  A conductivity inclusion, inserted in a homogeneous background, induces a
perturbation in the background potential. This perturbation admits a multipole
expansion whose coefficients are the so-called generalized polarization tensors
(GPTs). GPTs can be obtained from multistatic measurements. As a modification
of GPTs, the Faber polynomial polarization tensors (FPTs) were recently
introduced in two dimensions. In this study, we design two novel analytical
non-iterative methods for recovering the shape of a simply connected inclusion
from GPTs by employing the concept of FPTs. First, we derive an explicit
expression for the coefficients of the exterior conformal mapping associated
with an inclusion in a simple form in terms of GPTs, which allows us to
accurately reconstruct the shape of an inclusion with extreme or near-extreme
conductivity. Secondly, we provide an explicit asymptotic formula in terms of
GPTs for the shape of an inclusion with arbitrary conductivity by considering
the inclusion as a perturbation of its equivalent ellipse. With this formula,
one can non-iteratively approximate an inclusion of general shape with
arbitrary conductivity, including a straight or asymmetric shape. Numerical
experiments demonstrate the validity of the proposed analytical approaches.
