The generalized polarization tensors for resolved imaging Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements

Authors:
Habib Ammari, Hyeonbae Kang, Eunjoo Kim and June-Yub Lee

Journal:
Math. Comp. **81** (2012), 839-860

MSC (2010):
Primary 35R30, 35B30, 78M50

DOI:
https://doi.org/10.1090/S0025-5718-2011-02534-2

Published electronically:
September 20, 2011

MathSciNet review:
2869039

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi-static response matrix measurements at a single frequency. The target is of characteristic size less than the operating wavelength. Using small-volume asymptotic expansions of the measurements of high-order, we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed. The asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors. Moreover, we extend the optimization approach proposed in Part I to image geometric details of an electromagnetic target that are finer than the equivalent ellipse. The equivalent ellipse still provides a good initial guess for the optimization procedure. However, compared to the conductivity case in Part I, the cost functional measures the discrepancy between the computed and measured high-order frequency dependent polarization tensors rather than between the generalized polarization tensors. The main reason for such a modification of the cost functional is the fact that the (measured) frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment. The proposed methods are numerically implemented to demonstrate their validity and efficiency.

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Additional Information

**Habib Ammari**

Affiliation:
Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France

Email:
habib.ammari@ens.fr

**Hyeonbae Kang**

Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, Korea

Email:
hbkang@inha.ac.kr

**Eunjoo Kim**

Affiliation:
Institute of Mathematical Sciences and Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea

Email:
kej@ewha.ac.kr

**June-Yub Lee**

Affiliation:
Institute of Mathematical Sciences and Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea

Email:
jyllee@ewha.ac.kr

DOI:
https://doi.org/10.1090/S0025-5718-2011-02534-2

Keywords:
Generalized polarization tensors,
frequency dependent polarization tensors,
asymptotic expansions,
shape recovery,
target reconstruction,
wave imaging

Received by editor(s):
August 18, 2010

Received by editor(s) in revised form:
December 2, 2010

Published electronically:
September 20, 2011

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.