  The factorization method by Kirsch (1998) provides a necessary and sufficient
condition for characterizing the shape and position of an unknown scatterer by
using far-field patterns of infinitely many time-harmonic plane waves at a
fixed frequency. This paper is concerned with the factorization method with a
single far-field pattern to recover a convex polygonal scatterer/source. Its
one-wave version relies on the absence of analytical continuation of the
scattered/radiated wave-fields in corner domains. It can be regarded as a
domain-defined sampling method and does not require forward solvers. In this
paper we provide a rigorous mathematical justification of the one-wave
factorization method and present some preliminary numerical examples. In
particular, the proposed scheme can be interpreted as a model-driven and
data-driven method, because it essentially depends on the scattering model and
a priori given \emph{sample data}.
