  Within the context of upcoming full-sky lensing surveys, the edge-preserving
non- linear algorithm Aski is presented. Using the framework of Maximum A
Posteriori inversion, it aims at recovering the full-sky convergence map from
surveys with masks. It proceeds in two steps: CCD images of crowded galactic
fields are deblurred using automated edge-preserving deconvolution; once the
reduced shear is estimated, the convergence map is also inverted via an edge-
preserving method. For the deblurring, it is found that when the observed field
is crowded, this gain can be quite significant for realistic ground-based
surveys when both positivity and edge-preserving penalties are imposed during
the iterative deconvolution. For the convergence inversion, the quality of the
reconstruction is investigated on noisy maps derived from the horizon N-body
simulation, with and without Galactic cuts, and quantified using one-point
statistics, power spectra, cluster counts, peak patches and the skeleton. It is
found that the reconstruction is able to interpolate and extrapolate within the
Galactic cuts/non-uniform noise; its sharpness-preserving penalization avoids
strong biasing near the clusters of the map; it reconstructs well the shape of
the PDF as traced by its skewness and kurtosis; the geometry and topology of
the reconstructed map is close to the initial map as traced by the peak patch
distribution and the skeleton's differential length; the two-points statistics
of the recovered map is consistent with the corresponding smoothed version of
the initial map; the distribution of point sources is also consistent with the
corresponding smoothing, with a significant improvement when edge preserving
prior is applied. The contamination of B-modes when realistic Galactic cuts are
present is also investigated. Leakage mainly occurs on large scales.
