The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al (2013 Inverse Problems 29 075012) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (2017 J. Integral Equ. Appl. 29 271–95). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region.
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Direct sampling methods for isotropic and anisotropic scatterers with point source measurements
In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.
more » « less- NSF-PAR ID:
- 10333091
- Date Published:
- Journal Name:
- Inverse Problems and Imaging
- Volume:
- 0
- Issue:
- 0
- ISSN:
- 1930-8337
- Page Range / eLocation ID:
- 0
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/ipi.2022015
