Acoustic inverse scattering using topological derivative of far-field measurements-based L ² cost functionals

Abstract

Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for wave-based identification of finite-sized objects. This approach remains however largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort towards bridging this gap, this study focuses on a topological derivative approach applied to the L 2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full scattering model. Topological derivativebased imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other wellknown qualitative methods.

Analysis of topological derivative as a tool for qualitative identification

“…The expected decay of z → |T (z)| is as a result observed for far-field data (leading-order asymptotics), as expected from e.g. [6], but also on the next-order asymptotic contribution.…”

“…Towards this aim, we formulate the forward scattering problem as a volume integral equation, and take advantage of a recently-proposed reformulation of such volume integral equation [7] which allows to express T (z) separately in terms of the material contrast and a contrastindependent normalized integral operator; this in particular facilitates the handling of material anisotropy. Some of our main findings are similar in nature to those of [6]; in particular the sign component of the TD heuristic is again found to be valid within a "moderate enough scatterer" condition, here expressed in terms of the norm of the normalized integral operator. We emphasize that this condition is less stringent than a requirement that the Born approximation be valid.…”

Locating Multiple Multiscale Acoustic Scatterers

“…The introduction of a known admissible class A in our algorithm is related to the dictionary matching algorithms that have been recently investigated in a series of works by Ammari and his collaborators [2,3,13], where some a priori known base shapes form a dictionary for the reconstruction. We also note that comparable indicator functions are used in a recent work [15] for reconstructing the acoustic scatterers at small scale and regular scale, respectively.…”

“…We would like to point out that many numerical reconstruction methods have been developed for inverse scattering problems in various scenarios, such as the linear sampling method, factorization method, MUSIC-type methods, time reversal, and topological-optimization-type methods; we refer the reader [1,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,24,25,27,28,40,41,42,43] and the references therein for these methods and some other related developments. Compared with most of the existing methods, which rely on multiple scattered field measurements, the methods developed in this work are new and more general in the sense that they combine all of the following features: only one single far-field measurement is used; the scatterers are allowed to be a multiscale mixed set of inhomogeneous media and impenetrable obstacles; rigorous mathematical justifications are established under general settings; some iterative-type refining and local tuning strategies are introduced for quantitatively improving the reconstructions.…”

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

“…As shown in a number of recent studies, however, sampling-based non-iterative techniques [5,6] that reconstruct obstacles 2 H. Yuan et al from the distribution of an established indicator function provide a computationally effective preliminary imaging tool. The topological sensitivity method, [7,8] as one of its variants and the focus of this study, formulates the indicator function from a physical standpoint. The concept of topological sensitivity (TS), since its inception in the context of structural shape optimization, [9,10] has been generalized and applied to deal with inverse scattering problems in acoustics, [7,[11][12][13][14][15][16][17] electromagnetism, [18][19][20][21][22][23] and elastodynamics.…”