A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation

Abstract

In this work we develop a new reproducing kernel Hilbert space (RKHS) framework for inverse scattering problems using the Born approximation. We assume we have backscattered data of a field that is dependent on an unknown scattering potential. Our goal is to reconstruct or image this scattering potential. Assuming the scattering potential lies in a RKHS, we find that the imaging equation can be rewritten as the inner product of the desired unknown function with the adjoint of the forward operator applied to the kernel of the imaging operator. We therefore may choose the kernel of the imaging operator such that the adjoint applied to this kernel is precisely the reproducing kernel of the Hilbert space the reflectivity function lies in. In this way we are able to obtain an alternative definition of an ideal image. We will demonstrate this theory using synthetic aperture radar imaging as an example, though there are other applicable imaging modalities i.e. inverse diffraction and diffraction tomography [1,6]. We choose SAR as it was the motivating application for this work. We will compare the RKHS ideal imaging technique to the standard microlocal analytic ideal image from backprojection theory. Note this method requires a variation of the standard SAR data model with the assumption of a full two dimensional data collection surface as opposed to a one dimensional flight path, however we are able to perform imaging with a single frequency and avoid the approximations made in the backprojection imaging operator derivation.1. Introduction. In this paper we develop a new reproducing kernel Hilbert space framework for inverse scattering problems using the Born approximation. We assume we have backscattered data of some field that is dependent on an unknown scattering potential. Our goal is to reconstruct or image this scattering potential. We will use synthetic-aperture radar imaging as an example to demonstrate the theory developed in this work in terms of a real application.There are two main mathematical tasks to consider for imaging, first developing a forward model and then creating a method to invert that model to obtain information about the unknown function of interest. We will assume our collected data is in the form of a scattered field which is related to an unknown function to be reconstructed. We will therefore assume our field of interest satisfies the inhomogeneous scalar wave equation in the time domain and the inhomogeneous Helmholtz equation in the frequency domain. Thus we may use a Green's function