  Strong gravitational lensing offers a wealth of astrophysical information on
the background source it affects, provided the lensed source can be
reconstructed as if it was seen in the absence of lensing. In the present work,
we illustrate how sparse optimisation can address the problem. As a first step
towards a full free-form lens modelling technique, we consider linear inversion
of the lensed source under sparse regularisation and joint deblending from the
lens light profile. The method is based on morphological component analysis,
assuming a known mass model. We show with numerical experiments that
representing the lens and source light using an undecimated wavelet basis
allows us to reconstruct the source and to separate it from the foreground lens
at the same time. Both the source and lens light have a non-analytic form,
allowing for the flexibility needed in the inversion to represent arbitrarily
small and complex luminous structures in the lens and source. in addition,
sparse regularisation avoids over-fitting the data and does not require the use
of any adaptive mesh or pixel grid. As a consequence, our reconstructed sources
can be represented on a grid of very small pixels. Sparse regularisation in the
wavelet domain also allows for automated computation of the regularisation
parameter, thus minimising the impact of arbitrary choice of initial
parameters. Our inversion technique for a fixed mass distribution can be
incorporated in future lens modelling technique iterating over the lens mass
parameters. The python package corresponding to the algorithms described in
this article can be downloaded via the github platform at
https://github.com/herjy/SLIT.
