The generalized linear sampling and factorization methods only depends on the sign of contrast on the boundary

We extend the applicability of the Generalized Linear Sampling Method (GLSM) [2] and the Factorization Method (FM)[16] to the case of inhomogeneities where the contrast changes sign. Both methods give an ex-act characterization of the target shapes in terms of the farfield operator (at a fixed frequency) using the coercivity property of a special solution opera-tor. We prove this property assuming that the contrast has a fixed sign in a neighborhood of the inhomogeneities boundary. We treat both isotropic and anisotropic scatterers with possibly different supports for the isotropic and anisotropic parts. We finally validate the methods through some numerical tests in two dimensions.