  We combine two Gaussianization techniques - Wavelet Non-Linear Wiener Filter
(WNLWF) and density reconstruction - to quantify the recovery of Fisher
information that is lost in the gravitational collapse. We compute a
displacement fields, in analogy with the Zel'dovich approximation, and apply a
Wavelet Non-Linear Wiener Filter that decomposes the reconstructed density
fields into a Gaussian and a non-Gaussian component. From a series of 200
realizations of N-body simulations, we compute the recovery performance for
density fields obtained with both dark matter particles and haloes. We find
that the height of the Fisher information trans-linear plateau is increased by
more than an order of magnitude at k > 1.0h/Mpc for particles, whereas either
technique alone offers an individual recovery boost of only a factor of three
to five. We conclude that these two techniques work in a symbiosis, as their
combined performance is stronger than the sum of their individual contribution.
When applied to the halo catalogues, we find that the reconstruction has only a
weak effect on the recovery of Fisher Information, while the non-linear wavelet
filter boosts the information by about a factor offive. We also observe that
non-Gaussian Poisson noise saturates the Fisher information, and that shot
noise subtracted measurements exhibit a milder information recovery.
