  The direct sampling method (DSM) has been introduced for non-iterative
imaging of small inhomogeneities and is known to be fast, robust, and effective
for inverse scattering problems. However, to the best of our knowledge, a full
analysis of the behavior of the DSM has not been provided yet. Such an analysis
is proposed here within the framework of the asymptotic hypothesis in the 2D
case leading to the expression of the DSM indicator function in terms of the
Bessel function of order zero and the sizes, shapes and permittivities of the
inhomogeneities. Thanks to this analytical expression the limitations of the
DSM method when one of the inhomogeneities is smaller and/or has lower
permittivity than the others is exhibited and illustrated. An improved DSM is
proposed to overcome this intrinsic limitation in the case of multiple incident
waves. Then we show that both the traditional and improved DSM are closely
related to a normalized version of the Kirchhoff migration. The theoretical
elements of our proposal are supported by various results from numerical
simulations with synthetic and experimental data.
