Application of the Inhomogeneous Lippmann--Schwinger Equation to Inverse Scattering Problems

Abstract

In this paper we present a hybrid approach to numerically solve two-dimensional electromagnetic inverse scattering problems, whereby the unknown scatterer is hosted by a possibly inhomogeneous background. The approach is 'hybrid' in that it merges a qualitative and a quantitative method to optimize the way of exploiting the a priori information on the background within the inversion procedure, thus improving the quality of the reconstruction and reducing the data amount necessary for a satisfactory result. In the qualitative step, this a priori knowledge is utilized to implement the linear sampling method in its near-field formulation for an inhomogeneous background, in order to identify the region where the scatterer is located. On the other hand, the same a priori information is also encoded in the quantitative step by extending and applying the contrast source inversion method to what we call the 'inhomogeneous Lippmann-Schwinger equation': the latter is a generalization of the classical Lippmann-Schwinger equation to the case of an inhomogeneous background, and in our paper is deduced from the differential formulation of the direct scattering problem to provide the reconstruction algorithm with an appropriate theoretical basis. Then, the point values of the refractive index are computed only in the region identified by the linear sampling method at the previous step. The effectiveness of this hybrid approach is supported by numerical simulations presented at the end of the paper.

Inverse medium scattering problems with Kalman filter techniques II. Nonlinear case

“…Existing methods for solving nonlinear inverse problem can be roughly categorized into two groups: iterative optimization methods and qualitative methods. The iterative optimization method (see e.g., [3,12,15,20,27]) does not require many measurements, however it require the initial guess which is the starting point of the iteration. It must be appropriately chosen by a priori knowledge of the unknown function q, otherwise, the iterative solution could not converge to the true function.…”

Inverse medium scattering problems with Kalman filter techniques I. Linear case

“…Existing methods for solving nonlinear inverse problem can be roughly categorized into two groups: iterative optimization methods and qualitative methods. The iterative optimization method (see e.g., [4,12,15,19,25]) does not require many measurements, however it require the initial guess which is the starting point of the iteration. It must be appropriately chosen by a priori knowledge of the unknown function q, otherwise, the iterative solution could not converge to the true function.…”

Limited-angle tomographic reconstruction of dense layered objects by dynamical machine learning

“…Often, solving the Lippmann-Schwinger equation is the most robust approach to account for multiple scattering, but even then the solution is iterative and requires excessive amount of computation especially for complex 3D geometries. The inversion of these forward models to obtain the refractive index in 3D is referred to as inverse scattering, also a well studied topic [29][30][31][32][33][34][35][36].…”