A Factorization Method and Monotonicity Bounds in Inverse Medium Scattering for Contrasts with Fixed Sign on the Boundary

TL;DR

This paper extends the factorization method in inverse medium scattering to compute boundary bounds of the medium's contrast using a new monotonicity principle, enabling shape and boundary value identification with proven numerical feasibility.

Contribution

It introduces a novel factorization and monotonicity approach that allows boundary contrast bounds to be determined from far field data, even with interior obstacles.

Findings

The method can compute bounds for boundary contrast values.

The approach characterizes the support of the medium based on contrast sign.

Numerical experiments demonstrate the algorithm's feasibility.

Abstract

We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. Whilst the factorization method 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, independent 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.