We study the morphology of convergence maps by perturbatively reconstructing their Minkowski Functionals (MFs). We present a systematics study using a set of three generalised skew-spectra as a function of source redshift and smoothing angular scale. Using an approach based on pseudo-$S_{ell}$s (PSL) we show how these spectra will allow reconstruction of MFs in the presence of an arbitrary mask and inhomogeneous noise in an unbiased way. Our theoretical predictions are based on a recently introduced fitting function to the bispectrum. We compare our results against state-of-the art numerical simulations and find an excellent agreement. The reconstruction can be carried out in a controlled manner as a function of angular harmonics $ell$ and source redshift $z_s$ which allows for a greater handle on any possible sources of non-Gaussianity. Our method has the advantage of estimating the topology of convergence maps directly using shear data. We also study weak lensing convergence maps inferred from Cosmic Microwave Background (CMB) observations; and we find that, though less significant at low redshift, the post-Born corrections play an important role in any modelling of the non-Gaussianity of convergence maps at higher redshift. We also study the cross-correlations of estimates from different tomographic bins.
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