  In this paper, we consider the inverse shape problem of recovering isotropic
scatterers with a conductive boundary condition. Here, we assume that the
measured far-field data is known at a fixed wave number. Motivated by recent
work, we study a new direct sampling indicator based on the Landweber iteration
and the factorization method. Therefore, we prove the connection between these
reconstruction methods. The method studied here falls under the category of
qualitative reconstruction methods where an imaging function is used to recover
the absorbing scatterer. We prove stability of our new imaging function as well
as derive a discrepancy principle for recovering the regularization parameter.
The theoretical results are verified with numerical examples to show how the
reconstruction performs by the new Landweber direct sampling method.
