In this contribution a new two-step algorithm for the imaging of scatterers embedded in an inhomogeneous background is presented, which is very suitable for the reconstruction of inhomogeneities inside a known structure, such as defects in dielectric products, or objects inside plastic boxes. Namely, in the first step, a new formulation of the linear sampling method, called the no-sampling linear sampling method (nLSM), is applied to obtain information on the supports and the locations of the targets in the inhomogeneous background. Afterwards, the dielectric permittivity and the electric conductivity only of the region determined by the nLSM are retrieved by solving the Lippmann-Schwinger integral equation involving the Green's function of the inhomogeneous background by means of an inexact Newton method. The proposed method is then applied to two different scenarios: the imaging of objects inside a plastic box and of defects inside a cement-like slab.
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