Regularization of the Factorization Method with Applications to Inverse Scattering

Abstract

Here we discuss a regularized version of the factorization method for positive operators acting on a Hilbert Space. The factorization method is a qualitative reconstruction method that has been used to solve many inverse shape problems. In general, qualitative methods seek to reconstruct the shape of an unknown object using little to no a priori information. The regularized factorization method presented here seeks to avoid numerical instabilities in the inversion algorithm. This allows one to recover unknown structures in a computationally simple and analytically rigorous way. We will discuss the theory and application of the regularized factorization method to examples coming from acoustic inverse scattering. Numerical examples will also be presented using synthetic data to show the applicability of the method.

Regularization of the factorization method with applications to inverse scattering

“…We are motivate by the work in [24,27] where regularized variants of the factorization method were developed. This method has been applied to diffuse optical tomography [24], electrical impedance tomography [22] and inverse scattering [25]. In the aforementioned papers, the results hold when one has the unperturbed data operator whereas we wish to extend the results when one only has access to the perturbed data operator.…”

“…This is due to the fact that, in shape reconstruction problems, using (9) could result in some numerical instabilities (see for e.g. [25]). With the above assumptions, it is shown that…”

“…) is the radiating solution to (25). Then provided that (n − 1) and (η) are both uniformly positive (or negative) definite we have that:…”