Skip to main content
SpringerLink
Log in

Generalized polarization tensors for shape description

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

With each domain, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper (Ammari et al. in Math. Comput. 81, 367–386, 2012), a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that high-frequency oscillations of the boundary of a domain are only contained in its high-order GPTs. Indeed, we provide an original stability and resolution analysis for the reconstruction of small shape changes from the GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We also propose an indicator of topology which could be used in some particular cases to test whether we have the correct number of connected components in the reconstructed image. We provide analytical and numerical evidence that GPTs can capture topology and high-frequency shape oscillations. The results of this paper clearly show that the concept of GPTs is a very promising new tool for shape description.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Ammari, H., Boulier, T., Garnier, J., Jing, W., Kang, H., Wang, H.: Target identification using dictionary matching of generalized polarization tensors. (Submitted)

  2. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.: Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Rat. Mech. Anal. 208, 667–692 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Yun, K.: Spectral analysis of the Neumann–Poincaré operator and characterization of the gradient blow-up. Arch. Rat. Mech. Anal. 208, 275–304 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ammari, H., Garnier, J., Sølna, K.: Resolution and stability analysis in full-aperature, linearized conductivity and wave imaging. Proc. Am. Math. Soc. (To appear)

  5. Ammari, H., Kang, H.: High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. SIAM J. Math. Anal. 34, 1152–1166 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ammari, H., Kang, H.: Properties of generalized polarization tensors. SIAM Multiscale Model. Simul. 1, 335–348 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ammari, H., Kang, H.: Reconstruction of small inhomogeneities from boundary measurements. In: Lecture Notes in Mathematics, vol. 1846. Springer, Berlin (2004)

  8. Ammari, H., Kang, H.: Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities. J. Math. Anal. Appl. 296, 190–208 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ammari, H., Kang, H.: Polarization and moment tensors with applications to inverse problems and effective medium theory. In: Applied Mathematical Sciences, vol. 162. Springer, New York (2007)

  10. Ammari, H., Kang, H., Kim, E., Lim, M.: Reconstruction of closely spaced small inclusions. SIAM J. Numer. Anal. 42, 2408–2428 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ammari, H., Kang, H., Lim, M.: Polarization tensors and their applications. In: Proceedings of the second International Conference on Inverse Problems: recent developments and numerical approaches, Shanghai, 2004. J. Phy. Conf. Ser. 12, 13–22 (2005)

  12. Ammari, H., Kang, H., Lim, M., Lee, H.: Enhancement of near-cloaking using generalized polarization tensors vanishing structures. Part I: the conductivity problem. Comm. Math. Phys. 317, 253–266 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ammari, H., Kang, H., Lim, M., Zribi, H.: The generalized polarization tensors for resolved imaging. Part I: shape reconstruction of a conductivity inclusion. Math. Comp. 81, 367–386 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ammari, H., Kang, H., Kim, E., Lee, J.-Y.: The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp. 81, 839–860 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ammari, H., Kang, H., Nakamura, G., Tanuma, K.: Complete asymptotic expansions of solutions of the system of elastostatics in the presence of inhomogeneities of small diameter. J. Elast. 67, 97–129 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ammari, H., Kang, H., Touibi, K.: Boundary layer techniques for deriving the effective properties of composite materials. Asymp. Anal. 41, 119–140 (2005)

    MATH  MathSciNet  Google Scholar 

  17. Burger, M., Osher, S.J.: A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16, 263–301 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Capdeboscq, Y., Karrman, A.B., Nédélec, J.-C.: Numerical computation of approximate generalized polarization tensors. Appl. Anal. 91, 1189–1203 (2012)

    Google Scholar 

  19. Capdeboscq, Y., Vogelius, M.S.: A general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Num. Anal. 37, 159–173 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Cedio-Fengya, D.J., Moskow, S., Vogelius, M.S.: Identification of conductivity imperfections of small diameter by boundary measurements: continuous dependence and computational reconstruction. Inverse Probl. 14, 553–595 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dassios, G., Kleinman, R.: Low frequency scattering. In: Oxford Mathematical Monographs. Oxford University Press, New York (2000)

  22. Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  23. Friedman, A., Vogelius, M.S.: Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Rat. Mech. Anal. 105, 299–326 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Goldenshluger, A., Spokoiny, V.: On the shape-from-moments problem and recovering edges from noisy Radon data. Prob. Theory Relat. Fields 128, 123–140 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Goldenshluger, A., Zeevi, A.: Recovering convex boundaries from blurred and noisy measurements. Ann. Stat. 34, 1375–1394 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Khavinson, D., Putinar, M., Shapiro, H.S.: Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185, 143–184 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Loncaric, S.: A survey of shape analysis techniques. Pattern Recognit. 31, 983–1001 (1998)

    Article  Google Scholar 

  28. Milton, G.W.: The theory of composites. In: Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2001)

  29. Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. In: Annals of Mathematical Studies, vol. 27. Princeton University Press, Princeton (1951)

  30. Santosa, F.: A level-set approach for inverse problems involving obstacles. ESAIM COCV 1, 17–33 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Teague, M.R.: Image analysis via the general theory of moments. J. Opt. Soc. Am. 70, 920–930 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Habib Ammari

  2. Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 75205, Paris Cedex 13, France

    Josselin Garnier

  3. Department of Mathematics, Inha University, Incheon, 402-751, Korea

    Hyeonbae Kang

  4. Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

    Mikyoung Lim & Sanghyeon Yu

Authors
  1. Habib Ammari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josselin Garnier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hyeonbae Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mikyoung Lim
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sanghyeon Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Habib Ammari.

Additional information

This work was supported by ERC Advanced Grant Project MULTIMOD-267184, National Research Foundation of Korea through Grants No. 2009-0070442, 2010-0017532 and 2010-0004091, and Posco TJ Park foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ammari, H., Garnier, J., Kang, H. et al. Generalized polarization tensors for shape description. Numer. Math. 126, 199–224 (2014). https://doi.org/10.1007/s00211-013-0561-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-013-0561-5

Mathematics Subject Classification (2000)

Search

Navigation