  Weak gravitational lensing provides a unique method to map directly the dark
matter in the Universe. The majority of lensing analyses uses the two-point
statistics of the cosmic shear field to constrain the cosmological model
yielding degeneracies, such as that between sigma_8 and Omega_M respectively
the r.m.s. of the mass fluctuations at a scale of 8 Mpc/h and the matter
density parameter both at z = 0. However, the two-point statistics only measure
the Gaussian properties of the field and the weak lensing field is
non-Gaussian. It has been shown that the estimation of non-Gaussian statistics
on weak lensing data can improve the constraints on cosmological parameters. In
this paper, we systematically compare a wide range of non-Gaussian estimators
in order to determine which one provides tighter constraints on the
cosmological parameters. These statistical methods include skewness, kurtosis
and the Higher Criticism test in several sparse representations such as wavelet
and curvelet; as well as the bispectrum, peak counting and a new introduced
statistic called Wavelet Peak Counting (WPC). Comparisons based on sparse
representations show that the wavelet transform is the most sensitive to
non-Gaussian cosmological structures. It appears also that the best statistic
for non-Gaussian characterization in weak lensing mass maps is the WPC.
Finally, we show that the sigma_8 -Omega_m degeneracy could be even better
broken if the WPC estimation is performed on weak lensing mass maps filtered by
the wavelet method, MRLens.
