Convergence and stability of the inverse scattering series for diffuse waves

Abstract

We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the series.

Inverse Born series for diffuse waves

“…In this paper we extend the results of [4] in two regards. First, we obtain a stronger result on the convergence of the inverse Born series.…”

“…Thus an ill-posed nonlinear inverse problem is reduced to an ill-posed linear inverse problem plus a well-posed nonlinear problem, namely the computation of the higher order terms in the series. 1 We note that in [4], Lemma 2.2 is formulated in terms of the space…”

“…In previous work, we have shown that, to some extent, it is possible to fill this gap by employing methods which invert the Born series [3]. The resulting image reconstruction algorithms are fast, direct, and have analyzable convergence, stability and approximation error [4].…”

Inverse Born Series for Scalar Waves

“…In previous work, we have proposed a direct method to solve the inverse problem of optical tomography that is based on inversion of the Born series [11][12][13]. In this approach, the solution to the inverse problem is expressed as an explicitly computable functional of the scattering data.…”

“…Assume we measure data on the boundary of a ball of radius R, ∂B R . By proceeding with the same approach as found in [13], we find that the operators K j are bounded in L ∞ :…”