Difference factorizations and monotonicity in inverse medium scattering for contrasts with fixed sign on the boundary

We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. While the factorization method has been used so far mainly to identify the shape of a scatterer's support, we show that factorizations based on Dirichlet-to-Neumann operators can be used to compute bounds for numerical values of the medium on the boundary of its support. To this end, we generalize ideas from inside-outside duality to obtain a monotonicity principle that allows for alternative uniqueness proofs for particular inverse scattering problems (e.g., when obstacles are present inside the medium). This monotonicity principle indeed is our most important technical tool: It further directly shows that the boundary values of the medium's contrast function are uniquely determined by the corresponding far field operator. Our particular factorization of far field operators additionally implies that the factorization method rigorously characterizes the support of an inhomogeneous medium if the contrast function takes merely positive or negative values on the boundary of its support independently of the contrast's values inside its support. Finally, the monotonicity principle yields a simple algorithm to compute upper and lower bounds for these boundary values, assuming the support of the contrast is known. Numerical experiments show feasibility of a resulting numerical algorithm

Monotonicity in inverse scattering for Maxwell’s equations

We consider the inverse scattering problem to recover the support of penetrable scattering objects in three-dimensional free space from far field observations of scattered time-harmonic electromagnetic waves. The observed far field data are described by far field operators that map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered waves. We discuss monotonicity relations for the eigenvalues of linear combinations of these operators with suitable probing operators. These monotonicity relations yield criteria and algorithms for reconstructing the support of scattering objects from the corresponding far field operators. To establish these results we combine the monotonicity relations with certain localized vector wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy on some other prescribed region. Throughout we suppose that the relative magnetic permeability of the scattering objects is one, while their real-valued relative electric permittivity maybe inhomogeneous and the permittivity contrast may even change sign. Numerical examples illustrate our theoretical findings