  Stage IV lensing surveys promise to make available an unprecedented amount of
excellent data which will represent a huge leap in terms of both quantity and
quality. This will open the way to the use of novel tools, which go beyond the
standard second order statistics probing the high order properties of the
convergence field. We discuss the use of Minkowski Functionals (MFs) as
complementary probes to increase the lensing Figure of Merit (FoM), for a
survey made out of a wide total area $A_{\rm{tot}}$ imaged at a limiting
magnitude $\rm{mag_{W}}$ containing a subset of area $A_{\rm{deep}}$ where
observations are pushed to a deeper limiting magnitude $\rm{mag_{D}}$. We
present an updated procedure to match the theoretically predicted MFs to the
measured ones, taking into account the impact of map reconstruction from noisy
shear data. We validate this renewed method against simulated data sets with
different source redshift distributions and total number density, setting these
quantities in accordance with the depth of the survey. We can then rely on a
Fisher matrix analysis to forecast the improvement in the FoM due to the joint
use of shear tomography and MFs under different assumptions on
$(A_{\rm{tot}},\,A_{\rm{deep}},\,\rm{mag_{D}})$, and the prior on the MFs
nuisance parameters. It turns out that MFs can provide a valuable help in
increasing the FoM of the lensing survey, provided the nuisance parameters are
known with a non negligible precision. What is actually more interesting is the
possibility to compensate for the loss of FoM due to a cut in the multipole
range probed by shear tomography, which makes the results more robust against
uncertainties in the modeling of nonlinearities. This makes MFs a promising
tool to both increase the FoM and make the constraints on the cosmological
parameters less affected by theoretical systematic effects.
