  In this paper, we are concerned with the 2D and 3D geometric shape generation
by prescribing a set of characteristic values of a specific geometric body. One
of the major motivations of our study is the 3D human body generation in
various applications. We develop a novel method that can generate the desired
body with customized characteristic values. The proposed method follows a
machine-learning flavour that generates the inferred geometric body with the
input characteristic parameters from a training dataset. One of the critical
ingredients and novelties of our method is the borrowing of inverse scattering
techniques in the theory of wave propagation to the body generation. This is
done by establishing a delicate one-to-one correspondence between a geometric
body and the far-field pattern of a source scattering problem governed by the
Helmholtz system. It in turn enables us to establish a one-to-one
correspondence between the geometric body space and the function space defined
by the far-field patterns. Hence, the far-field patterns can act as the shape
generators. The shape generation with prescribed characteristic parameters is
achieved by first manipulating the shape generators and then reconstructing the
corresponding geometric body from the obtained shape generator by a stable
multiple-frequency Fourier method. Our method is easy to implement and produces
more efficient and stable body generations. We provide both theoretical
analysis and extensive numerical experiments for the proposed method. The study
is the first attempt to introduce inverse scattering approaches in combination
with machine learning to the geometric body generation and it opens up many
opportunities for further developments.
