On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media - Archive ouverte HAL Access content directly
Journal Articles Engineering Computations Year : 2022

On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media

Abstract

The concept of topological derivative (TD) is known to provide, through its heuristic interpretation involving its sign and its spatial decay away from the true anomaly, a basis for the qualitative imaging of finite-sized anomalies. The TD imaging heuristic is currently partially backed by conditional mathematical justifications. Continuing earlier efforts towards the justification of TD-based identification, this work investigates the acoustic wave-based imaging of finite-sized (i.e. not necessarily small) medium anomalies embedded in bounded domains and affecting the leading-order term of the acoustic field equation. Both the probing excitation and the measurement are assumed to take place on the domain boundary. We extend to this setting the analysis approach previously used for unbounded media with either refraction-index anomalies and far-field measurements (Bellis et al., \emph{Inverse Problems} \textbf{29}:075012, 2013) or mass-density anomalies and meaurements at finite distance (Bonnet, Cakoni, \emph{Inverse Problems} \textbf{35}:104007, 2019). Like in the latter work, TD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, facilitated by a volume integral formulation. Our results, which echo corresponding results of our earlier investigations, conditionally validate the TD imaging heuristic. Moreover, we show on a geometrically simple configuration that the spatial behavior of the TD associated with standard $L^2$ cost functionals is degraded by ``echoes'' of the true anomaly, an aspect specific to the present bounded-domain framework. This undesirable effect is removed by a combination of (i) post-processing the measurements by application of a suitable integral operator (a treatment introduced by Ammari et al., 2011, for the analysis of TD-based imaging involving true flaws modelled using small-anomaly asymptotics), and (ii) expressing the background field as an incoming single-layer potential defined in the full space (after an idea used in Bonnet, Cakoni, 2019). Finally, we also show that selecting eigenfunctions of the source-to-measurement operator as excitations enhances the spatial decay properties of the TD functionals
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Dates and versions

hal-03319821 , version 1 (13-08-2021)

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Marc Bonnet. On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media. Engineering Computations, 2022, 39 (1), pp.313-336. ⟨10.1108/EC-08-2021-0471⟩. ⟨hal-03319821⟩
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