Two direct factorization methods for inverse scattering problems

Abstract

In this paper, we propose and analyze two new direct factorization methods for solving inverse scattering problems. Both direct factorization methods are built upon the mathematically justified factorization method developed by Kirsch. The first one is naturally derived from a recent direct sampling method by replacing the corresponding far-field operator F in the indicator function by the factorized far-field operator (F * F) 1/4 . The second one is based on a truncated Neumann series approximation of the inverse of an appropriately scaled factorized far-field operator. Both direct factorization methods are shown to be stable with respect to noise and mathematically equivalent. Numerical results with both synthetic and real experimental data are presented to illustrate the promising accuracy of our proposed direct factorization methods in comparison to Kirsch's factorization method and the direct sampling method.

Analysis of new direct sampling indicators for far-field measurements

“…First, we provide a precise problem statement in section 2 and give some preliminary result. In section 3 we analyze the first indicator which is defined by an inner-product involving |F| 1/2 and φ z which was originally considered in [21]. The authors of [21] did not succeed in showing that the indicator is equivalent to the one proposed in [22] and here we will show that the indicators are equivalent.…”

“…Lately, there has been some interest in analyzing a so-called direct sampling method (DSM) using the theoretical framework used in the FM for far-field data (see for e.g. [22] and [21]). These methods have also been studied for there applicability to reconstruct scatterers from near-field data [9].…”

“…Here we propose the use of two indicator functions based on the FM. This research is motivated by the recent paper [21] where similar indicators are introduced and analyzed for the multi-static response matrix which is the discrete version of the far-field operator. The indicators in [21] are based on the theoretical setting of the FM and one of the proposed indicators is based on a Neumann series approximation to the FM equation.…”

“…This research is motivated by the recent paper [21] where similar indicators are introduced and analyzed for the multi-static response matrix which is the discrete version of the far-field operator. The indicators in [21] are based on the theoretical setting of the FM and one of the proposed indicators is based on a Neumann series approximation to the FM equation. In this paper, we will analyze the indicator function which uses the FM operator and prove that the indicator is equivalent to DSM studied in [22] and derive a new method based on the Tikhonov regularization.…”

Direct Sampling for Recovering Sound Soft Scatterers from Point Source Measurements

“…From this, we have the following result by the analysis in [2]. The imaging functional given in (19) was motivated by the work in [23], and we extend that work to the case of near-field data. There are some interesting questions when considering the implementation of the W TDSM (z) such as the following: how to pick α and which polynomial approximation method works best for constructing P α,ε (t).…”