  We present a novel, fast method to recover the density field through the
statistics of the transmitted flux in high redshift quasar absorption spectra.
The proposed technique requires the computation of the probability distribution
function of the transmitted flux (P_F) in the Ly-alpha forest region and, as a
sole assumption, the knowledge of the probability distribution function of the
matter density field (P_Delta). We show that the probability density
conservation of the flux and matter density unveils a flux-density (F-Delta)
relation which can be used to invert the Ly-alpha forest without any assumption
on the physical properties of the intergalactic medium. We test our inversion
method at z=3 through the following steps: [i] simulation of a sample of
synthetic spectra for which P_Delta is known; [ii] computation of P_F; [iii]
inversion of the Ly-alpha forest through the F-Delta relation. Our technique,
when applied to only 10 observed spectra characterized by a signal-to noise
ratio S/N >= 100 provides an exquisite (relative error epsilon_Delta <~ 12 % in
>~ 50 % of the pixels) reconstruction of the density field in >~ 90 % of the
line of sight. We finally discuss strengths and limitations of the method.
