Overview of so-called qualitative methods in inverse scattering theory
We shall give first a unified presentation of so-called sampling methods to solve the inverse shape problem, where one would like to reconstruct the geometry of an inclusion from multi-static measurements at a fixed frequency. We put a focus on three methods: the Linear Sampling Method, the Factorization Method and the Generalized Linear Sampling Method. Extension of these methods to handle the case of unknown backgrounds will then be discussed. We finally present how these methods can be used to construct various spectral signatures of the material properties. We end up with some applicative perspectives and open questions.