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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: 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.