where, under the alternative
hypothesis, the signal admits a low complexity model and, under both the null
and the alternative hypotheses, the distribution of the background noise is
{unknown}. We present several detection strategies for such cases, whose design
relies on exogenous or on endogenous data. These testing procedures have been
inspired by and are applied to two specific problems in Astrophysics, namely
the detection of exoplanets from radial velocity curves and of distant galaxies
in hyperspectral datacubes.
