  The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger
Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS)
system that defines the direct and inverse scattering transforms (IST). In this
paper, we derive explicit expressions for the leading order terms of these
transforms, which are called semiclassical limits of the direct and inverse
scattering transforms. Thus, we establish an explicit connection between the
decaying initial data of the form $q(x,0)=A(x)e^{iS(x)}$ and the leading order
term of its scattering data. This connection is expressed in terms of an
integral transform that can be viewed as a complexified version of an Abel type
transform. Our technique is not based on the WKB analysys of the ZS system, but
on the inversion of the modulation equations that solve the inverse scattering
problem in the leading order. The results are illustrated by a number of
examples.
